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knowlewj01
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Homework Statement
Show that internal energy U = U(T) only for an ideal gas who'se equation of state is:
[itex]P(V-b) = RT[/itex]
(the claussius equation for n moles of gas)
Homework Equations
Thermodynamic Equation of state:
[itex]\left(\frac{\partial U}{\partial V}\right)_T = T\left(\frac{\partial P}{\partial T}\right)_V - P[/itex]
The Attempt at a Solution
so, basically we need to prove that for this gas the following criteria are met:
[itex]\left(\frac{\partial U}{\partial V}\right)_T = \left(\frac{\partial U}{\partial P}\right)_T = 0 [/itex]
internal energy U does not change with respect to P or V.
using the equation of state of the gas, we can differentiate P with respect to T at constant V:
[itex]\left(\frac{\partial P}{\partial T}\right)_V = \frac{R}{V-b}[/itex]
now substitute this into the thermodynamic equation of state:
[itex]\left(\frac{\partial U}{\partial V}\right)_T = \frac{RT}{V-b} - P[/itex]
and from the equation of state of the gas, we can obtain P:
[itex] P = \frac{RT}{V-b}[/itex]
so the result is that [itex] \left(\frac{\partial U}{\partial V}\right)_T = 0[/itex]
thats half the work done, I am not sure how to prove [itex] \left(\frac{\partial U}{\partial P}\right)_T = 0[/itex]
any pointers? Thanks