## Sunday, August 18, 2019

### Unizor - Physics4Teens - Energy - Energy of Nucleus - Fusion

Notes to a video lecture on http://www.unizor.com

Nucleus Fission

Fusion is a nuclear reaction, when light nuclei are brought together and combined into a heavier ones.

The reason for this reaction to release the energy is the difference
between amount of energy needed to overcome the repulsion between nuclei
because they have the same positive electric charge (this energy is
consumed by fusion) and the potential energy released by strong forces, when the formation of a combined nucleus occurs (this energy is released by fusion).

The former is less than the latter.

When the light nuclei are fused into a heavier one, the excess of potential energy of strong forces, released in the process of fusion,
over the energy needed to squeeze together protons against their
repulsion is converted into thermal and electromagnetic field energy.

Analogy to this process can be two magnets separated by a spring.

The magnets represent two separate protons, the magnetic force of attraction between them represents the strong force
that is supposed to hold the nucleus together, when these particles are
close to each other, the spring represents the electrical repulsive
force between them, acting on a larger distance, as both are positively
charged.

It's known that magnetic force is inversely proportional to a square of a
distance between objects, while the resistance of a spring against
contraction obeys the Hooke's Law and is proportional to the length of
contraction.

On the picture magnets are separated. To bring them together, we have to
spend certain amount of energy to move against a spring that resists
contraction. But the magnetic attraction grows faster then the
resistance of the spring, so, at some moment this attraction will be
greater than the resistance of a spring. At this moment nothing would
prevent magnets to fuse.

As is in the above analogy, if we want to fuse two protons, we have to bring them together sufficiently close for strong forces to overtake the repulsion of their positive charges.

Consider the following nuclear reaction of fusion.

One nucleus of hydrogen isotope deuterium 1H2 with atomic mass 2 contains one proton and one neutron.

One nucleus of hydrogen isotope tritium 1H3 with atomic mass 3 contains one proton and two neutrons.

If we force these two nuclei to fuse, they will form a nucleus of helium 2He4 and releasing certain amount of energy:

1H2 + 1H3 = 2He4 + 0n1

It's not easy to overcome the repulsion of protons. High temperature and
pressure, like in the core of our Sun, are conditions where it happens.
On Earth these conditions are created in the nuclear bomb, using the
atomic bomd to achieve proper amount of heat and pressure, thus creating
an uncontrlled fusion.

Controlled nuclear reaction of fusion is what scientists are working on right now. So far, it's still in the experimental stage.

## Monday, August 12, 2019

### Unizor - Physics4Teens - Energy - Energy of a Nucleus - Fission

Notes to a video lecture on http://www.unizor.com

Nucleus Fission

Fission, first of all, is a nuclear reaction, when heavier nuclei are split into lighter ones.
The reason for this reaction to release the energy is the difference between amount of energy needed to break strong forces that hold the nucleus together (this energy is consumed by fission) and amount of potential energy in positively charged and repelling protons inside nucleus (this energy is released by fission).
The former is less than the latter.

When the heavy nucleus is broken into parts, the excess of potential energy of squeezed together protons against their repelling force over the energy of strong forces that keep nucleus together is converted into thermal and electromagnetic field energy.

Analogy of this is a spring squeezed tightly and held in this position by a thread. A thread plays the role of strong forces, while a potential energy of a squeezed spring plays the role of protons kept close to each other by a this force. When you cut a thread, the spring will release the potential energy, similarly to protons repelling from each other.

Electrically positively charged protons repel each other and, at the same time, are bonded together by strong forces inside a nucleus. At the same time neutrons are also bonded by strong forces among themselves and with protons without any repulsion.
So, the more neutrons the nucleus has - the stronger it is. Neutrons only add "bonding material" to a nucleus without adding any repelling forces that work against the nucleus' stability.

Uranium-238 with 92 protons and 146 neutrons (92U238) naturally occurs on Earth and is relatively stable.
Uranium-235 with the same 92 protons and 143 neutrons (92U235) has less "bonding material" (less neutrons) and is more susceptible to fission.

All it takes to break the nucleus of 92U235 is a little "push" from outside, which can be accomplished by bombarding it with neutrons. In the process of fission, caused by hitting a nucleus of 92U235 with a neutron, it can transforms into Barium-141 with 56 protons and 85 neutrons 56Ba141, Krypton-92 with 36 protons and 56 neutrons 36Kr92 and 3 free neutrons.
As we see, the numbers of protons is balanced (input: 92, output: 56 and 36), as well as a number of neutrons (input: 1 free hitting neutron and 143 in a nucleus of 92U235 total 144, output: 85 in a nucleus 56Ba141, 56 in a nucleus of 36Kr92 and 3 new free neutrons total 144).

Let's express this reaction in a formula (letter n denotes a neutron):
0n1 + 92U235 =
56Ba141 + 36Kr92 + 3·0n1

What's interesting in this reaction is that it not only produces energy because we break a heavy nucleus into lighter ones, but also that it produces 2 new neutrons that can bombard other atoms, causing a chain reaction and, potentially, an explosion (atomic bomb). However, if we absorb extra neutrons, it will allow to slowly release of nuclear energy (nuclear power stations).

## Monday, August 5, 2019

### Unizor - Physics4Teens - Energy - Energy of a Nucleus

Notes to a video lecture on http://www.unizor.com

Energy of Nucleus

In this lecture we will analyze the energy aspect of nucleus - the central part of an atom.

By now we have built a pyramid of energy types, related to the depth of our view inside the matter.

First, we analyzed the mechanical energy - the energy of moving macro-objects.

Our next view deep into the world of macro-objects uncovered the molecules - the smallest parts of macro-objects that retain their characteristics. The movement of these molecules was the source of thermal energy, which we often call the heat.

Next step inside the molecules uncovered atoms, as the molecules'
components. There are about 100 types of atoms and their composition
inside the molecules creates all the thousands of different molecules. Chemical reactions
change the composition of atoms in molecules, thereby creating new
molecules from the atoms of old molecules. This process broke some
inter-atomic bonds and created the new ones and is the source of chemical energy.

Now we look deep inside the atoms and find there 3 major elementary particles - electrically positively charged protons and electrically neutral neutrons inside a small but heavy nucleus and electrically negatively charged electrons,
circulating around nucleus on different orbits. For electrically
neutral atoms the numbers of protons and electrons are equal. Nuclear energy is hidden inside the nucleus and is the subject of this lecture.

The first question we would like to answer is "What holds nucleus, its
protons and neutrons, together, considering protons, as electrically
positively charged particles must repel each other?"

The answer is simple. There are other forces in the Universe, not only
electrostatic ones, that act in this case. These intra-nucleus forces
that hold the nucleus together are called strong forces. They are strong
because they are the source of attraction between the protons that is
stronger than electrostatic repelling. However, these strong forces act
only on a very small distance, comparable to the size of a nucleus
inside an atom. For example, at a distance 10−15m the strong force is more than 100 times stronger than electrostatic one.

If, by regrouping protons and neutrons, we will be able to create different atoms (inasmuch as regrouping atoms in chemical reaction we create new molecules), a new source of energy, based on strong forces, the nuclear energy, can be uncovered in the course of nuclear reaction.

There is another form of nuclear reaction related to
transformation of elementary particles. Under certain circumstance a
neutron inside a nucleus can transform into proton and, to keep the
total electrical charge in balance, it emits an electron. This reaction
is called beta-decay and it also produces energy in the form of electromagnetic waves of very high frequency (gamma-rays).

Nuclear reactions are a very powerful source of nuclear energy, which is
so much more powerful than other types of energy, that, if misused, it
might represent a danger for life on our planet.

There is a clear analogy between nuclear and chemical reactions.

What happens with atoms in the chemical reaction, happens with protons
and neutrons in nuclear reaction. Some atomic bonds break in a chemical
reaction, some are created. Some nuclear bonds between protons and
neutrons break in a nuclear reaction, some are created.

Sometimes the chemical reaction happens by itself, as long as
participating substances are close together, but sometimes we have to
initiate it, like lighting methane gas with a spark or a flame of a
match to initiate continuous burning.

Similar approach is valid for nuclear reaction. Sometimes it happens by
itself, but sometimes it should be started, like bombarding the nucleus
with neutrons, after which it continues by itself.

Here is an interesting fact.

Physicists have measured the masses of protons, neutrons and many
different nuclei that contain these protons and neutrons and have
discovered that the sum of masses of individual protons and neutrons is
greater than the mass of a nucleus that contain these exact particles.

For example,

mass of proton is 1.0072766 atomic mass units or 1.6726·10-27kg,

mass of neutron is 1.0086654 atomic mass units or 1.6749·10-27kg.

At the same time, mass of deuterium nucleus, that contains 1 proton and 1
neutron is 2.0135532 atomic mass units, which is smaller than the sum
of masses of proton and neutron (1.0072766 + 1.0086654 = 2.015942).

This so-called "mass defect" is directly related to nuclear energy - the energy of strong forces that hold the nucleus together.

A simplified explanation of this effect is based on the law of energy
conservation. Consider the force of gravity between a planet and an
object above its surface. The object has certain potential energy and,
if dropped to the ground, this potential energy transforms into other
forms, like kinetic, thermal etc.

Similarly, if we consider two independent neutrons (or neutron and
proton, or two protons) on a very small distance from each other, but
not forming a nucleus, there is a potential energy of the strong forces
acting between them. If we let these two particles to form a nucleus,
analogously to an object falling towards the surface of a planet, this
potential energy should be transformed into other forms, like thermal.

Now the Theory of Relativity comes to play, that has established the equivalence of mass and energy by a famous formula E=m·c².
According to this equivalence, if some energy is released during the
formation of a nucleus from individual protons and neutrons, there must
be certain amount of mass released associated with this energy. That is
the explanation of "mass defect".

It should be noted that to form a nucleus of deuterium from 1 proton and
1 neutron is easier than to form a nucleus that contains more than one
proton, because electrostatic repulsion between positively charged
protons prevents their bonding. So, to bring protons sufficiently close
to each other for strong forces to overcome the electrostatic
repulsion, we have to spend some energy. The net energy released by
forming a nucleus from protons and neutrons is the difference between
the energy released from strong forces taking hold of these particles inside a nucleus and the energy consumed to overcome repulsion of protons.

Actually, as we attempt to form bigger nuclei, the energy we have to
spend to overcome electrostatic repulsion forces become greater than
amount of energy released by forming a nucleus. This border line is
approximately around the nucleus of iron Fe. Forming iron
and heavier elements from protons and electrons is a process that
consumes more energy than releases. These heavier nuclei will produce
energy, if we reverse the procedure, breaking them into individual
protons and neutrons.

The mechanisms described above are used in nuclear reactors and atomic
bomb, where heavier elements are broken into lighter ones (fission),
releasing energy, and in hydrogen bomb, where lighter elements are
bonded together to release the energy (fusion).

## Monday, July 29, 2019

### Unizor - Physics4Teens - Energy - Atoms and Chemical Reactions - Interat...

Notes to a video lecture on http://www.unizor.com

Interatomic Bonds

Atoms in a molecule are bonded together to form a stable chemical substance or compound.

The mechanism of bonding is quite complex and different for different
molecules. In fact, the complexity of these bonds is outside of the
scope of this course. However, certain basic knowledge about molecular
bonding and molecular structure is necessary to understand the following
lecture, where we will make certain calculations related to energy
produced or consumed in chemical reactions.

The key to a mechanism of bonding atoms into molecules lies in an internal structure of atoms.

For our purposes we can consider the orbital model of atom as consisting
of electrically positive nucleus and electrically negative electrons
circulating on different orbits around a nucleus. This is only a model,
not an exact representation of what's really happening inside the atom,
but this model gives relatively good results that correspond to some
simple experiments.

Two different particles can be found in a nucleus - positively charged
protons and electrically neutral neutrons. The number of protons inside a
nucleus and electrons circulating on different orbits around a nucleus
should be the same for electrically neutral atoms in their most common
state.

For reasons not well understood by many physicists, each orbit can have
certain maximum number of electrons that can circulate on it without
"bumping" into each other. The higher the orbit - the more electrons it
can hold. The lowest orbit can hold no more than 2 electrons, the next -
no more than 8, the next - no more than 14 etc.

Consider a few examples.

1. Let's consider the structure of a simplest molecule - the molecule of
hydrogen, formed by two atoms of hydrogen. Each hydrogen atom has one
electron on the lowest orbit around a nucleus. The maximum number of
electrons on this orbit is two, in which case the compound becomes much
more stable. So, two atoms of hydrogen grab each other and the two
electrons, each from its own atom, are shared by a couple of atoms, thus
creating a stable molecule of hydrogen with symbol H2. The bond between two atoms of hydrogen is formed by one pair of shared electrons, so structurally the molecule of hydrogen H2 can be pictured as

H−H.

2. Atom of oxygen has 8 electrons - 2 on the lowest orbit and 6 on the
next higher one. The next higher orbit is stable when it has 8
electrons. So, two atoms of oxygen are grabbing each other and share 2
out of 6 electrons on the outer orbit with another atom. So, each atom
has 4 "personal" electrons, 2 electrons that it shares with another atom
and 2 electrons that the other atom shares with it. Thus, the orbit
becomes full, all 8 spots are filled. The bond between two atoms of
oxygen is formed by two pairs of shared electrons, so structurally the
molecule of oxygen O2 can be pictured as

O=O

(notice double link between the atoms).

3. Our next example is gas methane. Its molecule consists of one atom of
carbon (6 electrons, 2 of them on the lowest orbit, 4 - on the next
one) and 4 atoms of hydrogen (1 electron on the lowest orbit of each
atom). Obviously, having only 4 electrons on the second orbit, carbon is
actively looking for electrons to fill the orbit. It needs 4 of them to
complete an orbit of 8 electrons. Exactly this it finds in 4 atoms of
hydrogen that need to complete their own lowest orbit. Sharing
electrons, one atom of carbon and 4 atoms of hydrogen fill their
corresponding orbits, thus creating a molecule of methane CH4 with can be pictured as

H

|

H−C−H

|

H

4. Carbon dioxide molecule contains 1 atom of carbon, that needs 4
electrons to complete its orbit, and 2 atoms of oxygen, each needs 2
electrons to complete its orbit: CO2. By
sharing 2 electrons from each atom of oxygen with 4 electrons from atom
of carbon they all fill up their outer orbit of electrons and become a
stable molecule, pictured as

O=C=O

(notice double link between the atoms).

5. Ethanol molecule contains 2 atoms of carbon, 1 atom of oxygen and 6 atoms of hydrogen connected as follows

H   H

|     |

H−C−C−O−H

|     |

H   H

(notice single bond between atoms of carbon and oxygen in ethanol, while
the bond between them in carbon dioxide has double link)

6. Hydrogen peroxide molecule contains 2 atoms of hydrogen and 2 atoms of oxygen connected as follows

H−O−O−H

(notice single bond between atoms of oxygen, not like in a molecule of oxygen)

Numerous examples above illustrate that bonds between atoms can be
different, even between the same atoms in different molecules. That's
why it is important to understand the structure of molecules, how
exactly the atoms are linked and what kind of links exist between them.
This is the basis for calculation of the amount of energy produced or
consumed by chemical reactions that rearrange the atoms from one set of
molecules to another.

Obviously, bonds O−O and O=O are different.
The first one is facilitated by one shared electron, the second one - by
two. The amounts of energy, needed to break these bonds, are different
too. Therefore, when calculating the energy of chemical reaction, it's
important to understand the kind of bond between atoms in each separate
case.

## Tuesday, July 23, 2019

### Unizor - Physics4Teens - Energy - Chemical Energy of Atomic Bonds

Notes to a video lecture on http://www.unizor.com

Energy of Atomic Bonds

in Molecules

In this lecture we will analyze the energy aspect of chemical reactions.

Consider the reaction of burning of methane. This gas is used in regular
gas stoves, so the reaction happens every time we cook something.

A molecule of methane consists of one atom of carbon C and four atoms of hydrogen H, the chemical formula of methane is CH4.

You can imagine a molecule of methane as a tetrahedron, in its center is
an atom of carbon and on each of its four vertices is an atom of
hydrogen.

A molecule of oxygen, as we know, consists of two atoms of oxygen and has a chemical formula O2.

As a result of the reaction of burning of methane, water and carbon dioxide are produced, according to the following equation:

CH4 + 2O2 = 2H2O + CO2

So, during this reaction

(a) four atomic bonds between carbon and hydrogen in one molecule of methane are broken,

(b) one atomic bond in each molecule of oxygen (out of two) are broken,

(c) two atomic bonds between hydrogen and oxygen in each molecule of water (out of two) are created,

(d) two atomic bonds between carbon and oxygen in a molecule of carbon dioxide are created.

Amounts of potential energy of the different atomic bonds are
experimentally determined, which would lead to calculation of the amount
of chemical energy released (for exothermic) or consumed (by
endothermic) reaction.

To make experiments to determine potential energy of the bonds inside a
molecule, we have to make experiments with known amounts of components
in chemical reaction. The reaction above includes one molecule of
methane and two molecules of oxygen. Obviously, we cannot experiment
with one or two molecules. The solution is to experiment with proportional amounts of components, say, 1 million of molecules of methane and 2 million of molecules of oxygen.

To explain how to do this, we have to get deeper into atoms. Physics
models atoms as consisting of three kinds of elementary particles -
protons (electrically positively charged), neutrons (electrically
neutral) and electrons (electrically negatively charged). This is a
relatively simple model, that corresponds to most of experiments, though
the reality is more complex than this. For our purposes we can view
this model of atom as a nucleus, that contains certain number of protons
and neutrons, and a number of electrons circulating the nucleus on
different orbits.

Electrons are very light relatively to protons and neutrons, so the mass
of an atom is concentrated, mostly, in its nucleus. Protons and neutron
have approximately the same mass, which is called atomic mass unit. So, the mass of an atom in atomic mass units
("atomic weight") is equal to the number of protons and neutrons in its
nucleus. This mass is known for each element of the Periodic Table of
Mendeleev, that is for each known atom.

For example, it is determined that atom of hydrogen H has atomic weight of 1 atomic unit, atom of carbon C has atomic weight of 12, atom of oxygen O has atomic weight 16.

Knowing atomic weights of atoms, we can calculate atomic weight of molecules. Thus, the atomic weight of a molecule of methane CH4 is 12+4=16. Atomic weight of a molecule of oxygen O2 is 16+16=32. Atomic weight of water H2O is 2+16=18.

Now we can take components of any chemical reaction proportional to the
atomic weight of corresponding molecules, which will result in
proportional number of molecules. For example, not being able to
experiment with one molecule of methane CH4 and two molecules of oxygen O2, we can experiment with 16 gram of methane and 64 gram
of oxygen, and the proportionality of the number of molecules will be
preserved - for each molecule of methane there will be two molecules of
oxygen.

As you see, taking amount of any mono-molecular substance in grams
equaled to the atomic weight of the molecules of this substance (called a
mole) assures taking the same number of molecules, regardless of the substance. This number is the Avogadro Number and is equal to N=6.02214076·1023.

Thus, one mole of methane CH4 (atomic weight of C is 12, atomic weight of H is 1) weighs 16g, one mole of silicon Si2 (atomic weight of Si is 14) weighs 28g, one mole of copper oxide CuO (atomic weight of Cu is 64, atomic weight of O
is 16) weighs 80g etc. And all those amounts of different substances
have the same number of molecules - the Avogadro number (approximately,
of course).

The theory behind the atomic bonds inside a molecule is quite complex
and is beyond the scope of this course. Based on this theory and
experimental data, for many kinds of atomic bonds there had been
obtained an amount of energy needed to break these bonds, that is its
inner chemical energy.

Thus, chemical energy of atomic bonds inside a mole of methane CH4
is 1640 kilo-joules (because a molecule of methane has 4 bonds between
carbon and each atom of hydrogen, each bond at 410KJ), inside a molecule
of oxygen O2 - 494 kilo-joules (1 bond between 2 atoms oxygen at 494KJ), inside a molecule of carbon dioxide CO2 is 1598 kilo-joules (2 bonds between carbon and each atom of oxygen, each 799KJ), inside a molecule of water H2O is 920 kilo-joules (2 bonds between oxygen and each atom of hydrogen, each 460KJ).

Let's go back to methane burning:

CH4 + 2O2 = 2H2O + CO2

This chemical reaction converts 1 mole of methane (16g) and 2 moles of
oxygen (64g) into 1 mole of carbon dioxide (44g) and 2 moles of water
(36g).

The energy we have to spend to break the atomic bonds of 1 mole of methane and 2 moles of oxygen, according to above data, is

Ein = 1640 + 2·494 = 2628 KJ

The energy we have to spend to break atomic bonds of 2 moles of water and 1 mole of carbon dioxide, according to above data, is

Eout = 2·920 + 1598 = 3438 KJ

The net energy is

Enet = 2628 − 3438 = −810 KJ

This net energy is the amount of thermal energy released by burning 16g
of methane, using 64g of oxygen, obtaining as a result 44g of carbon
dioxide and 36g of water.

## Thursday, July 18, 2019

### Unizor - Physics4Teens - Energy - Atoms

Notes to a video lecture on http://www.unizor.com

Atoms and Chemical Reaction

Discussing mechanical energy, we analyzed the movement of objects.

When talking about thermal energy (heat), we had to go deeper inside the objects and analyzed the movement of molecules, the smallest parts of objects that retained the properties of objects themselves.

Now we go even deeper, inside the molecules, in search of new kinds of energy.

The components of molecules are called atoms. Currently there are more than 100 kinds of atoms, classified in the Mendeleev's Periodic Table.

Different combinations of these atoms in different quantities make up all kinds of molecules, each with its own properties.

In some cases a single atom makes up a molecule. For example, a single atom of iron (denoted by symbol Fe) makes up a molecule of iron.

In some other cases a pair of atoms of the same type makes up a molecule. For example, two atoms of oxygen (denoted by symbol O) make up a molecule of oxygen (denoted by symbol O2).

In more complicated cases a few atoms of different types make up a molecule. For example, two atoms of hydrogen (denoted by symbol H) and one atom of oxygen (O) make up a molecule of water (H2O).

One of the most complicated molecules that contains many elements in
different quantities is a molecule of protein that has about half a
million of atoms.

Chemical energy is a potential energy of bonds between atoms that hold them together in a molecule.

Chemical reaction is a process of re-arranging of atoms in a group of molecules, getting, as a result, a group of other molecules.

During chemical reactions some bonds between atoms are broken and some are created. Therefore, the energy might be either released or consumed in the process of chemical reaction.
This energy, stored in the molecules as potential energy of atomic
bonds and released or consumed during chemical reaction, is classified
as chemical energy.

Let's consider a few examples of chemical energy.

1. Coal burning

One molecule of carbon, that consists of one carbon atom C, and one molecule of oxygen, that consists of two oxygen atoms O2, when brought together and lit up, will join into one molecule of carbon dioxide CO2
in the process of burning. After the chemical reaction of burning is
initiated, it will maintain itself, as the process of burning produces a
flame that lights up new molecules of carbon, joining them with oxygen.

The chemical reaction

C + O2 = CO2

is endothermic (consumes heat energy) in the very beginning, when we
have to light up the carbon, but, as soon as the reaction started, it
becomes exothermic, that is it produces heat energy, because the
potential energy of atoms inside molecules of carbon and oxygen together
is greater than potential energy of atoms inside a molecule of carbon
dioxide.

2. Making water from hydrogen and oxygen

Two molecules of hydrogen, each consisting of two hydrogen atom H2, and one molecule of oxygen, that consists of two oxygen atoms O2, when brought together and lit up, will join into two molecules of water H2O
in the process of hydrogen burning. After the chemical reaction of
burning is initiated, it will maintain itself, as the process of burning
produces a flame that lights up new molecules of hydrogen, joining them
with oxygen.

The chemical reaction

2H2 + O2 = 2H2O

is endothermic (consumes heat energy) in the very beginning, when we
have to light up the hydrogen, but, as soon as the reaction started, it
becomes exothermic, that is it produces heat energy, because the
potential energy of atoms inside two molecules of hydrogen and one
molecule of oxygen together is greater than potential energy of atoms
inside two molecules of water.

3. Photosynthesis

This is a complicated process, during which the light from sun, air
components (such as carbon dioxide, nitrogen and oxygen), water and
whatever is in the soil are converted by the plants into chemical energy
that maintains their life. This is an endothermic process, and, as its
result, plants grow. In most cases they consume carbon dioxide from the
air, break it into carbon and oxygen, consume the water from the soil,
break it into hydrogen and oxygen (they need sun's radiation energy to
break the molecules of CO2 and H2O),
use the carbon, hydrogen and part of oxygen to produce new organic
molecules they consist of and release the unused oxygen back into
atmosphere.

4. Battery

Battery consists of three major components: anode, cathode and electrolyte
in-between anode and cathode. As a result of a chemical and
electro-magnetic reaction between the molecules of anode and
electrolyte, some electrons are transferred from anode to electrolyte.
Then, as a result of a chemical and electro-magnetic reaction between
the molecules of cathode and electrolyte, some electrons are transferred
from electrolyte to cathode. As the result, there are extra electrons
on the cathode, which were taken from the anode, thus creating
electrical potential.

These simple examples explain the general mechanism of chemical energy,
released or consumed in the course of chemical reaction, that transforms
molecules by rearranging their atoms' composition. As a result of a
chemical reaction and change in the atomic composition of molecules,
potential energy of bonds between atoms in molecules is changing. If the
total potential energy of the resulting molecules is greater than the
potential energy of the bonds inside original molecules, the process is
endothermic, it consumes energy. In an opposite case the process is
exothermic, it produces energy.

The exothermic process of extracting chemical energy using chemical
reaction is the key to getting energy from gasoline in the car engine,
producing heat and light in the fireplace by burning wood, it's the
source of energy in all living organisms, including humans. We exist
because our body knows how to extract chemical energy from the food.

## Monday, July 8, 2019

### Unizor - Physics4Teens - Energy - Heat Transfer Problems

Notes to a video lecture on http://www.unizor.com

Heat Transfer - Problems

Problem 1

Determine the power of the heat source inside the room required to
maintain a certain difference between inside and outside air
temperature, given the following:

(a) the difference between inside and outside temperature δ=Tout−Thome

(b) the room has only one wall facing the outside air, and the area of this wall is A

(c) the thickness of the wall is L

(d) the wall is made of solid material with a coefficient of thermal conductivity k.

Solution

Let x be the distance from a point inside the wall to its surface facing the room.

The T(x) is a temperature at this point as a function from this distance.

To be equal to Thome at the surface facing the room and to be Tout at the surface facing the outside and to be linearly changing from one value to another inside the wall of the thickness L, the temperature inside the wall at distance x from the surface facing the room should be

T(x)=Thome+(Tout−Thome)·x/L

From this we can determine the heat flux through the wall at a distance x, using the Fourier's law of thermal conduction:

Q(x,A) = −k·A·dT(x)/dx =

= −k·A·(Tout−Thome)/L =

= −k·A·δ/L

That is exactly how much heat we need to maintain the difference between temperature in the room and outside.

Problem 2

Calculate the heating requirement of the room with only one concrete
wall facing outside with no windows, assuming the following:

(a) the temperature in the room must constantly be Troom=20°C

(b) the temperature outside is also constant Tout=5°C

(c) the thickness of the concrete wall facing outside is L=0.2m

(d) the area of the wall facing outside is A=12m²

(e) heat conductivity of concrete is k=0.6W/(m·°K)

Solution

Using the above, we can determine the heat flux through the wall at a distance x:

Q(x,A) = −k·A·δ/L

(for any distance x from the surface of the wall that faces the room)

Outside surface of the wall is at 5°C, inside is at 20°C.

So, δ=15°.

Therefore, the heat flux through the wall (at any distance from the inside surface) will be

Q = 0.6·12·15/0.2 = 540W

That is exactly how much heat we need to maintain the temperature in the room.

Problem 3

Our task is to determine the law of cooling of a relatively small hot
object immersed in the cool infinitely large reservoir with liquid or
gaseous substance.

We assume the volume of substance this object is immersed in to be
"infinite" to ignore its own change of temperature related to heat
emitted by our hot object.

This law of cooling should be expressed in terms of object's temperature T as a function of time t, that is, we have to find the function T(t).

Assumptions:

(a) the initial temperature of the object at time t=0 is assumed to be T0

(b) the object has a shape of a thin flat square (so, its temperature is changing simultaneously in all its volume) of size LxL and mass m

(c) the specific heat capacity of the object's material is C

(d) the temperature of the substance surrounding our object is constant and equals Ts

(e) the convective heat transfer coefficient of the substance around our object is h.

Solution

Consider a small time interval from t to t+Δt.

The temperature of an object is T(t), while the temperature of the substance around it is constant Ts.
Then the amount of heat transferred from our object to the substance
around it per unit of time per unit of its surface area is proportional
to the difference in temperatures with the convective heat transfer coefficient as a factor:

q = −h·[T(t)−Ts]

Since total surface area of our thin flat square, ignoring its thickness, is 2L² and the time interval we consider is Δt, the total amount of heat transferred by our object through its surface during this time period is

ΔQ(t) = −2L²·h·[T(t)−Ts]·Δt

This is amount of heat taken away by the substance from our object during a time interval Δt.

The same amount of heat is lost by the object, taking its temperature from T(t) to T(t+Δt).

As we know, changes of heat and temperature are proportional and related to the object mass and specific heat capacity:

ΔQ(t) = C·m·ΔT

where ΔT = T(t+Δt)−T(t).

Equating the amount of heat lost by our object to the amount of heat
carried away by convection of the substance around it, we have come to
an equation

−2L²·h·[T(t)−Ts]·Δt =

= C·m·
[T(t+Δt)−T(t)]

Dividing both parts by Δt, diminishing this time interval to zero and using a derivative by time t to express the limit, we get the following differential equation

−2L²·h·[T(t)−Ts] =

= C·m·
dT(t)/dt

To solve it, let's make two simple substitutions:

(a) A = 2h·L²/(C·m)

(b) X(t) = T(t) − Ts

Then, since Ts is constant,

dT(t)/dt = dX(t)/dt

and our differential equation looks like

dX(t)/dt = −A·X(t)

To solve this, we convert it as follows:

dX(t)/X(t) = −A·dt

d[ln(X(t))] = −A·dt

Integrating:

ln(X(t))] = −A·t + B,

where B is any constant, defined by initial condition

X(0)=T(0)−Ts=T0−Ts.

From the last equation:

X(t) = eB·e−A·t

Using initial condition mentioned above,

X(t) = (T0−Ts)·e−A·t

Since X(t)=T(t)−Ts

T(t) − Ts = (T0−Ts)·e−A·t

or

T(t) = Ts + (T0−Ts)·e−A·t

where A = 2h·L²/(C·m)

So, the difference in temperature between a hot object and infinitely
large surrounding substance is exponentially decreasing with time.

## Tuesday, June 25, 2019

### Unizor - Physics4Teens - Energy - Heat Transfer - Radiation

Notes to a video lecture on http://www.unizor.com

Heat radiation IS NOT the same as radioactivity. Though, under certain circumstances (like an explosion of an atomic bomb) the heat radiation and radioactivity are both present. When we discuss the heat radiation we talk about a process that occurs in any object with a temperature greater than absolute zero, while radioactivity occurs in extreme cases of very high energy output.

Heat transfer through radiation is totally different from conduction and convection. The most important property of heat transfer by radiation is that heat transfer occurs without any visible material conduit that carries the heat, like molecular movement in two other cases.

Let's start from an example.
The brightest example is our Sun, as a source of heat energy. Between Sun and Earth there is no visible material conduit, yet the heat comes to Earth and is a source of life on our planet.

The fundamental concept that lies in the foundation of a process of heat radiation is a concept of a field.
The field is a region of space, where certain forces act on certain objects without visible material medium.
As an example of the field, consider gravity. The Sun keeps planets on their orbits, the Earth keeps the Moon circling around, people are walking on the ground without flying away to stars etc. We did not know much about WHY the gravitational field exist, yet we did study its behavior, the forces involved and the laws of motion in this field.

There are other fields.
Magnetic field around our planet, acting on a compass, forces the arrow to point North.
Electric field exists around electrically charged objects, so other electrically charged objects are attracted to or repulsed from it.

In Physics we successfully study these fields, but complete understanding of WHY they have the properties that we observed is not completely clear. So, we will concentrate on properties, answering the question HOW?, not on a more fundamental question WHY?.

Let's start with a particular field called electro-magnetic. Very simplified description of this field is as follows.

Any electron creates an electric field around itself. Moving electrons, which we call electric current or electric field that changes in time, also create a magnetic field around them. So, changing in time electric field creates magnetic field that changes in space. It's an experimental fact, and we have the whole theory about properties of these fields.

Consider an experiment, when you move a metal rod or any other electrical conductor in a magnetic field or change a magnetic field around any electrical conductor, thus creating a magnetic field that changes in time. You will observe that there is an electric current in the conductor, thus creating an electric field that changes in space. It's an experimental fact, and we also have the whole theory about properties of this process.

So, electric field creates magnetic field, which, in turn, creates electric field etc. This is a loop of energy conversion that propagates extremely fast, with a speed of light, about 3·108m/sec.

The combination of electric and magnetic forces form electro-magnetic field that propagates much faster than its physical medium - electrons. So, the propagation of the electro-magnetic field seems to be a self-sufficient process, occurring without the medium. This is a very brief and unsatisfactory explanation of the nature of the electro-magnetic field. We will not go much further in this explanation, but rather concentrate on the properties of the electro-magnetic field.

Assuming that we accept the existence of the electro-magnetic field and, however uncomfortably we feel about it, but accept that there is no need for a medium to propagate this field, we can talk about frequency of electro-magnetic transformations, that can be considered similar to oscillation of molecules in a solid. Inasmuch as the oscillations of molecules in a metal are propagated, thus transferring heat energy from hot area to cold one, oscillations of the electric and magnetic components of the electro-magnetic field transfers energy.
This energy transfer by electro-magnetic field is called radiation.

As in a case of oscillating molecules in a solid, carrying more heat energy when oscillation is more intense (higher frequency), the electro-magnetic field oscillation carries energy with higher frequencies being more "energetic" than lower.

Interestingly, receptors in our skin feel the temperature of a solid object, that is, we feel the intensity of oscillation of its molecules. Similarly, we feel the warm rays of Sun on our skin, that is, we feel the intensity of electro-magnetic oscillation of the electro-magnetic field.
What's more remarkable, we see the light. Apparently, electro-magnetic oscillations in certain frequency range act upon censors in our eyes, thus we see the light. Moreover, in this visible range of frequencies different frequencies of electro-magnetic oscillation produce effect of different colors in our eyes.

As you see, the light and heat of radiation have the same source - the oscillation of electro-magnetic field, the only difference is the frequency. In other words, the light and heat radiation are manifestations of the same process of transferring energy by the oscillations of the components of the electro-magnetic field.

An object does not have to have a temperature of the Sun to emit heat radiation. All objects that have temperature higher than absolute zero emit thermal radiation of some frequencies. Usually, the whole spectrum of frequencies of electro-magnetic oscillations is emitted by objects. Lower frequencies (usually called infrared) are felt by skin receptors, higher frequencies are visible by an eye. Frequencies higher than those visible by a human eye are called ultraviolet. Even higher frequencies are called X-rays, which can be produced by special equipment and, depending on intensity and time of exposure, can represent a health hazard. Even higher intensity and high frequencies are called gamma rays, and they are produced in extreme cases like nuclear explosion or nuclear reactor meltdown, and they are extremely dangerous and are usually meant, when the term radioactivity is used. All frequencies can be observed using some scientific instruments.

Any object, placed in the outer space will emit its heat energy through radiation until its temperature will reach absolute zero. Our Sun emits huge amounts of energy in all spectrum of frequencies in all directions and, eventually, run out of heat energy and go dark.

The intensity of radiation, that is amount of heat radiated per unit of time per unit of area of an object depends, as in other cases of heat transfer, on the temperature of an object and temperature of surrounding environment.
In the complete vacuum with no other source of energy the radiation intensity of an object is proportional to the fourth degree of its absolute temperature in °K:
q = σ·T4 where
σ = 5.67·10−8 W/(m2·°K4)
is the Stefan-Boltzmann constant.
This is the Law of Stefan-Boltzmann. Its derivation is complex and is outside of the scope of this course.

Radiation is not only emitted by objects with temperatures above absolute zero, but also can be absorbed by them and even reflected. While ability to absorb the heat is common for other heat transfer types (conduction and convection), reflection is a specific property of heat radiation. More precisely, it's a specific property of oscillations of the electro-magnetic field.
Obvious application of this property is the usage of mirrors that reflect the oscillations of the electro-magnetic field in a very broad spectrum of frequencies, including the visible light.
An example of absorbed radiation is a slice of bread toasted in the electric toaster. It absorbs the thermal radiation emitted by electric coils, that changes the bread's structure.

## Friday, June 21, 2019

### Unizor - Physics4Teens - Energy - Heat Transfer - Convection

Notes to a video lecture on http://www.unizor.com

Heat Transfer - Convection

As in a case of conduction, we start with a statement: heat is a form of internal energy that is related to molecular movement.
However, while heat transfer during the process of conduction occurs between molecules oscillating around their relatively fixed positions and transferring their internal energy by "shaking" the neighboring molecules, convection occurs when molecules are free to travel in different directions and carry their internal energy with them.

In other words, conduction is a pure transfer of energy on a micro level from one oscillating molecule in a relatively fixed position to another such molecule, while convectionoccurs when molecules freely fly away from their positions, carrying their internal energy with themselves, thus transferring energy on a macro level.

It should be noted that, when dealing with solid objects, conduction is a prevailing way of heat transfer, while in liquids and gases the main way of heat transfer is convection. It does not mean that conduction does not occur in liquids or gases, it does, but it does not constitute the major way of heat transfer. Much more heat is transferred through the mechanism of convection

Here are a few examples of heat transfer through convection:
(a) heating up water in a pot; heat is carried from hot bottom of a pot up by hot (fast moving with high kinetic energy) molecules;
(b) circulation of air in the atmosphere from hot places to cold;
(c) circulation of water in oceans from hot places to cold.

Describing convectionmathematically is not a simple task.
While in case of conduction we can use a relatively simple Fourier's Law of Thermal Conduction
q(x) = −k·dT(x)/dx
that describes the heat flow as a function of how fast the temperature between the layers of conducting material changes (dT(x)/dx) and properties of the material itself (conductivity coefficient k), the process of convection is significantly more complex, described by convection-diffusion differential equations that are beyond the scope of this course.

However, for practical purposes we can use a similar formula that puts the amount of heat transferred by convectionprocess in a liquid or gas during a unit of time through a unit of area as proportional to a difference of temperatures between the layers of liquid or gas and a convective heat transfer coefficient h that depends on the physical properties of this liquid or gas:
q = −h·(T2−T1)

This formula puts amount of heat q going through a layer of a unit area of liquid or gas during a unit of time as proportional to a difference of temperatures between bounding surfaces of this layer T2−T1 and some physical properties of liquid or gas expressed in convective heat transfer coefficient h that, in turn, depend on such properties as viscositydensity, the type of flow (turbulent or laminar) etc.

Consider an example.
A round steam pipe of temperature 100°C goes through a room with air temperature 25°C. We have to calculate the amount of heat from the pipe to select an air conditioner required to neutralize the heat from a pipe and keep the room temperature at that level.
Assume that the pipe's length is 4m, diameter 0.2m and the convective heat transfer coefficient of air is 40J/(sec·m²·°C). As we know, J/sec is a unit called "watt", so we will use W instead of J/sec.

The heat transfer per unit of time through a unit of area of a pipe is, therefore,
q = 40·(100−25) = 3000(W/m²)
The pipe's area is
A = π·0.2·4 = 2.512(m²)
Therefore, the pipe is producing the following amount of heat:
Q = 3000·2.512 = 7536(W)

So, we need an air conditioner that can extract 7536W of heat from the room to maintain stable temperature of 25°C.
Usually, the power of air conditioners is measured in BTU/hr (1 watt = 3.41 BTU/hr). So we need an air conditioner of approximately 2200 BTU/hr - a relatively small one.

Another example.
Outside temperature is 40°C, inside a room we want temperature 25°C. The glass wall between a room and outside air has an area of 20m². What kind of air conditioner is needed to maintain the room temperature at 25°C, assuming the convective heat transfer coefficient of air is 40W/(m²·°C)?

Q = 40·(40−25)·20 = 12000(W)
This is equivalent to about 3500 BTU/hr.

## Thursday, June 20, 2019

### Unizor - Physics4Teens - Energy - Heat Transfer - Conduction

Notes to a video lecture on http://www.unizor.com

Heat Transfer - Conduction

As we know, heat is a form of internal energy that is related to molecular movement.

For solids the molecular movement is usually restricted to molecules' oscillation around some neutral positions.

For liquids the freedom of molecular motion is greater, but still restricted by external forces, like gravity, and surface tension. The average distance between molecules of liquids is relatively constant.

Gas molecules are usually taking all the space available for them. Such forces as gravity also restrict their movement (otherwise, the air molecules would fly away from our planet), but still allow substantial freedom. The average distance between molecules of gas mostly depends on a reservoir the gas is in, the larger the reservoir - the larger average distance between molecules.

Transfer of heat is transfer of molecular movement from one object or part of an object to another object or part of an object, from an object or part of an object with more intense molecular movement (relatively warmer) to an object or part of an object with less intense movement (relatively cooler).

There are three major ways to transfer heat from a hot object to a cold one:
Conduction,
Convection,
This lecture explains a concept of conduction.

Conduction

Conduction of heat energy is a transfer of molecular movement mostly applicable to solid objects - between two solid objects that touch each other, having an area of a contact, or within one object, one part of it having different temperature than another.
The conductivity is present in heat transfer in liquids and gases, but there it's usually combined with another form of heat transfer - convection, while in solids it's not the case, and we can study conductivity by itself.

An example is a building wall, one side of which towards the outside having temperature of the air outside, while inner surface of the wall having room temperature. The heat energy constantly flows from a warmer surface of the wall to the opposite cooler one with some rate of flow that depends on the thermal conductivity of the wall material. The wall material with higher level of thermal conductivity will transfer more heat energy to a cooler side within unit of time per unit of area, which is usually not a desirable property of the building walls.

The mechanism of heat transfer through conductivity can be explained as follows.
Imagine two objects (or two parts of the same object), a hot one with higher intensity of molecular movement and a cold one with lower intensity level of molecular movement, that touch each other along some surface, while completely insulated from heat around them. For example, you put a cold metal spoon into a styrofoam cup with hot tea.

Molecules of a hot object are hitting the molecules of a cold one, thus forcing the molecules of a cold object to move faster. These faster molecules of a cold object, in turn, hit their neighbors, forcing them to move faster. This process of transferring heat energy through contacting surfaces continues until the intensity of molecular movement gradually equalizes on average. A hot object will lose some energy of molecular movement, while a cold one will gain it. As a result, the temperatures of both will equalize.

Since the heat energy is a kinetic energy of molecular movement, that is a sort of mechanical energy, we expect that the total amount of energy for an isolated system will remain constant, whatever a hot object loses in its kinetic energy of molecular movement will be gained by a cold object. The total amount of heat energy will remain the same.

If you put a silver spoon into a cup of hot tea, it will heat up faster than a spoon made of steel, which, in turn, will heat up much faster then a spoon made of wood. The reason for this is that the thermal conductivity of different materials is different.

We can experimentally measure the thermal conductivity of different solids by having a standard rod of any solid material at certain starting temperature and heating its one end by bringing it to contact with some hot object. Measuring the temperature on the other end after different time intervals will give us a picture of growing temperature.

Some materials with higher thermal conductivity will have the temperature at the opposite end of a rod growing faster than in case of other materials.
Metals have much higher heat conductivity then plastic or wood, for example. That's why the handle of a tea kettle is usually made of plastic or wood. Diamonds have one of the highest thermal conductivity, even higher than silver.

More precise definition of thermal conductivity is related to a concept of heat flux(sometimes, called heat flow density or thermal flux, or thermal flow density). Heat flux is an amount of heat energy flowing through a unit of area during a unit of time.

Let's examine how heat flows through a building wall made of some uniform material from a warm room to cold air outside the building.
Assume, the room temperature is Troom and the cold air outside the building has temperature Tair . If the thickness of a wall is L, the temperature inside the wall T(x), as a function of the distance x from the surface facing outside, gradually changes from T(0)=Tair to T(L)=Troom.

It is intuitively understandable and experimentally confirmed that amount of heat energy flowing through a unit of area of such a wall during a unit of time (heat conductivity of a wall) is proportional to a difference between temperatures Troomand Tair and inversely proportional to a thickness of a wall L:
q = −k·(Troom − Tair / L
(negative sign is used because the flow of heat is opposite to a direction of temperature growth).

The situation with heat fluxmight be compared with a water flow down a river between two points A and B. The difference in levels above the sea level of these points is similar to a difference in temperature between the inside and outside walls of a building. The distance between points A and B is similar to a thickness of a wall. It's reasonable to assume that amount of water flowing through a unit of area in a unit of time will be proportional to a difference between the levels of points A and B above the sea level and inversely proportional to a distance between these two points.

To make this definition of the heat flux more precise and independent of the way how the heat flows inside the wall, let's consider a thin slice of wall parallel to both sides from a point at distance x from the outdoor cold side to a point at distance x+Δx.
The heat flow through this thin slice of a wall, as a function of distance x, can be expressed similarly to the above:
q(x)=−k·[T(x+Δx)−T(x)] /Δx

Next step is, obviously, to reduce the thickness of the slice by making Δx infinitesimal, that is Δx→0, which leads to the following definition of the heat flux:
q(x) = −k·dT(x)/dx
This definition was formulated by Fourier in 1822 and is called Fourier's law of thermal conduction.
The coefficient k is called thermal conductivity.

To find the amount of heat Q(x,A) going through an area Aat distance x from the outside wall during a unit of time, we have to multiply the heat flux by an area:
Q(x,A) = −k·A·dT(x)/dx
As in a case of water flow along the river, if the temperature is linearly dependent on the distance from the outside wall, the derivative is constant and the flow of heat is constant. But, if the wall material is uneven, like in case of a river bed not being a straight line down, the heat flow rate will change.

## Wednesday, June 12, 2019

### Unizor - Physics4Teens - Energy - Measuring Heat - Problems

Notes to a video lecture on http://www.unizor.com

Measuring Heat - Problems

Problem 1

How much heat energy is required to raise the temperature of 1 kg of water from 20°C to a boiling point of 100°C?
Assume the specific heat capacity of water is
Cw = 4184 J/(kg·°C).

Q = C·m·(Tend−Tbeg) =
= 4184·1·(100−20) =
= 334,720 J

Problem 2

A piece of unknown metal of mass Mm and temperature Tm was put into an isolated reservoir filled with Mw mass of water at temperature Tw. After the system of water and metal came to thermal equilibrium, its temperature became T.
Assume that the metal is not too hot (so, water will not vaporize) and not too cold (so, the water will not freeze).
Assuming that the water's specific heat capacity is known and equals to Cw, what is the specific heat capacity Cm of the unknown metal?

Cw·Mw·(T−Tw) =
= Cm·Mm·(Tm−T)

from which Cm equals to
Cw·Mw·(T−Tw)/[Mm·(Tm−T)]

Problem 3

A burger has about 300 kcal of energy in it.
1 kcal = 4184 J.
A person, who ate it, wants to spend this energy by climbing up the stairs. A person's mass is 75 kg, the height between the floor is 3 m.
Assume that only 25% of energy in the food can be used for climbing, while the other 75% is needed to maintain our body's internal functions.
Counting from the ground floor as floor #0, to what floor can a person climb using that energy from a burger?

Ebur = 300kcal · 4184J/kcal =
= 1,255,200
J

Eclimb = 0.25·Ebur =
= 313,800
J

Efloor = 75kg · 9.8m/sec² · 3m =
= 2205
J

N = Eclimb / Efloor ≅ 142 floors

Problem 4

An ice of mass 0.1kg has temperature −10°C.
What's the minimum amount of water M at temperature 20°C needed to melt it?
Assume, specific heat capacity of water is 4183J/(kg·°C) and that of ice is 2090J/(kg·°C). Assume also that the amount needed to melt ice at 0°C is 333,000J/kg.

Ewarm = 2090·0.1·10 = 2090J
Emelt = 333000·0.1 = 33300J
Eneed = 2090 + 33300 = 35390J
Ewater = 4183·M·20 = 83660·M
Ewater = Eneed
35390 = 83660·M
M = 0.423kg

## Monday, June 10, 2019

### Unizor - Physics4Teens - Energy - Heat - Heat & Temperature

Notes to a video lecture on http://www.unizor.com

Heat and Temperature

Specific Heat Capacity

Let's discuss the relationship between heat and temperature.
From the unscientific standpoint these two concepts are almost identical. Heating an object results in increase of its temperature. Increasing a temperature of an object constitutes its heating.
Yet, from the strictly scientific viewpoint these two concepts are different.

Heat is energy of some specific type, that can be transferred from one object to another, while temperature is an average kinetic energy of molecules of an object.
The same amount of heat, transferred into different objects, will result in different growth of their temperatures, depending on many factors, like mass, chemical composition, state etc. of these objects. Analogy of this is that same amount of fuel in different cars results in achieving very different speeds and, therefore, different kinetic energies in different cars, even if the gas pedal is pushed all the way down for all cars. It's just because cars are different and their internal structure converts fuel into movement differently.

It has been experimentally observed that to increase a temperature of an isolated object of a unit mass by a unit of temperature is independent (within reasonable level of precision) of the initial temperature of an object, but depends only on the type of object's material, composition, state etc. In other words, amount of heat needed to increase a temperature of 1 kg of water from 20°C to 21°C is the same as from 50°C to 51°C. If, instead of water, we take copper, the amount of heat, needed to increase its temperature from 20°C to 21°C, will be the same as to increase it from 50°C to 51°C, but different than that for water.

The above experimental fact allowed to establish a concept of specific heat capacity for each material as an amount of heat required to increase the temperature of a unit of mass of this material (1 kg in SI) by a unit of temperature (1°C or 1°K in SI).

Thus, specific heat capacity of water is, as we know, one kilocalorie per kilogram per degree - 1 kcal/(kg·°K), that is about 4184 joules per kilogram per degree - 4183 J/(kg·°K).
For copper the specific heat capacity is 385 J/(kg·°K).
Gold has the specific heat capacity of 129 J/(kg·°K).
Uranium's specific heat capacity is 116 J/(kg·°K).
Cotton's specific heat capacity is 1400 J/(kg·°K).
Hydrogen's specific heat capacity is 14304 J/(kg·°K).
Generally speaking, but not always, more dense, more solid materials have less specific heat capacity than less dense or liquids, which have, in turn, less specific heat capacity than gases.

Knowing specific heat capacityC of material of an object and its mass m, we can easily determine amount of energy ΔQneeded to heat it up by ΔTdegrees:
ΔQ = C·m·ΔT
Inversely, knowing the amount of heat supplied, we can determine an increase in temperature:
ΔT = ΔQ/(C·m)
Notice, that increment of temperature ΔT and increment of heat energy ΔQ can be both positive or both negative, which means that an object, that has increased its temperature, has increased (gained, consumed) energy, and the object that decreased its temperature, has decreased (lost, released) energy.

Change of State

Consider specific heat capacity of ice and water:
Ice: 2090 J/(kg·°K)
Water: 4183 J/(kg·°K)
Both these substances exist at temperature about 0°C=273°K. That means that we can heat 1 kg of ice up to 0°C spending 2090 joules per each degree of temperature, but to increase the temperaature of the water around 0°C we have to spend 4183 joules per each degree. But, while ice is melting into water, which takes some time, both states, solid and liquid, exist side by side and the temperature of the water will not rise while the ice is not completely melted, we have to spend heat energy just on melting without actually changing the temperature of a substance, that remains around 0°C during the melting process.

This experimental observation leads us to believe that, while graduate change of temperature for any specific state of matter linearly depends on the amount of heat supplied, change of state (like melting or freezing, or evaporating etc.) brings an element of non-linearity to this dependency.
More precisely, the temperature, as a function of amount of heat supply, in case of an object going through transformation of state from solid to liquid, looks like this (for better view, right click on the picture and open it in another tab of a browser):

As seen on this graph, supplying heat to ice will increase its temperature proportionally to amount of heat supplied.
Then, when the temperature reaches 0°C, ice will start melting and new heat will not change the temperature of ice/water mix, but will be used to change the state of matter from solid (ice) to liquid (water).

This process of melting consumes heat without increasing the temperature until all ice is melted. Such a transformation of state that requires supply of heat energy is called endothermic.
Then, when the process of melting is complete, and all ice is transformed into water, the temperature of water will start increasing, as the new heat is supplied, but with a different coefficient of linearity relatively to the heat than in case of ice, since water's specific heat capacity is different than that of ice.
The graph above is characteristic to any heating process, where the transformation of the state of matter is involved. It is relatively the same for transforming liquid to gas (evaporation) or solid to gas (sublimation).

In case of a reversed transformation from liquid to solid (freezing) or gas to liquid (condensation) or gas to solid (deposition) the heat energy must be taken away from a substance. It is the same amount in absolute value as was needed to supply to solid to melt it into liquid or to liquid to vaporise it into gas, or to solid to vaporize it into gas. So, in case of freezing, condensation or deposition we deal with the process of decreasing heat energy of a substance. Such a process is called exothermic.

Amount of heat energy needed to transform a substance from one state to another is also experimentally determined and, obviously, depends on a substance, its mass and a kind of transformation it undergoes.
For example, melting of 1 kg of ice requires 333,000 joules of heat energy to be supplied. The same amount of heat energy should be extracted from the water at 0°C to freeze it into ice.

### Unizor - Physics4Teens - Energy - Heat - Measuring

Notes to a video lecture on http://www.unizor.com

Measuring Heat - Calorie

Heat and temperature were known and researched by scientists long before the molecular movement was determined as their essence. As a result, attempts of measuring the amount of heat were unrelated to kinetic energy of the molecules.

The unit of amount of heat was defined as amount of heat needed to increase the temperature of one gram of water by one degree Celsius (or Kelvin), and this unit of amount of heat was called a calorie (cal).

Obviously, this definition has its flaws. For example, the amount of heat needed to heat the same amount of water by the same temperature depends where exactly on Earth we are. The higher we are above the sea level - the less heat is needed. It also depends on the chemical purity of water. Also it's not obvious that the amount of heat needed to heat the water from 1°C to 2°C is the same as amount of heat needed to heat it from 88°C to 89°C, though within reasonable level of precision we do assume that this is true.

With the development of molecular theory of heat and establishing relationship between heat and kinetic energy of molecules there was a need to put into correspondence existing units of heat (calories) and units of mechanical energy (joules). A simple experiment allowed to do just that.

Imagine a standing on the ground large reservoir with known amount of water M of depth H from the surface to the bottom, kept at certain known temperature T°, and a relatively small stone of known mass m, having the same temperature T°, kept on the level of the surface of the water in a reservoir. This stone has certain known amount of potential energy E=m·g·Hrelatively to the bottom level of a reservoir.
Now we let the stone go down the reservoir to its bottom. Its potential energy relatively to the bottom of a reservoir decreases to zero. Where did the potential energy go? It's used to stir the water, thereby increasing the kinetic energy of its molecules.

Because of this more intense movement of molecules, the water will increase its temperature. Potential energy of a stone will turn into kinetic energy of the molecular movement of water. If the temperature of the water has risen by ΔT°, potential energy of a stone m·g·H (in joules) equals to M·ΔT (in calories).

Precise experiments like the above allowed to determine the correspondence between historical measure of the amount of heat in calories under different conditions and contemporary one in units of energy in SI - joules. This correspondence had been established approximately as
1 calorie = 4.184 joules
but under different conditions (initial temperature, air pressure etc.) it might be equal to a slightly different value.

Besides calorie, which is a relatively small amount of heat, the unit kilocalorie (kcal) had been introduced.
As is obvious from its name, 1 kilocalorie = 1000 calories.

The amount of energy contained in food and in some other practical cases very often is measured in kilocalories, which sometimes are called large calories, while a calorie is sometimes called small calorie. Unfortunately, the word "large" in many cases is omitted, which might cause misunderstanding.
From the experimental viewpoint, 1 kcal is amount of heat needed to heat 1 kg of water by one degree Celsius (or Kelvin).

To avoid problems with the definition of calorie as amount of heat needed to warm up one gram of water by one degree Celsius, the contemporary scientific definition of thermocalorie is
1 thermocalorie = 4.1833 joules

## Monday, June 3, 2019

### Unizor - Physics4Teens - Energy - Ideal Gas Kinetics - Problems

Notes to a video lecture on http://www.unizor.com

Problem 1

How much kinetic energy have all the molecules in the room?
How fast should an average size car move to have this amount of kinetic energy?

Assume the following:
(a) the room dimensions are 4x4x3 meters (that is, V=48m³);
(b) normal atmospheric pressure is 100,000 Pascals (that is, p=100,000N/m²);
(c) a mass of an average size car is 2,000 kg (that is, M=2,000kg).

Solution

From the lecture on Kinetics of Ideal Gas we know the relationship between the pressure on the walls of a reservoir, volume of a reservoir and total kinetic energy of gas inside this reservoir:
p = (2/3)Etot /V

From this we derive a formula for total kinetic energy:
Etot = (3/2)·p·V

Substituting the values for pressure and volume, we obtain
Etot = (3/2)·100,000·48 =
= 7,200,000
(joules)

The kinetic energy of a car is
E = M·v²/2
Therefore, given the kinetic energy and mass, we can determine the car's speed:
v = √2·E/M
Substituting calculated above Etot=7,200,000(J) for E and the value for mass M=2,000(kg), we obtain
v = √2·7,200,000/2,000
≅ 85
(m/sec)
≅ 306
(km/hour)
≅ 190
(miles/hour)

Problem 2

Given the temperature, pressure and volume of the air in a room, determine the number of gas molecules in it.

Assume the following:
(a) the room dimensions are 4x4x3 meters (that is, V=48m³);
(b) normal atmospheric pressure is 100,000 Pascals (that is, p=100,000=105N/m²);
(c) temperature is 20°C (that is, T=20+273=293°K).

Solution
Recall the combined law of ideal gas
p·V/T = kB·N = const
where
kB = 1.381·10−23 (J/°K) is Boltzmann's constant and
N is the number of gas molecules in a reservoir.
From this we derive the number of molecules
N = p·V/(kB·T)
Substituting the values,
N = 105·48/(1.381·10−23·293) = 0.12·1028
It's a lot!

## Monday, May 6, 2019

### Unizor - Physics4Teens - Energy - Heat - Temperature, Pressure, Volume o...

Notes to a video lecture on http://www.unizor.com

Temperature, Pressure
and Volume of Ideal Gas

Temperature is an observable macro property of an object. It's related to a particular instrument we use to measure this property.
Let's examine the mechanism of this measurement using a classic mercury-based or alcohol-based thermometer.

Our first step in measurement is to make a physical contact between a thermometer and an object of measurement (for example, a human body or air in a room). When accomplished, we expect that the measurement of a thermometer would correspond to a state of average kinetic energy of molecules of an object. The reason it happens (and it takes some time to happen) is that on a micro level molecules at the surface of an object are colliding with molecules at the surface of a thermometer, and exchange the kinetic energy, eventually equalizing it. The molecules close to a surface, in turn, collide with surface molecules and also eventually equalize their average kinetic energies. This process continues until the average level of kinetic energy in all parts of an object and in a thermometer equalize.

What is important in this case is that the total amount of kinetic energy of all molecules of an object and a thermometer remains the same. So, if an object has more intense movement of molecules and a thermometer's molecules are moving slower, the kinetic energy is transferred from an object to a thermometer. If the molecules of a thermometer are, on average, faster, then the exchange of kinetic energy will be from a thermometer to an object.
In any case, the average kinetic energy of molecules of both an object and a thermometer equalize.

An important consideration is that the contact between an object and a thermometer changes the average level of kinetic energy in both. The process of measuring, therefore, is not completely neutral towards an object. However, what happens in most cases is that the number of molecules inside an object we measure is usually significantly greater that the number of molecules in a thermometer. As a result, equalizing the average kinetic energy of all molecules does not significantly change the level of average kinetic energy of molecules of an object, and the level of average kinetic energy of the molecules of a thermometer is a good representation of this characteristic of an object.

As explained in the Heat and Energy lecture of this course, the temperature in mercury or alcohol thermometers is an observable expansion of the volume of liquid inside a thermometer. We also indicated in that lecture that this thermo-expansion is proportional to an average of squares of velocities of molecules, that is proportional to average kinetic energy of the molecules of a thermometer, which, in turn, is equalized with average kinetic energy of molecules of an object.

Thus, by observing the expansion of liquid in a thermometer we measure the average kinetic energy of molecules of an object, which allows us to write the following equation:
T ≅ AVE(Ekin) = Eave
where T is an observable level of liquid inside a thermometer in some units (that is, temperature) and
AVE(Ekin) = Eave is average kinetic energy of molecules of an object.
This looks more natural in a form
AVE(Ekin) = Eave ≅ T
since average kinetic energy of molecules Eave (micro characteristic) cannot be easily observed, while the temperature T (macro characteristic) can.

So, we have established that the temperature (in some units, starting from absolute zero) and average kinetic energy of molecules are proportional. The coefficient of proportionality between the temperature and average kinetic energy of molecules remains unknown and is different for different substances.

Obviously, the measure of liquid in a thermometer should be calibrated and, for this equation to be true, we have to assign the zero temperature to a state of an object when all its molecules are at rest, which happens when there is no source of energy around, like in open space far from stars.
The convenient scale is the Kelvin scale with zero temperature on this scale corresponding to this state of molecules at complete rest and a unit of measurement of the temperature is a degree with the distance between the temperature of melting ice and boiling water assigned as 100 degrees on this scale.

From the previous lecture about kinetics of ideal gas we know the relationship between the pressure of ideal gas on the walls of a reservoir, the volumeof a reservoir and average kinetic energy of the molecules
p = (2/3)N·Eave /V
where
p is the gas pressure against the walls of a reservoir,
N is the number of gas molecules in a reservoir,
V volume of a reservoir,
Eave is average kinetic energy of molecules,

Alternatively, it can be written as
Eave = (3/2)·p·V/N

Comparing this with a derived above relationship between temperature (counting from the absolute zero) and average kinetic energy of molecules, we have established a relationship
Eave = (3/2)·p·V/N ≅ T
and
p·V/N = k·T
where k is an unknown coefficient of proportionality.

The only thing that prevents us from determining average kinetic energy of molecules by its temperature is an unknown coefficient of proportionality k.

Now we will concentrate attention on gases, as an object of measuring temperature and average kinetic energy of molecules.
Different gases have different molecules and different molecular mass. Using certain theory and using chemical and physical experiments, we can compare the masses of different molecules and even measure this mass in certain "units of mass" called atomic mass units.
For such a unit of molecular mass scientists used 1/12 of a mass of a single atom of carbon. So, in this system of units hydrogen molecule of 2 hydrogen atoms H2 has approximate mass of 2, oxygen molecule of two oxygen atoms O2 - approximately 32, carbon dioxide molecule CO2 had a measure of, approximately, 44 atomic mass units, etc.

Using this measurement, we can always establish experiments with the same number of molecules of different gases. For example, if the mass of certain amount of oxygen (atomic mass of molecules O2 is 32) is 16 times greater than the mass of certain amount of hydrogen (atomic mass of molecules H2 is 2), we can assume that the number of molecules in both cases is the same.

It has been experimentally established that, if the same number of different gas molecules are placed in reservoirs of the same volume and hold them at the same temperature, the pressure on the walls in both cases will be the same. Alternatively, if the pressure is the same, the temperature will be the same too.
In other words, the coefficient kin formula
p·V/N = k·T
does not depend on the type of gas we deal with, it's a universal constant called the Boltzman's constant, which is equal to
kB = 1.381·10−23 (J/°K)
This was the reason to introduce a concept of ideal gas. All gases are, approximately, ideal to a certain degree of precision. This is related to the fact that molecules of the gas are flying with high speeds and on large distances from each other, much larger than their geometric sizes.

Now we can write the equation between the temperature T (in degrees °K from absolute zero), average kinetic energy of molecules (in units of SI joules J), pressure p (in units of SI newton/m²), volume V (in ) and number of molecules in a reservoir for ideal gas:
Eave = (3/2)·p·V/N = (3/2)·kB·T

Consequently, if we are dealing with certain fixed amount of gas (N molecules) then
p·V/T = kB·N = const
That means that changing the pressure, volume and temperature of the same amount of gas preserves the expression p·V/T
which is called the Combined Ideal Gas Law.

For example, if the absolute temperature remains the same, but volume taken by certain amount of gas increases (decreases) by some factor, the pressure will decrease (increase) by the same factor, that is pressure and volume are inversely proportional to each other (Boyle-Mariotte's Law).

If the pressure remains the same, but the volume taken by certain amount of gas increases (decreases) by some factor, the absolute temperature will increase (decrease) by the same factor, that is volume and absolute temperature are proportional to each other (Charles' Law).

If the volume taken by certain amount of gas remains the same, but the absolute temperature increases (decreases) by some factor, the pressure will increase (decrease) by the same factor, that is absolute temperature and pressure are proportional to each other (Gay-Lussac's Law).

Out of curiosity, let's use the formula
Eave = (3/2)·kB·T
to calculate how fast the molecules of oxygen are flying in the room at some normal temperature.
Assume the pressure at the ground level is about 100,000N/m² and the temperature in the room is about 20°C=293°K. Then the average kinetic energy of a molecule of oxygen is
Eave = (3/2)·1.381·10−23·293 =
= 6.06·10−21 J

Mass of a molecule of oxygen O2 is m=5.31·10−26 kg
From the formula for kinetic energy E=m·v²/2 we derive the average of squares of velocities of oxygen molecules as
AVE() = 2·Eave /m =
= 2·6.06·10−21/5.31·10−26 =
= 2.28·105

Therefore, the average speed of oxygen molecule will be equal to a square root of this number:
AVE(v) = 478 m/sec
Pretty fast moving! Take into consideration, however, that real oxygen molecules, as molecules of any real gas, are chaotically colliding with other and change the direction all the time.