Question

The change in the internal energy of 3 moles of a gas heated at constant volume from 20 $^\circ$C to 40 $^\circ$C is 1080 J. The molar specific heat of the gas at constant volume in J mol$^{-1}$ K$^{-1}$ is

• A. 21
• B. 18
• C. 24
• D. 12

Hint:

Change in internal energy at constant volume: $\Delta U = n C_V \Delta T$.
A change in temperature $\Delta T$ in Celsius is numerically equal to $\Delta T$ in Kelvin.
Ensure units are consistent.

Answer 1

The Correct Option is B

The change in internal energy ($\Delta U$) of an ideal gas at constant volume is given by: $\Delta U = n C_V \Delta T$ where $n$ is the number of moles, $C_V$ is the molar specific heat at constant volume, and $\Delta T$ is the change in temperature. Given values: Number of moles $n = 3 \text{ moles}$. Change in internal energy $\Delta U = 1080 \text{ J}$. Initial temperature $T_1 = 20 ^\circ\text{C}$. Final temperature $T_2 = 40 ^\circ\text{C}$. Change in temperature $\Delta T = T_2 - T_1 = (40 - 20) ^\circ\text{C} = 20 ^\circ\text{C}$. Since this is a temperature difference, $\Delta T = 20 \text{ K}$. We need to find $C_V$. Rearranging the formula: $C_V = \frac{\Delta U}{n \Delta T}$. Substitute the given values: $C_V = \frac{1080 \text{ J}}{(3 \text{ moles}) \times (20 \text{ K})}$. $C_V = \frac{1080}{60} \text{ J mol}^{-1}\text{K}^{-1}$. $C_V = \frac{108}{6} \text{ J mol}^{-1}\text{K}^{-1}$. $C_V = 18 \text{ J mol}^{-1}\text{K}^{-1}$. \[ \boxed{18 \text{ J mol}^{-1}\text{K}^{-1}} \]