Kinetic Theory of Gases

Specific Heat of an Ideal Gas

We have written

Q = c m T

for heat added to liquids and solids. We did not ask for additional information. However, we do need additional information when we talk about heat added to a gas. Remember the First Law of Thermodynamics,

Q = W + U

and the work W done by the gas depends -- very strongly! -- on how the process is carried out. That means the heat Q will also depend upon how the process is carried out.

Consider these two isotherms at temperatures T and T + T. The "area under the curve" is drastically different for different paths. So the amount of heat necessary to change the temperature by an amount T depends upon the process (or path) taken. In particular, we will look at two common processes:

Q = n C_V T (for constant volume)

Q = n C_P T (for constant pressure)

n is the number of moles and C_V is the molar specific heat at constant volume and C_P is the molar specific heat at constant pressure. T, of course, is the change in temperature.

On the previous web page we found

that is, the kinetic energy of a gas molecule is directly related to the temperature. For a monatomic gas, the total internal energy U must be the sum of the translational kinetic energy of all of the gas molecules (or atoms). We can write this as

U = N [ KE ] = N [(¹/₂) m <v²>]

U = (³/₂) N k T = (³/₂) n R T

If we now add heat at constant volume -- so that no work is done -- all of that heat Q does only one thing. It increases the internal energy U,

Q = U = (³/₂) n R T

Q = n C_V T = (³/₂) n R T

n C_V = (³/₂) n R

C_V = (³/₂) R

That means that

C_V = (³/₂) R = 12.5 J/mol-K

for all monatomic gases -- primarily the noble gases, He, Ne, Ag, Xe, etc. And this is in good agreement with what we find experimentally (as it ought to be!).

Now, look at a constant pressure process,

Q = n C_PT

W = P V

U = Q - W

U = n C_P T - P V

For the same change in temperature, T, the change in the internal energy U must be the same for both processes,

n C_V T = n C_P T - P V

C_V = C_P - P V/(n T)

C_V = C_P - R

C_P - C_V = R

C_P = (⁵/₂) R

It is often useful to describe, calculate, predict, or measure the ratio of these two specific heats,

These values for C_V, C_P, and are in good agreement with experimental results.

Molecular Model			Adiabatic Processes
Return to Ch 21 ToC

(c) Doug Davis, 2002; all rights reserved