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,
Radiation.
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 T_room and the cold air outside the building has temperature T_air . 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)=T_air to T(L)=T_room.

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 T_roomand T_air and inversely proportional to a thickness of a wall L:
q = −k·(T_room − T_air ) / 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.

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
C_w = 4184 J/(kg·°C).

Answer
Q = C·m·(T_end−T_beg) =
= 4184·1·(100−20) =
= 334,720 J

Problem 2

A piece of unknown metal of mass M_m and temperature T_m was put into an isolated reservoir filled with M_w mass of water at temperature T_w. 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 C_w, what is the specific heat capacity C_m of the unknown metal?

Answer

C_w·M_w·(T−T_w) =
= C_m·M_m·(T_m−T)
from which C_m equals to
C_w·M_w·(T−T_w)/[M_m·(T_m−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?

Answer
E_bur = 300kcal · 4184J/kcal =
= 1,255,200J
E_climb = 0.25·E_bur =
= 313,800J
E_floor = 75kg · 9.8m/sec² · 3m =
= 2205J
N = E_climb / E_floor ≅ 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.

Answer
E_warm = 2090·0.1·10 = 2090J
E_melt = 333000·0.1 = 33300J
E_need = 2090 + 33300 = 35390J
E_water = 4183·M·20 = 83660·M
E_water = E_need
35390 = 83660·M
M = 0.423kg

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

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)E_tot /V

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

Substituting the values for pressure and volume, we obtain
E_tot = (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 E_tot=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 = k_B·N = const
where
k_B = 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/(k_B·T)
Substituting the values,
N = 10⁵·48/(1.381·10⁻²³·293) = 0.12·10²⁸
It's a lot!

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(E_kin) = E_ave
where T is an observable level of liquid inside a thermometer in some units (that is, temperature) and
AVE(E_kin) = E_ave is average kinetic energy of molecules of an object.
This looks more natural in a form
AVE(E_kin) = E_ave ≅ T
since average kinetic energy of molecules E_ave (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·E_ave /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,
E_ave is average kinetic energy of molecules,

Alternatively, it can be written as
E_ave = (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
E_ave = (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
k_B = 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 m³) and number of molecules in a reservoir for ideal gas:
E_ave = (3/2)·p·V/N = (3/2)·k_B·T

Consequently, if we are dealing with certain fixed amount of gas (N molecules) then
p·V/T = k_B·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
E_ave = (3/2)·k_B·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
E_ave = (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(v²) = 2·E_ave /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.

Unizor - Physics4Teens - Energy - Heat - Kinetics of Ideal Gas

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

Kinetics of Ideal Gas

In this lecture we will discuss a concept of pressure of gas, enclosed in some reservoir, against the walls of this reservoir.

Usually, speaking about pressure, we have in mind an object of certain weight on a horizontal table, in which case the constant force of its weight exerts a pressure on a table equal to this weight divided by an area of the table occupied by an object or, as it's sometimes presented, the weight per unit of area.

As an introduction to kinetics of gases, let's consider a reservoir shaped as a cube with sides aligned parallel to coordinate planes and with the length of each edge L.
Assume that there is a single molecule of gas of mass mflying between the opposite walls perpendicularly to them, elastically reflecting off these walls.
Keep in mind that, according to the Third Newton's Law, the force exerted by the wall towards the molecule, which causes the reflection of a molecule from the wall, is equal to the force exerted by a molecule towards the wall, which causes pressure.

The area of each of these opposite walls equals to L², which will be used to calculate the pressure.

In this case of a single gas molecule, flying between the opposite walls of a reservoir, the force is variable. It exists during the time when a molecule hits the wall and then the force becomes zero until the next time this molecule hits the same wall.
A proper definition of pressurein this case is based on a concept of the average force during certain amount of time.

Firstly, let's evaluate the average force during a single period of oscillation of a molecule between walls from a moment of time it's near one wall to a next moment it's at this position after flying to the opposite wall and returning back.

As we know, a change of a momentum of an object equals to an impulse of the force that caused this change
m·v_end − m·v_beg = F·τ
where
m is a mass of an object
v_beg is the velocity of an object in the beginning of the action of the force
v_end is the velocity of an object at the end of the action of the force
F is the force, acting on an object
τ is the time duration of the force, acting on an object.
Keep in mind that velocity and force are vectors in the above equation.

In our case we consider one period of oscillation of a molecule as the time period of average force acted on it.
So, τ is the time between two consecutive events when a molecule is at the wall opposite to the one it collides with:
τ = 2L/|v|.
where |v| is now an absolute speed of a molecule (scalar).

Since we are considering elastic reflection of a molecule in the opposite direction to its initial trajectory,
v_end = −v_beg
and the equation above can be written in scalar form
2m·|v| = |F|·τ = |F|·2L/|v|
from which we determine the absolute average force
|F| = m·|v|²/L
Mass m of a molecule is, obviously, a constant during this process.
The average pressure of a molecule on a wall is the average force divided by the area of a wall:
p = m·|v|²/L³ = m·|v|²/V =
= 2E_kin /V
where V is the volume of a reservoir and E_kin is a kinetic energy of a molecule.

Our first step towards a comprehensive theory is to consider a case of many molecules flying parallel to each other in the same reservoir as above, but with, generally speaking, different speeds. Obviously, the pressure against the wall will be greater because each molecule contributes its own force against the wall during a collision.
If v_i is a speed of the i^thmolecule, the combined pressure will be
p = Σp_i = Σm·v_i²/V =
= 2E_tot /V
where E_tot is a total kinetic energy of all molecules moving parallel to each other.

Considering the number of molecules N remains the same, it's more convenient to express this formula in terms of an average of squares of molecular speed and average kinetic energy of molecules:
v²_ave = (1/N)Σv_i²
E_ave = (1/N)ΣE_i
Using this average of squares of molecular speed, the pressure is
p = N·m·v²_ave /V = 2N·E_ave /V
which brings us to a principle of proportionality between pressure of the gas against the walls of a reservoir and an average of squares of the molecular speed or to an average of kinetic energy of the molecules.

Granted, we showed this only for a case of all molecules moving between two opposite walls parallel each other and perpendicularly to the walls they collide with.

Let's make another step towards comprehensive theory and consider the chaotic movement of all gas molecules.
To quantitatively approach molecular movement, we will use a model of an ideal gas.
This model assumes that molecules of ideal gas are point-mass objects completely chaotically moving in all directions with equal probabilities and elastically colliding at random times among themselves or with all the walls of a reservoir that contains this gas.

We further assume that no interacting forces (like gravity or electromagnetic forces) exist between them. So, only kinetic energy of these molecules plays the role in evaluation of the characteristics of the ideal gas.

Every vector of velocity of any molecule can be represented as a sum of three vectors along the XYZ-coordinates
v = v_x+v_y+v_z

So, every movement of a molecule can be represented as simultaneous movement in three different directions. Depending on which wall of the reservoir is hit by a molecule, its pressureagainst the wall is determined only by one component - the one that is perpendicular to the surface of a wall.

If molecules inside the reservoir are moving completely chaotically, they are hitting all walls with approximately the same frequency. So, for each of the six walls of a cube we can use the same logic as above for one side, except for the walls parallel to YZ coordinate plane the pressures p_x will be proportional to the average of v_x², for the walls parallel to XZ coordinate plane the pressures p_y will be proportional to the average of v_y² and for the walls parallel to XY coordinate plane the pressures p_z will be proportional to the average of v_z².

Considering completely chaotic character of the molecular motion in the ideal gas, the three pressures p_x, p_y and p_zmust be the same. Similarly, averages of v_x², v_y² and v_z² must be also the same.

Since v = v_x+v_y+v_z, according to three-dimensional equivalent of Pythagorean Theorem,
v² = v_x² + v_z² + v_z²
and, similarly, for averages
AVE(v²) = v²_ave =
=AVE(v_x²)+AVE(v_y²)+AVE(v_z²)

From the equality of averages for the ideal gas we come to the following equalities:
v²_ave = 3·AVE(v_x²)
AVE(v_x²) = (1/3)·v²_ave

Therefore,
p_x = N·m·AVE(v_x²)/V =
= (1/3)N·m·v²_ave /V =
= (2/3)N·E_ave /V

Similarly,
p_y = N·m·AVE(v_y²)/V =
= (1/3)N·m·v²_ave /V =
= (2/3)N·E_ave /V
and
p_z = N·m·AVE(v_z²)/V =
= (1/3)N·m·v²_ave /V =
= (2/3)N·E_ave /V

And, since pressures on all walls are the same in the ideal gas,
p = (1/3)N·m·v²_ave /V =
= (2/3)N·E_ave /V = (2/3)E_tot /V

The expression of a pressure in terms of average kinetic energy and volume is more general than in terms of mass, average of squares of molecular speed and volume, as it encompasses a case with molecules of different masses.