- #1
Mikhail_MR
- 17
- 0
Hello,
I have some trouble understanding the virial expansion of the ideal gas.
1. Homework Statement
I have given the state equation:
$$ pV = N k_b T \left(1+\frac{A\left(T\right)}{V}\right) $$
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and a hint how to calculate the caloric equation of state $$ \left(\frac{\partial U}{\partial V}\right)_T = T^2 \left(\frac{\partial}{\partial T}\left(\frac{p}{T}\right)\right)_V $$
[/B]
I calculated ## \frac{\partial}{\partial T} \left(\frac{p}{T}\right)_V = \frac{N k_b}{V^2} \left(\frac{\partial A\left(T\right)}{\partial T}\right)## first. So using the hint $$ U = \int_0^V \frac{N k_b}{V^2} T^2 \frac{\partial A\left(T\right)}{\partial T} dV = -N k_b T^2 \frac{\partial A\left(T\right)}{\partial T} \frac{1}{V}$$
If my calculations are correct, why is my internal energy negative? Unfortenately, I don't know which sign does ## \frac{\partial A\left(T\right)}{\partial T} ## have. My next task is to calculate ## c_V ## as a function of ## T ## and ## V ## and determine when it is volume independent.
$$ c_V = \left(\frac{\partial U}{\partial T}\right)_V = - \frac{1}{V} N k_B \left(2 T \frac{\partial A\left(T\right)}{\partial T} + T^2 \frac{\partial^2 A\left(T\right)}{\partial^2 T}\right) $$
If I look at this equation I cannot say when it does not depend on ## V ##. Where have I made a mistake?
Any help would be greatly appreciated
I have some trouble understanding the virial expansion of the ideal gas.
1. Homework Statement
I have given the state equation:
$$ pV = N k_b T \left(1+\frac{A\left(T\right)}{V}\right) $$
Homework Equations
[/B]
and a hint how to calculate the caloric equation of state $$ \left(\frac{\partial U}{\partial V}\right)_T = T^2 \left(\frac{\partial}{\partial T}\left(\frac{p}{T}\right)\right)_V $$
The Attempt at a Solution
[/B]
I calculated ## \frac{\partial}{\partial T} \left(\frac{p}{T}\right)_V = \frac{N k_b}{V^2} \left(\frac{\partial A\left(T\right)}{\partial T}\right)## first. So using the hint $$ U = \int_0^V \frac{N k_b}{V^2} T^2 \frac{\partial A\left(T\right)}{\partial T} dV = -N k_b T^2 \frac{\partial A\left(T\right)}{\partial T} \frac{1}{V}$$
If my calculations are correct, why is my internal energy negative? Unfortenately, I don't know which sign does ## \frac{\partial A\left(T\right)}{\partial T} ## have. My next task is to calculate ## c_V ## as a function of ## T ## and ## V ## and determine when it is volume independent.
$$ c_V = \left(\frac{\partial U}{\partial T}\right)_V = - \frac{1}{V} N k_B \left(2 T \frac{\partial A\left(T\right)}{\partial T} + T^2 \frac{\partial^2 A\left(T\right)}{\partial^2 T}\right) $$
If I look at this equation I cannot say when it does not depend on ## V ##. Where have I made a mistake?
Any help would be greatly appreciated