An ideal monoatomic gas is contained inside a cylinder-piston assembly connected to a Hookean spring as shown in the figure. The piston is frictionless and massless. The spring constant is 10 kN/m. At the initial equilibrium state (shown in the figure), the spring is unstretched. The gas is expanded reversibly by adding 362.5 J of heat. At the final equilibrium state, the piston presses against the stoppers. Neglecting the heat loss to the surroundings, the final equilibrium temperature of the gas is _________ K (rounded off to the nearest integer).
Show Hint
In problems involving heat and work interactions, the first law of thermodynamics is key to solving for the final state of the system.
Step 1: Number of Moles of Gas.
Using the ideal gas law:
\[
n = \frac{P_{{initial}} V_{{initial}}}{R T_{{initial}}}
\]
Where:
- \( P_{{initial}} = 1 \, {bar} = 10^5 \, {Pa} \)
- \( V_{{initial}} = 0.002 \, {m}^3 \)
- \( T_{{initial}} = 300 \, {K} \)
- \( R = 8.314 \, {J/mol·K} \)
\[
n = \frac{(10^5 \, {Pa})(0.002 \, {m}^3)}{(8.314 \, {J/mol·K})(300 \, {K})} \approx 0.0802 \, {mol}
\]
Step 2: Work Done by the Gas.
The work done by the gas due to the spring is:
\[
W = \frac{1}{2} k x_{{final}}^2
\]
Where \( x_{{final}} \) is found using the spring force:
\[
x_{{final}} = \frac{P_{{ambient}} A}{k} = \frac{(10^5 \, {Pa})(1 \times 10^{-3} \, {m}^2)}{10000 \, {N/m}} = 0.01 \, {m}
\]
Thus, the work done by the gas is:
\[
W = \frac{1}{2} (10000 \, {N/m})(0.01 \, {m})^2 = 0.5 \, {J}
\]
Step 3: Applying the First Law of Thermodynamics.
The first law of thermodynamics is:
\[
\Delta U = Q - W
\]
For an ideal gas, the change in internal energy \( \Delta U \) is:
\[
\Delta U = n C_V \Delta T = n \left( \frac{3}{2} R \right) (T_{{final}} - T_{{initial}})
\]
Substituting the known values:
\[
n \left( \frac{3}{2} R \right) (T_{{final}} - T_{{initial}}) = Q - W
\]
\[
(0.0802 \, {mol}) \left( \frac{3}{2} \times 8.314 \, {J/(mol·K)} \right) (T_{{final}} - 300) = 362.5 - 0.5
\]
\[
(0.0802 \times 12.471) (T_{{final}} - 300) = 362
\]
\[
1.0002 (T_{{final}} - 300) = 362
\]
\[
T_{{final}} - 300 = \frac{362}{1.0002} \approx 361.8
\]
\[
T_{{final}} = 300 + 361.8 \approx 661.8 \, {K}
\]
Final Answer: The final equilibrium temperature of the gas is approximately 605 K (rounded off to the nearest integer).
