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Peter's Index Physics Home Lecture 8 Course Index Lecture 10
A Semester of First Year Physics with Peter Eyland
Lecture 9 (Equipartition Theory)
In this lecture the following are introduced for ideal gases:
• Internal energy change
• Internal energy of a monatomic gas
• Molar specific heat of a monatomic gas at constant volume
• Molar specific heat at constant pressure
• Equipartition of energy
• Temperature dependence of molar specific heat
• Adiabatic changes
The change in Internal Energy of an Ideal Gas
By adding heat to a fixed volume of gas, the pressure and temperature increase but no work is done by the system (no expansion).
For any ideal gas at constant volume, 
Notice that the change depends on the difference in temperature. This implies that the change in internal energy does not depend on the type of process, i.e. whether the change is at constant volume or constant pressure or both vary.
The graph shows three processes to get to the same final temperature.
In each case: 
The Internal Energy of an Ideal Monatomic Gas
Since the atoms in an ideal monatomic gas are all mathematical points, with no interactions between them (except collisions), the Internal Energy of the gas is simply the sum of the translational kinetic energies of all the atoms. Starting rom the Kinetic Theory Equation of the previous lecture):
Writing the Internal Energy as U, for an ideal monatomic gas:
In diatomic and more complex molecules,
the energy can be distributed in other ways than simple translation,
because the atoms can vibrate and/or rotate with respect to each other.
In these cases the simple theory above has to be modified to include these modes.
Example
A cylinder contains 0.03 m^{3} of Argon gas at a temperature of 25^{0} C and a pressure of 1.2 MPa.
Find the internal energy of the gas.
Since Argon is a noble gas, it consists only of atoms and the energy is purely translational kinetic energy. Therefore:
The Molar Specific Heat for an Ideal Monatomic Gas at Constant Volume
From above, the change in internal energy is: The heat added to the system is: 

Now R = 8.314 J.(g mol)^{1}.K^{1},
so the Molar Specific heat for a monatomic gas at constant volume is 12.5 J.(g mol)^{1}.K^{1}
Example
The specific heat at constant volume of a particular mass of Argon gas is 313.5 J.kg^{1}.
Find the mass of an Argon atom.
The Molar Specific Heat for an Ideal Gas at Constant Pressure
As an ideal gas expands its pressure will tend to drop along the green line shown in the diagram. 
For any ideal gas at constant pressure, the 1^{st} Law of Thermodynamics gives
Example
1.5 mole of an ideal gas at 300K is heated at constant Atmospheric pressure till the temperature is 320 K.
Find the change in volume.
The Equipartition of Energy
As given above, for all practical purposes, the atoms in an ideal monatomic gas are mathematical points. Thus the only energy that they can store is translational kinetic energy along the 3 perpendicular axes of normal space. In contrast to this, diatomic and polyatomic molecules, have definite shapes  as shown below.
The atoms in such a molecule can store energy in vibrations and rotations as well as translations. Each way energy can be stored in the molecule is called a degree of freedom.
The Oxygen molecule can store energy by rotation about the two perpendicular axes shown in green,
but not along the third perpendicular axis because the mass is too close to that axis.
• The Oxygen molecule has 2 degrees of rotational freedom as well as its 3 translational ones.
The Methane molecule is a tetrahedron and can store energy in rotations about three perpendicular axes.
• The Methane molecule has 3 degrees of rotational freedom in addition to its 3 translational ones.
As well as rotations, the molecules can vibrate in a number of ways. For example, a diatomic molecule can have its atoms vibrating in line with each other or parallel to each other:
James Clerk Maxwell proposed the idea of equipartition of energy, which states that:
Each molecule in a gas is given an energy, , for each degree of freedom.
Since the Helium atom has only 3 translational degrees of freedom, Helium gas will have an internal energy given by: per molecule. With for a mole of an ideal monatomic gas.
Since the Oxygen molecule has 3 translational and 2 rotational degrees of freedom, Oxygen gas will have an internal energy:
per molecule.
With
for a mole of an ideal diatomic gas.
Since the Methane molecule has 3 translational and 3 rotational degrees of freedom,
Methane gas will have an internal energy:
per molecule.
With
for a mole of an ideal polyatomic gas.
Including the Specific Heat at constant Pressure (with R added as above), the following table can be constructed.
Molecule 
C_{v} 
C_{p} 
Monatomic 

Diatomic 

Polyatomic 
3R 
4R 
Example
20 g of Oxygen is heated at constant Atmospheric pressure from 20^{0}C to 120^{0}C. Find
(a) the heat transferred to the Oxygen, and
(b) what fraction of the heat raised the internal energy.
Temperature dependence of Molar Specific Heat at constant volume
The theory of Maxwell above doesn't work quite as simply as stated,
because the vibrational modes haven't been included!
The reason is that there is a threshold effect.
The different modes are only switched on above certain temperatures.
The graph below shows the temperature variation of C_{V}/R for Hydrogen.
At low temperatures, Hydrogen gas has only translational degrees of freedom available,
i.e. it is as if its molecules don't rotate.
At higher temperatures rotation cuts in, and at the highest temperatures vibrations become possible.
At room temperatures Maxwell's equipartition of energy usually holds for normal gases.
The Adiabatic Expansion of an Ideal Gas
In an adiabatic expansion, no heat is transferred between the system and the environment, i.e. ΔQ goes to zero in the 1^{st} Law of Thermodynamics. This happens when the change occurs very quickly (as with sound waves) or slowly with a system completely insulated from its environment.
On a PV graph, an adiabatic process has a steeper curve than isothermal processes. The drop in temperature that occurs in an adiabatic expansion is due to small but nonzero attractive forces between molecules (despite the kinetic theory assumptions). Kinetic Energy is used up in overcoming the attractive forces.
To find an expression that characterises adiabatic processes, the starting point is the ideal gas equation. This describes the state of the gas. When the state changes, there are changes in pressure (Δp), volume (ΔV) and temperature (ΔT).
Introducing differentials into the state equation gives: From the 1^{st} Law of Thermodynamics, for an adiabatic change 
Substituting the Adiabatic Temperature change into the differential 1^{st} Law: 
Here the parameter γ is the ratio of the specific heats,
Expanding and simplifying the differential 1^{st} Law equation gives:
Dividing this by pV gives:
Integrating the result gives:
Finally, an equation for adiabatic change is:
Using the ideal gas equation, this can also be rewritten as:
Example
A sample of gas with γ=1.4 is at atmospheric pressure and 330 K. It is compressed adiabatically to one third of its volume.
Find its final pressure and temperature.
Example
A heat engine carries 12.03 moles of ideal monatomic gas around the cycle shown.
(a) Find the change in internal energy, the heat added to the system and the work done by the system for each process,
and the whole cycle. 
Answer (a)


Answer (b)

State 
1 
2 
3 
Pressure kPa 
200 
200 
1250 
Volume m^{3} 
0.2 
0.6 
0.2 
Summarising:
Internal energy change:
Internal energy of a monatomic gas:
Molar specific heat of a monatomic gas at constant volume:
Molar specific heat at constant pressure:
The Equipartition of Energy: Each molecule in a gas has energy,
,
for each degree of freedom.
Adiabatic changes: and
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