Kinetic Theory of Gases

Specific Heat of an Ideal GasWe 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

howthe process is carried out. That means the heat Q will also depend uponhowthe 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

twocommon 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 <v^{2}>]U = (

^{3}/_{2}) N k T = (^{3}/_{2}) n R TIf we now add heat at constant volume -- so that

no workis done -- all of that heat Q does only one thing. It increases the internal energy U,Q = U = ( ^{3}/_{2}) n R TQ = n C

_{V}T = (^{3}/_{2}) n R Tn C

_{V}= (^{3}/_{2}) n RC

_{V}= (^{3}/_{2}) RThat means that

C_{V}= (^{3}/_{2}) R = 12.5 J/mol-Kfor

allmonatomic gases -- primarily the noble gases, He, Ne, Ag, Xe, etc. And this is in good agreement with what we findexperimentally(as it ought to be!).Now, look at a

constant pressureprocess,Q = n C

_{P}TW = P V

U = Q - W

U = n C

_{P}T - P VFor 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 VC

_{V}= C_{P}- P V/(n T)C

_{V}= C_{P}- RC

_{P}- C_{V}= RC

_{P}= (^{5}/_{2}) RIt is often useful to describe, calculate, predict, or measure the

ratioof these two specific heats,These values for C

_{V}, C_{P}, and are in good agreement withexperimentalresults.

Return to Ch 21 ToC(c) Doug Davis, 2002; all rights reserved