Unizor - Physics4Teens - Energy - Temperature and Kinetic Energy

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

Heat, Temperature &
Kinetic Energy of Molecules

The goal of this lecture is to demonstrate that temperature, as a measure of heat, is proportional to average of a square of molecular speed.

Temperature is what we see on a thermometer that measures the intensity of the molecular movement. Let's examine this process of measurement using the classical mercury-based thermometer.

The thermometer shows different values of temperature based on physical property of mercury to expand, as its molecules are moving faster. The reason for this is that, as the molecules move faster, they cover longer distance before they are stopped by collision with other molecules or the walls or by a surface tension forces. The faster molecules push against obstacles with greater forces, and that is the reason why the volume of hot mercury is larger than the volume of the same mass of cold mercury.

Let's quantify this increase in volume.
There is a force of resistance that acts against the movement of the molecule, as it pushes its way through, that stops this movement. In some way it's analogous to throwing a stone vertically up against the force of gravity. The force of gravity eventually stops the movement upward, and the stone falls back on the ground.

The height of the trajectory of a stone depends on the initial speed it has at the beginning of movement on the ground level. Assume, the initial speed is V. Then the kinetic energy in the beginning of motion is M·V²/2, where M is a mass of a stone. The potential energy relative to the ground in the beginning of motion is zero.
At the highest point at height Hthe kinetic energy is zero but potential is M·g·H, where g is a free fall acceleration.

As we know, the total mechanical energy is constant in this case, from which follows that kinetic energy in the beginning of motion (with potential energy zero) equals to potential energy at the highest point (with kinetic energy zero):
E_kin = M·V²/2 = M·g·H
From this follows
H = V²/(2g)
So, the height of the rise and the square of initial velocity are proportional to each other, independently of mass.

Obviously, one mercury molecule does not do much, but when the heat is relatively evenly distributed among all molecules, their combined efforts push the surface of the mercury in the tube sufficiently noticeably, and the height it rises is proportional to average of a square of molecular speed.

The thermometers are different, their design and construction differ, they all must be calibrated, but one thing remains - a principle of proportionality between geometrical expansion (that is, temperature, as we read it on a thermometer's scale) and an average of squares of the molecular speed.

It is important to point out that this proportionality should be understood in absolute terms. That is, if the speed is zero, the temperature is zero as well. So, we are talking about temperature scales with zero representing absolute zero on the Kelvin's scale, like in space far from any source of energy.
The coefficient of proportionality, of course, depends on the units of measurement.

So, if the average speed of molecules doubles, the temperature on the Kelvin's scale rises by a factor of 4.

Inversely, if an object is at temperature 0°K (absolute zero), the kinetic energy of its molecule is zero, that is no molecular movements.
If we observe that the temperature has risen by a factor of 4, we can safely assume that the average speed of molecules doubles.

Unizor - Physics4Teens - Energy - Heat - Molecular Movement

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

Molecular Movement

Molecules

In order to understand the nature of heat and temperature, we have to go inside the objects we experiment with.
If we divide a drop of water into two smaller drops, each of these smaller drops will still be water. Let's continue dividing a smaller drop into even smaller and smaller. There will be a point in this process of division, when further division is not possible without changing a nature of the object, it will no longer be water.

From what we know now, the water consists of hydrogen and oxygen - two gases, connected in some way. In our process of division of a drop of water, we will reach such a point that, if we divide this tiniest drop of water, the result will be certain amount of hydrogen and certain amount of oxygen, but no water.
That tiniest amount of water, that still preserves the quality of being water, is called a molecule.
Similarly, tiniest amount of any substance, that retains the qualities of this substance, is called a molecule.
Incidentally, if we divide a molecule of any substance, the result will be certain atoms, that possess completely different qualities.

In our discussion about heat we will not go deeper than the molecules because our purpose at this stage is to study the nature of heat as it relates to different objects and substances, so the preservation of the qualities of these objects and substances is important. That's why in this part of a course we will not cross the border between molecular and more elementary atomic level.

Any object or substance we are dealing with consists of certain number of molecules - the smallest particles that retains the qualities of this object or substance. This number of molecules, by the way, for regular objects we see and use in practical life, is extremely large because the size of molecules is extremely small. We cannot see individual molecules with a naked eye. Only special equipment, different in different cases, can help us to see individual molecules. And, being so tiny, molecules of different substances are different in size among themselves.
And not only in size. Since the molecules contain different, more elementary particles called atoms, the configuration of these atoms that form a molecule is different for different molecules. Thus, a molecule of water contains two atoms of hydrogen and one atom of oxygen that connects hydrogen atoms into some three-dimensional construction. A molecule of protein consists of many different types of atoms and its structure and size are quite different from the molecule of water.

States of Matter

The next topic we would like to address is the states of matter.
When the word matter is used in physics, it means any object or substance that occupies certain space and has certain mass, thus consisting of certain molecules interacting among themselves.

There are three major states of matter: solid, liquid and gas.

When an object is solid or is in solid state, it means that it retains its shape and form regardless of surrounding environment, not intended to change its form. The molecules of this object are strongly connected to each other. Their movement relative to each other is rather restricted. This movement can be oscillating around some point, maintaining an orderly three-dimensional structure, for crystal (or crystalline type of) solids or just slow movement, changing their relative position, but not changing the overall form for amorphous (non-crystalline) type.
Examples of solid objects are ice (crystal), steel (crystal), plastic cup (amorphous).

When an object or substance is liquid or is in liquid state, it takes the shape of a vessel or reservoir it's in. The connection between the molecules in case of liquid is strong enough to hold the molecules together, but not strong enough to preserve the overall shape.
Examples of liquid substances are water, mercury ("quick silver"), oil.

When an object or substance is gas or is in gaseous state, it takes as much space as it is available. Connections between the molecules are weak and they fly in all directions in completely chaotic fashion. Examples of gases are air, helium, oxygen.

Some examples above represent objects or substances that contain only one type of molecules, like ice or mercury, or helium. Some other examples are objects or substances that contain more than one type of molecules mixed together, like steel, oil or air.

Nature of Heat

Now we are in position to talk about heat.
Heat is the energy of molecular motion inside any object or substance.
As we mentioned, molecules are in constant motion inside any object. The more intense this motion is - the more heat this object possesses. This implies that heat is mechanical energyof molecules inside the object or internal energy of the object or substance.

As we know, mechanical energy can be transferred from one object to another, like during the collision of two billiard balls. Similarly, mechanical energy of one molecule can be transferred to its neighbors, from them - to their neighbors etc. This is a process of dissipation of heat. All what's necessary for this is the relative proximity of the molecules. This is exactly the way how heat is transferred from one body to another, from flame to pot, from pot to water, from water to vegetables in it, making soup.

Since heat or internal energy of an object is related to motion of its molecules, and increased heat means faster movement of the molecules, and, as we see, different states of matter are related to the strength of connection between the molecules, we can expect that the state of an object (solid, liquid, gas) might change with increasing or decreasing its internal energy by supplying or taking away the source of energy.
Indeed, it's true. Heat the ice - it will transform into water. Heat the water - it will transform into vapor. Heat the steel - it will melt. Freeze the helium - it will transform into liquid helium. Freeze the mercury - it will solidify.

As we see, the same molecules can form objects in different states. It only depends on the amount of internal energy, that is amount of heat, the object possesses.

Even without transformation from one state to another, heat causes certain changes in the object visible without any special instruments. We all know that mercury thermometer is working based on the property of mercury to expand as the temperature is rising.
This is a general property of most of the objects - to change physical dimensions with increase or decrease of amount of heat (internal energy) carried by their molecules during their constant motion.
This property is the principle, on which measuring of the intensity of molecular movement is based.

Heat and Temperature

Now let's address the issue of measuring the heat, that is amount of internal energy inside any object.
The term temperature is related to average intensity of the molecular movement inside an object or a substance. So, when we say that the temperature of an object has increased or decreased, we mean that average intensity of the molecular movement in it has increased or decreased correspondingly.

Our obvious task now is to quantitatively evaluate the temperature, thus measuring the intensity of the molecular movement inside an object.

It would be great, if we knew kinetic energy of each molecule at each moment of time and average it up to get the temperature in the units of energy. Alas, it's impossible. We have to find some easier method, not necessarily 100% accurate, but sufficient for day-to-day practical purposes.

Convenient instrument for this is a classic thermometer, whose indications are directly related to a change in physical size of objects with change of intensity of the molecular motion inside them.

A simple thermometer consists of a small reservoir with mercury and thin tube coming from it, so the mercury level in the tube will go up with increase of intensity of the molecular motion of the mercury or down, when the intensity decreases.

If we want to measure the temperature of any object, we bring it in contact with our thermometer and, when the temperatures equalize, which might take some time, the level of mercury in a tube of a thermometer will correspond to intensity of the molecular movement inside the object.

All, which is left to establish is the scale and units of measure.
There are three major systems of measurement of temperature: Celsius, Fahrenheit and Kelvin. Celsius system is used everywhere, except United States and its territories. Fahrenheit system is used in United States and its territories. Kelvin system is used everywhere in scientific research and equations of Theoretical Physics.

The unit of measurement in each system is called a degreeand the temperature is written with an indication of the system as follows:
0°C, 20°C, -40°C for temperatures in Celsius system;
32°F, 68°F, -40°F for temperatures in Fahrenheit system;
273.15°K, 293.15°K, 233.15°K for temperatures in Kelvin system.
Above are the examples of three different temperatures in three different measurement systems, correspondingly.

The conversion formulas are:
X°F = 5(X−32)/9°C
X°C = (X+273.15)°K

In the Celsius system the temperature 0°C corresponds to the temperature of melting ice at the sea level on Earth. Temperature of 100°C is the temperature of boiling water at the sea level on Earth. This range is divided into 100 degrees making up a scale.
Degrees in the Fahrenheit system are also connected to some natural processes. 0°F is the temperature of freezing of some chemical solution, while 100°F is approximately the temperature of a human body.
Finally, in Kelvin system 0°K is so-called absolute zerotemperature - the temperature of outer space far from any source of energy. The unit of one degree in Kelvin system equals to that of one degree in Celsius.

Unizor - Physics4Teens - Energy - Potential Energy - Gravity

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

Energy of Gravity

An object of mass m is raised on certain height h above the surface of the Earth.
Analyze its potential energy, as a function of height h.
Do not assume that the force of gravity is constant, but rather use the Newton's Law of Universal Gravitation.
Assume that the radius of Earth is R and its mass is M.

Solution:

If x is the distance from the center of the Earth to an object, the force of gravity equals to
F(x) = G·M·m/x²
(where G is a gravitational constant approximately equaled to 6.674·10^-11N·m²·kg⁻²).

Potential energy of an object at height h above the ground is the work performed by the force of gravity as an object falls down to the surface of the Earth from the distance R+h from the center to the distance R.

Since the force of gravity varies, we should use the calculus to calculate this work.
An infinitesimal increment of work dW(x), assuming the force F(x) acts at a distance dx, equals to
dW(x) = F(x)·dx

Integrating this by x from R+hto R, we get the full work and the potential energy of an object at height h above the ground:
E_pot = W(h) = ∫^R_R+hF(x)·dx =
= G·M·m·[1/R − 1/(R+h)]

Obviously, when an object is at the ground level (h=0), its potential energy equals to zero and, as it moves higher (h is increasing), its potential energy grows to its maximum value of G·M·m/R, as the object moves farther and farther from the ground.

When the height of the object his significantly smaller than the radius of the Earth (h << R), our formula for potential energy can be approximated with a simpler expression:
E_pot = W(h) =
= G·M·m·[1/R − 1/(R+h)] =
= G·M·m·h/[R·(R+h)] ≅
≅ G·M·m·h/R² = m·g·h,
where g=G·M/R² is the free fall acceleration at the ground level, approximately equaled to
g ≅ 9.81 m/sec²
and P = m·g is the weight of an object at the ground level, assumed constant for small heights h.
In this case the potential energy is simply a product of weight (force of gravity) and distance - a classical expression for work performed by a constant force:
E_pot = W(h) ≅ P·h

Let's also analyze the kinetic energy of the object.
Firstly, we assume that our object starts its free falling from the height H above the ground with zero initial velocity.

Its potential energy at any height h we know from the above calculation:
E_pot = G·M·m·[1/R − 1/(R+h)]
Its kinetic energy depends on its speed V:
E_kin = m·V²/2
The problem is, we don't know how the speed depends on the height h.

What we do know is the dependency of the force of gravity F on the height h:
F = G·M·m/(R+h)²
We also know that speed V is the first derivative of height hby time t and the acceleration is the second derivative:
V(t) = h'(t)
a(t) = h"(t)
Therefore, according to the Newton's Second Law,
F = m·a = G·M·m/(R+h)²
Now we have an expression for acceleration a:
a = G·M/(R+h)²
Let's underscore that acceleration is the second derivative of distance h, as a function of time t:
h"(t) = G·M/[R+h(t)]²

We will not attempt to solve this differential equation to get a function h(t), take its derivative to get the speed and substitute into a formula for kinetic energy.
Instead, we will apply a clever trick.
Let's multiply the last expression by 2h'(t):
2h'(t)h"(t) =
= 2G·M·h'(t)/[R+h(t)]²
The reason for this operation is the following.
The left side of this equation is the derivative of the square of the speed:
d/dt{[h'(t)]²} = 2h'(t)·h"(t)
and on the right side of this equation we find another derivative:d/dt{2G·M/[R+h(t)]} =
= 2G·M·h'(t)/[R+h(t)]²
Therefore, we have the equality of the derivatives:
d/dt{[h'(t)]²} =
= d/dt{2G·M/[R+h(t)]}

If derivatives are equal, the functions differ by a constant:
[h'(t)]² = 2G·M/[R+h(t)] + C
The constant C can be determined, using the conditions at the beginning of motion at time t=0:
h(0) = H and h'(0) = 0
Therefore,
[h'(0)]² = 2G·M/[R+h(0)] + C
[0]² = 2G·M/[R+H] + C
C = − 2G·M/[R+H]
Now we can write
[h'(t)]² =
= 2G·M/[R+h(t)]−2G·M/[R+H]
From this we can express the kinetic energy
E_kin = m·V²(t)/2 =
= m·[h'(t)]²(t)/2 =
= G·M·{1/[R+h(t)]−1/[R+H]}

Since our task was to express the energy in terms of height above the ground, we can simply write it as
E_kin = G·M·[1/(R+h)−1/(R+H)]

Finally, let's find the full mechanical energy of an object falling from the sky.
E_pot = G·M·m·[1/R−1/(R+h)]
E_kin = G·M·[1/(R+h)−1/(R+H)]
E_full = E_pot + E_kin =
= G·M·[1/R−1/(R+H)]
As we see, the full mechanical energy is constant, it does not depend on the height. As an object falls from the sky, its potential energy is decreasing, but kinetic energy increasing, and the sum of both types of energy remains constant.