- #1

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- Homework Statement
- Hi, I've been struggling to find the form of the adiabat of this problem:

Given the fundamental relation, find the form of the adiabats in the P-V plane

(problem 2.3-5. Thermodynamics and an Introduction to Thermostatistics, Callen, pp 42)

- Relevant Equations
- $${S \over R} = {UV \over N} - {N^3 \over UV}$$

I tried this... but I'm not sure if I'm doing it right or maybe there's a simpler way. Thanks for your time or help :)

The fundamental relation is:

$${S \over R} = {UV \over N} - {N^3 \over UV}$$

but I used $$s=S/N, u=U/N, v=V/N$$ to obtain

$${s \over R} = uv - {1 \over uv}$$

then I did some algebra...

$$u^2 - ({s \over Rv})u - {1 \over v^2} = 0$$

and used the quadratic formula

$$u = ({1 \over 2Rv})({s \pm \sqrt{s^2+4R^2}})$$

next I derived the previous expression

$$ (\frac{\partial u}{\partial v})_s = -P $$

$$ P = ({1 \over 2Rv^2})({s \pm \sqrt{s^2+4R^2}})$$

And my doubt is... how do I get rid of s?

I also tried

$$ (\frac{\partial s}{\partial v})_u = {P \over T} $$

and got

$$ {P \over T} = R(u+{1 \over uv^2})$$

but then how do I get rid of u?

The fundamental relation is:

$${S \over R} = {UV \over N} - {N^3 \over UV}$$

but I used $$s=S/N, u=U/N, v=V/N$$ to obtain

$${s \over R} = uv - {1 \over uv}$$

then I did some algebra...

$$u^2 - ({s \over Rv})u - {1 \over v^2} = 0$$

and used the quadratic formula

$$u = ({1 \over 2Rv})({s \pm \sqrt{s^2+4R^2}})$$

next I derived the previous expression

$$ (\frac{\partial u}{\partial v})_s = -P $$

$$ P = ({1 \over 2Rv^2})({s \pm \sqrt{s^2+4R^2}})$$

And my doubt is... how do I get rid of s?

I also tried

$$ (\frac{\partial s}{\partial v})_u = {P \over T} $$

and got

$$ {P \over T} = R(u+{1 \over uv^2})$$

but then how do I get rid of u?

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