Question

A rigid tank whose volume is unknown is divided into two parts by a partition. One side of the tank contains an ideal gas at \(927^{\circ} \mathrm{C}\). The other side is evacuated and has a volume twice the size of the part containing the gas. The partition is now removed and the gas expands to fill the entire tank. Heat is now applied to the gas until the pressure equals the initial pressure. Determine the final temperature of the gas. Answer: \(3327^{\circ} \mathrm{C}.\)

Short answer

Answer: The final temperature of the gas is approximately 3327°C.

Step-by-step solution

Analyze Initial State

Initially, one side of the tank contains an ideal gas at \(927^{\circ} \mathrm{C}\). Let the volume of this side be \(V_1\). The other side is evacuated and has a volume twice the size of the part containing the gas, which means its volume is \(2V_1\). The total volume of the tank is \(V_1 + 2V_1 = 3V_1\). Let the initial pressure be \(P_1\) and the initial temperature be \(T_1 = 927 + 273.15 = 1200.15\,\text{K}\). Since the other side is evacuated, the moles of gas (\(n\)) and the Gas constant (\(R\)) are constant.

Analyze Final State

After the partition is removed and the gas expands, it fills the entire tank, so the final volume \(V_2\) is \(3V_1\). Let the final pressure be \(P_2\) and the final temperature be \(T_2\). Since we are applying heat until the pressure equals the initial pressure, \(P_2 = P_1\).

Apply the Ideal Gas Law to the Initial State

For the initial state, we have: \(P_1V_1 = nRT_1\)

Apply the Ideal Gas Law to the Final State

For the final state, we have: \(P_2V_2 = nRT_2\)

Determine the Final Temperature

Since \(P_2 = P_1\) and \(V_2 = 3V_1\), we can rewrite the equation for the final state as: \(P_1(3V_1) = nRT_2\) Now, we can divide the equation for the final state by the equation for the initial state: \(\frac{nRT_2}{P_1V_1} = \frac{nRT_1}{P_1(3V_1)}\) Simplifying the equation, we get: \(T_2 = 3T_1\) Now, substitute the value of \(T_1\): \(T_2 = 3 \times 1200.15\,\text{K} \approx 3600.45\,\text{K}\) Finally, convert \(T_2\) back to Celsius: \(T_2 = 3600.45 - 273.15 = 3327^{\circ}\mathrm{C}\) The final temperature of the gas is \(3327^{\circ}\mathrm{C}\).