Question

A monatomic gas ($\gamma = \frac{5}{3}$) at a pressure of 4 atm is compressed adiabatically so that its temperature rises from 27 $^\circ$C to 327 $^\circ$C. The pressure of the gas in its final state is

• A. $2^{\frac{5}{3}}$ atm
• B. $2^{\frac{10}{3}}$ atm
• C. $2^{\frac{2}{3}}$ atm
• D. $2^{\frac{8}{3}}$ atm

Hint:

For an adiabatic process: $P V^\gamma = \text{const}$, $T V^{\gamma-1} = \text{const}$, $P^{1-\gamma}T^\gamma = \text{const}$.
From $P^{1-\gamma}T^\gamma = \text{const}$, derive $P_2 = P_1 (T_2/T_1)^{\gamma/(\gamma-1)}$.
Convert temperatures to Kelvin ($T(K) = T(^\circ C) + 273$).
Be careful with exponent arithmetic.

Answer 1

The Correct Option is D

For an adiabatic process, the relationship between pressure ($P$) and temperature ($T$) is given by: $P^{1-\gamma}T^\gamma = \text{constant}$, or $P_1^{1-\gamma}T_1^\gamma = P_2^{1-\gamma}T_2^\gamma$. This can be rewritten as $\frac{P_2}{P_1} = \left(\frac{T_2}{T_1}\right)^{\frac{\gamma}{\gamma-1}}$. So, $P_2 = P_1 \left(\frac{T_2}{T_1}\right)^{\frac{\gamma}{\gamma-1}}$. Given values: Initial pressure $P_1 = 4 \text{ atm}$. Adiabatic index for monatomic gas $\gamma = \frac{5}{3}$. Initial temperature $T_1 = 27 ^\circ\text{C} = 27 + 273 = 300 \text{ K}$. Final temperature $T_2 = 327 ^\circ\text{C} = 327 + 273 = 600 \text{ K}$. Calculate the exponent $\frac{\gamma}{\gamma-1}$: $\gamma-1 = \frac{5}{3} - 1 = \frac{5-3}{3} = \frac{2}{3}$. $\frac{\gamma}{\gamma-1} = \frac{5/3}{2/3} = \frac{5}{2}$. Calculate the temperature ratio: $\frac{T_2}{T_1} = \frac{600 \text{ K}}{300 \text{ K}} = 2$. Now calculate $P_2$: $P_2 = P_1 \left(2\right)^{5/2} = 4 \text{ atm} \times 2^{5/2}$. Since $4 = 2^2$: $P_2 = 2^2 \times 2^{5/2} = 2^{2 + 5/2} = 2^{4/2 + 5/2} = 2^{9/2} \text{ atm}$. $2^{9/2} = 2^{4.5} = \sqrt{2^9} = \sqrt{512} = 16\sqrt{2} \text{ atm}$. Numerically, $16\sqrt{2} \approx 16 \times 1.4142 = 22.627 \text{ atm}$. The calculated value $P_2 = 2^{9/2}$ atm does not match any of the options directly. Option (d) is $2^{8/3}$ atm $\approx 6.35$ atm. There is a significant discrepancy between the calculated result and the provided options/marked answer. The derived answer using standard physics formulas is $2^{9/2}$ atm. \[ \boxed{2^{9/2} \text{ atm (Calculated, does not match options/marked answer)}} \] (Note: Solution adheres to calculation. Marked answer (d) implies $P_2=2^{8/3}$ atm which is inconsistent with $P_1=4$ atm and the temperature change.)