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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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