Question

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

• A. ${\left(\frac{dV}{dT}\right)}_P=0$
• B. ${\left(\frac{dP}{dT}\right)}_V=0$
• C. ${\left(\frac{dU}{dV}\right)}_T=0$
• D. ${\left(\frac{dU}{dT}\right)}_V=0$

Answer 1

Answer: (C)

(a) From charle's law: $V\propto T$
$\Rightarrow V=k_{1}T$ (at cons. P)
$\Rightarrow {\left(\frac{dV}{dT}\right)}_P=k_1\ne 0$
(b) From GayLussac's law:
$P\propto T$ (at cons. $V$ )
$\Rightarrow P=K_{2}T$
$\Rightarrow {\left(\frac{dP}{dT}\right)}_V=k_2\ne 0$
(c) For an ideal gas, change in internal energy is zero at constant temperature. $(dU{)}_T=0$
$\Rightarrow {\left(\frac{dU}{dV}\right)}_T=0$
(d) $\Delta U=n{C}_V\Delta T$ (at cons. $V$ )
$\Rightarrow {\left(\frac{dU}{dT}\right)}_V=n{C}_V\ne 0$