- #1

EE18

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- Homework Statement
- I am trying to solve Problem 2a in Chapter 2 of Ashcroft and Mermin. I'm not asking about the whole problem, but just the start. Essentially, one needs to expression the entropy density ##s## in terms of the energy density ##u## and, as far as I can tell from other methods to solving this problem, I should arrive at ##s(T) = u(T)/T##.

- Relevant Equations
- ##s(T) = u(T)/T##

The starting point is the identity

$$\left(\frac{\partial u}{\partial T}\right)_n = T\left(\frac{\partial s}{\partial T}\right)_n.$$

I then try to proceed as follows:

Integrating both with respect to ##T## after dividing through by ##T##, we find

$$ \int_0^T \left(\frac{\partial s}{\partial T'}\right)_n = s(T) - s(0) = s(T) $$

$$ = \int_0^T \frac{1}{T'}\left(\frac{\partial u}{\partial T'}\right)_n = \frac{u}{T'}\biggr\rvert_0^T + \int_0^T \frac{1}{T'^2}u ...$$

where we have used an integration by parts, the fundamental theorem of calculus, and the third law of thermodynamics (in (1)), and where we recall that $u_0$ is the energy density in the ground state (see AM (2.79)).

But I can't seem to go any further. I found a solution which does something I totally can't understand in the last line, and I'm hoping someone can either clarify that solution and/or show me how to go further. The problem is obviously the last integral on the right-hand side.

$$\left(\frac{\partial u}{\partial T}\right)_n = T\left(\frac{\partial s}{\partial T}\right)_n.$$

I then try to proceed as follows:

Integrating both with respect to ##T## after dividing through by ##T##, we find

$$ \int_0^T \left(\frac{\partial s}{\partial T'}\right)_n = s(T) - s(0) = s(T) $$

$$ = \int_0^T \frac{1}{T'}\left(\frac{\partial u}{\partial T'}\right)_n = \frac{u}{T'}\biggr\rvert_0^T + \int_0^T \frac{1}{T'^2}u ...$$

where we have used an integration by parts, the fundamental theorem of calculus, and the third law of thermodynamics (in (1)), and where we recall that $u_0$ is the energy density in the ground state (see AM (2.79)).

But I can't seem to go any further. I found a solution which does something I totally can't understand in the last line, and I'm hoping someone can either clarify that solution and/or show me how to go further. The problem is obviously the last integral on the right-hand side.

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