Question

Monoatomic gas is kept under a piston of mass 3 kg in a vessel in equilibrium. Vessel & piston are thermally insulated. Piston can move vertically without friction. A block of mass 5 kg is put on top of piston and allowed to attain the equilibrium as in figure (b). The temperature of gas becomes $T_1$. After removing the block piston moves upward and when equilibrium is attained again temperature of gas becomes $T_2$. Neglect atmospheric pressure. Then-

• A. $T_1T_2 = \frac{35}{9}T_0^2$
• B. $T_1T_2 = \frac{25}{12}T_0^2$
• C. $T_1T_2 = \frac{25}{9}T_0^2$
• D. $T_1T_2 = \frac{35}{12}T_0^2$

Answer 1

None of the above. The correct relationship is $T_1T_2 = T_0^2 \left(\frac{8}{3}\right)^{2/5}$

Answer 2

The system undergoes adiabatic processes. First, a 5 kg block is added, compressing the gas and raising the temperature from $T_0$ to $T_1$. Then, the block is removed, allowing the gas to expand and cool to $T_2$.

Key relationships:

• Adiabatic process: $T^{\gamma}P^{1-\gamma} = constant$, where $\gamma = \frac{5}{3}$ for a monoatomic gas. This implies $T \propto P^{\frac{1-\gamma}{\gamma}}$ which simplifies to $T \propto P^{2/5}$.

• Pressure: Pressure is determined by the weight on the piston divided by its area (A).

  • $P_0 = \frac{3g}{A}$
  • $P_1 = \frac{8g}{A}$
  • $P_2 = \frac{3g}{A}$

Step 1: Process from state (a) to (b)

$\frac{T_1}{T_0} = \left(\frac{P_1}{P_0}\right)^{2/5} = \left(\frac{8g/A}{3g/A}\right)^{2/5} = \left(\frac{8}{3}\right)^{2/5}$ $T_1 = T_0 \left(\frac{8}{3}\right)^{2/5}$

Step 2: Process from state (b) to (c)

$\frac{T_2}{T_1} = \left(\frac{P_2}{P_1}\right)^{2/5} = \left(\frac{3g/A}{8g/A}\right)^{2/5} = \left(\frac{3}{8}\right)^{2/5}$ $T_2 = T_1 \left(\frac{3}{8}\right)^{2/5}$

Step 3: Calculate $T_1T_2$

$T_1 T_2 = T_0 \left(\frac{8}{3}\right)^{2/5} \times T_1 \left(\frac{3}{8}\right)^{2/5}= T_0 T_1 \left(\frac{8}{3}\right)^{2/5}\left(\frac{3}{8}\right)^{2/5} = T_0 T_1$

Substitute $T_1$:

$T_1 T_2 = T_0 \left(\frac{8}{3}\right)^{2/5} T_0 = T_0^2 \left(\frac{8}{3}\right)^{2/5}$

Therefore, $T_1T_2 = T_0^2 \left(\frac{8}{3}\right)^{2/5}$. None of the provided options match this result, suggesting a potential error in the question or answer choices.