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.