Chapter 15. Molar specific heat at constant volume for an ideal gas (15-18)

Question

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Question

Ga8PI+cWuZ0n7BavcLHkD451gPDVonmPqrXUrDazZbiwvM//+Zi3vLPcRd4OdC/61FAKZC4xQtTMLDMJ+j9+pA==
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Question

wtRAiJSdJqY9BBQst0hZPa7Zvo2EBLDaMFekFg==
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Review

We'll use the symbol \(D\) to represent the number of degress of freedom. Then the average energy per gas molecule is \((D/2)kT\). One mole of the gas contains Avogadro's number of molecules, so the energy per mole is \(N_A(D/2)kT = (D/2)\ (N_Ak)T = (D/2)RT\). (Again, recall that \(k = R/N_A\), so \(R = N_Ak)\).) Then the internal energy of \(n\) moles of ideal gas is

(15-16) \(U = n\left(\frac{D}{2}R\right)T\) (ideal gas, \(D\) degrees of freedom)

If the temperature of the gas changes by \(\Delta{T}\), it follows from Equation 15-16 that the internal energy change is

(15-17) \(\Delta{U} = n\left(\frac{D}{2}R\right)\Delta{T}\) (ideal gas, \(D\) degrees of freedom)

But, we saw above that for an ideal gas, \(\Delta{U} = nC_V\ \Delta{T}\). Comparing this to Equation 15-17, we see that