Two moles of a monoatomic ideal gas occupy volume V at 27^circtext C . It expands adiabatically to 2V. Calculate the final temperature and the internal energy change. (gamma = 5/3, 2^ 2/3 approx 1.587)

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EEJ MAIN Physics QUESTION #7060

Question 1

Two moles of a monoatomic ideal gas occupy volume $V$ at $27^\circ\text{C}$. It expands adiabatically to $2V$. Calculate the final temperature and the internal energy change. ($\gamma = 5/3$, $2^{2/3} \approx 1.587$)

(a) $189 \text{ K}$ (b) $-2.7 \text{ kJ}$✔️
(a) $195 \text{ K}$ (b) $2.7 \text{ kJ}$
(a) $189 \text{ K}$ (b) $2.7 \text{ kJ}$
(a) $195 \text{ K}$ (b) $-2.7 \text{ kJ}$

Correct Answer Explanation

For adiabatic expansion: $T_1 V_1^{\gamma-1} = T_2 V_2^{\gamma-1}$

$300 \times V^{2/3} = T_2 \times (2V)^{2/3} \implies T_2 = \frac{300}{2^{2/3}} \approx \frac{300}{1.587} \approx 189 \text{ K}$

Change in internal energy: $\Delta U = n C_v \Delta T = n \left(\frac{3}{2}R\right) (T_2 - T_1)$

$\Delta U = 2 \times 1.5 \times 8.31 \times (189 - 300) \approx 24.93 \times (-111) \approx -2767 \text{ J} \approx -2.7 \text{ kJ}$.

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