Question

Which of the following expressions is true for an ideal gas?

• A. ${\left(\frac{\partial V}{\partial T}\right)}_P=0$
• B. ${\left(\frac{\partial P}{\partial T}\right)}_V=0$
• C. ${\left(\frac{\partial U}{\partial V}\right)}_T=0$
• D. ${\left(\frac{\partial U}{\partial T}\right)}_V=0$

Answer 1

The correct option is C ${\left(\frac{\partial U}{\partial V}\right)}_T=0$

For an ideal gas

$△U=n{C}_v△T$

$PV=nRT$

Now

$P\frac{\partial V}{\partial T}=nR\frac{\partial T}{\partial T}$

$(\frac{\partial V}{\partial T}{)}_P=\frac{nR}{P}$

Again, $\frac{\partial P}{\partial T}V=nR\frac{\partial T}{\partial T}$

$(\frac{\partial P}{\partial T}{)}_V=\frac{nR}{V}$

Again, $△U=n{C}_v△T$

$(\frac{\partial U}{\partial V}{)}_T=\frac{\partial \left(n{C}_V△T\right)}{\partial V}=0$

$(\frac{\partial U}{\partial T}{)}_T=\frac{\partial \left(n{C}_V△T\right)}{\partial T}=n{C}_V$

Hence the correct option is [C]