Degrees of Freedom and Molar Specific Heats — Question 10

Question 10

A temperature-dependent effective count cannot simply multiply temperature

A supplied smooth activation model has molar $C_V(T)=R[\tfrac32+T/(T+\Theta)]$ with fixed $\Theta>0$. Define thermal energy by $U(0)=0$ for one mole. Let $f_C(T)=2C_V(T)/R$ and $f_U(T)=2U(T)/(RT)$. This is a phenomenological model, not a claim that partially active modes are classical quadratic terms.

1. Integrate $C_V(T)$ to find $U(T)$ and derive $f_C(T)$ and $f_U(T)$.

2. Evaluate both counts at $T=\Theta$ and find their low- and high-temperature limits. Why is substituting $f_C(T)$ into $U=fRT/2$ incorrect?

Question 10 - Solution

A temperature-dependent effective count cannot simply multiply temperature

(a) Method and calculation

$$U(T)=R\int_0^T\left(\frac32+\frac{t}{t+\Theta}\right)dt=R\left[\frac52T-\Theta\ln\left(1+\frac T\Theta\right)\right].$$$$f_C=3+\frac{2T}{T+\Theta},\quad f_U=5-\frac{2\Theta}{T}\ln(1+T/\Theta).$$

(b) Interpretation and further calculation

$$f_C(\Theta)=4,\quad f_U(\Theta)=5-2\ln2\approx3.61371.$$$$T/\Theta\to0:\ (f_C,f_U)\to(3,3);\qquad T/\Theta\to\infty:\ (f_C,f_U)\to(5,5).$$$$\frac d{dT}\left[\tfrac12Rf_C(T)T\right]=\tfrac12R[f_C+Tf_C^{\prime}]\ne C_V.$$Capacity is a local derivative; total thermal energy includes the full heating history from the specified reference.

Physical check

The familiar $U=f nRT/2$ shortcut assumes a temperature-independent set of fully active quadratic terms and a compatible energy reference.