10 mole of oxygen is heated at constant volume from $30^{\circ} \mathrm{C}$ to $40^{\circ} \mathrm{C

Question

10 mole of oxygen is heated at constant volume from $30^{\circ} \mathrm{C}$ to $40^{\circ} \mathrm{C}$. The change in the internal energy of the gas is $\_\_\_\_$ cal. (The molecular specific heat of oxygen at constant pressure, $C_p=7 \mathrm{cal} / \mathrm{mol} .{ }^{\circ} \mathrm{C}$ and $\mathrm{R}=2 \mathrm{cal} . / \mathrm{mol} .{ }^{\circ} \mathrm{C}$.)

Professor PC Thomas Classes · Answer

For a gas at constant volume, the change in internal energy is given by $\Delta U = n C_v \Delta T$. We are given $C_p = 7 	ext{ cal/mol}\cdot^\circ	ext{C}$ and $R = 2 	ext{ cal/mol}\cdot^\circ	ext{C}$. Using Mayer's relation, $C_v = C_p - R = 7 - 2 = 5 	ext{ cal/mol}\cdot^\circ	ext{C}$. The change in temperature is $\Delta T = 40^\circ	ext{C} - 30^\circ	ext{C} = 10^\circ	ext{C}$. Substituting values: $\Delta U = 10 	imes 5 	imes 10 = 500 	ext{ cal}$.