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Let's say we have an isolated system that contains two equally sized volumes of graphite that are in thermal contact with each other. At $t=0$ one graphite block has a temperature of 298 K, while the other block has a temperature of 498 K. Eventually a thermal equilibrium is reached between the two graphite blocks, and the equilibrium temperature is not simply the average temperature

$$ \langle T \rangle \neq \frac{298 + 498}{2} = 398 $$

but is instead 410 K. How explain this in therms of statistical thermodynamics?


Derivation of an expression for the heat capacity with constant volume

I have that the heat capacity for constant volume, $C_V$, is

$$ C_V = \left( \frac{\partial U}{\partial T} \right)_V $$

and that the internal energy, $U$, is

$$ U = kT^2 \left( \frac{\partial \ln Q}{\partial T} \right)_{N,V} $$

Assuming interacting and distinguishable particles (because of solid-state graphite) we have that the natural logarithm of the canonical partition function $Q$ is

$$ \ln Q = \ln q^N = N\ln q_{tr} + N\ln q_{rot} + N\ln q_{vib} + N\ln q_{el} $$

where $q_{tr}$, $q_{rot}$, $q_{vib}$, and $q_{el}$ are the translational, rotational, vibrational, and electronic contributions to the molecular partition function $q$, and $N$ is the number of particles. We obtain models for these contributions based on the particle in a box, rigid rotor, and harmonic oscillator, and neglecting the electronic contribution. We then get

$$ \ln Q = N \ln \frac{V\cdot e \left( \frac{2 \pi mkT}{h^2} \right)^{3/2}}{N} + N \ln \frac{T}{\theta_{rot}} + n \ln \frac{1}{1-e^{h\nu_0 /kT}} $$

This is the same as

$$ \ln Q = N \ln V + N \ln e + \frac{3}{2}N \ln \left( \frac{2\pi mkT}{h^2} \right) + \frac{3}{2}N \ln T + N \ln T - N \ln \theta_{rot} - N \ln (1 - e^{-h\nu_0/kT}) $$

To get an expression for the heat capacity, we need to take the partial time derivative of $\ln Q$. Only three terms depend on T and will be non-zero. Therefore

$$ \frac{\partial \ln Q}{\partial T} = \frac{\partial}{\partial T} \frac{3}{2}N\ln T + \frac{\partial}{\partial T}N\ln T - \frac{\partial}{\partial T}N\ln (1-e^{-h\nu_0/kT}) $$

which becomes

$$ \frac{\partial \ln Q}{\partial T} = \frac{3N}{2T} + \frac{N}{T} + \frac{h\nu_0 e^{-h\nu_0/kT}}{kT^2(1-e^{-h\nu_0/kT})} $$

Plugging this into the expression for $U$ gives

$$ U = \frac{3}{2}NkT + NkT + \frac{h\nu_0}{e^{h\nu_0/kT}} $$

Taking again the partial time derivative to get $C_V$, we finally get the following expression for the heat capacity

$$ \underline{C_V = \frac{3}{2}Nk + Nk + \frac{h^2\nu_0^2 e^{h\nu_0/kT}}{kT^2(e^{h\nu_0/kT}-1)^2}} $$

However, I am not sure how to use this to explain the non-linear heat transfer observed for the isolated system described at the beginning of the question. I see that $C_V$ is not linear in $T$, but I don't know whether using the heat capacity is the correct way of explaining this in the first place.


I made this simple sketch to try and understand why the final temperature is closer to 498 K than to 298 K. I found the temperature in the middle of $C_v(298)$ and $C_V(498)$. Is this a good way of thinking about this? For $T \rightarrow \infty$, the heat capacity will also go to infinity, so I assumed this shape of the graph:

enter image description here

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  • $\begingroup$ "and the equilibrium temperature is not simply the average temperature" Is this based on observations? $\endgroup$ – Gert Sep 6 '16 at 21:35
  • $\begingroup$ I don't know, the exercise text stated this. I suppose we can measure this. $\endgroup$ – Yoda Sep 7 '16 at 8:24
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You can see from your final equation that the heat capacity is not necessarily constant, but, instead, can depend on temperature. So, if $$\int_{298}^{T_f}C_v(T)dT=\int_{T_f}^{498}C_v(T)dT$$ there is no guarantee that $T_f=398$ (unless $C_v$ is constant, independent of temperature).

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  • $\begingroup$ I also see that only the vibrational contribution depends on temperature, so it is the vibrational (collisional) part of the heat transfer that results in this non-linear heat transfer? $\endgroup$ – Yoda Sep 7 '16 at 8:12

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