Question

If 300 J of heat energy is used to raise the temperature of one mole of an ideal gas by 10 K at constant pressure, then heat energy required to raise its temperature by 10 K at constant volume is $(R=8.3\text{ J mol}^{-1}\text{K}^{-1})$

• A. 117 J
• B. 150 J
• C. 183 J
• D. 217 J
• E. 300 J

Hint:

Logic Tip: You can calculate this even faster without finding $C_p$ explicitly. The difference in heat is simply the work done: $Q_p - Q_v = nR\Delta T$. Therefore, $Q_v = Q_p - nR\Delta T = 300 - (1)(8.3)(10) = 300 - 83 = 217$ J.

Answer 1

The Correct Option is D

Concept:
The heat energy required to change the temperature of a gas is given by: $Q_p = n C_p \Delta T$ (at constant pressure) $Q_v = n C_v \Delta T$ (at constant volume) The molar specific heats are related by Mayer's relation: $C_p - C_v = R$, where $R$ is the universal gas constant.
Step 1: Calculate the molar heat capacity at constant pressure ($C_p$).
We are given: $Q_p = 300\text{ J}$ $n = 1\text{ mole}$ $\Delta T = 10\text{ K}$ Using the formula $Q_p = n C_p \Delta T$: $$300 = (1) \cdot C_p \cdot (10)$$ $$C_p = \frac{300}{10} = 30\text{ J mol}^{-1}\text{K}^{-1}$$
Step 2: Calculate the molar heat capacity at constant volume ($C_v$).
Using Mayer's relation $C_p - C_v = R$: $$C_v = C_p - R$$ Substitute $C_p = 30$ and $R = 8.3$: $$C_v = 30 - 8.3$$ $$C_v = 21.7\text{ J mol}^{-1}\text{K}^{-1}$$
Step 3: Calculate the heat required at constant volume ($Q_v$).
We need the heat to raise the temperature of the same 1 mole of gas by 10 K at constant volume: $$Q_v = n C_v \Delta T$$ $$Q_v = (1) \cdot (21.7) \cdot (10)$$ $$Q_v = 217\text{ J}$$