Question

An insulated box is initially divided into halves by a frictionless, thermally conducting piston. On one side of the piston is 1.5 m³ of air at 400 K, 4 bar.. On the other side is 1.5 m³ of air at 400 K, 2 bar. The piston is released and equilibrium is attained, with the piston experiencing no change of state. Employing the ideal gas model for the air, determine:

a. The final temperature, in K.
b. The final pressure, in bar.
c. The amount of entropy produced, in kJ/K.

Answer 1

a. The final temperature after equilibrium is 400 K.

b. The final pressure after equilibrium is 3 bar.

c. The amount of entropy produced during this process is 0 kJ/K.

Step-by-step explanation:

Initially, the two chambers contain air at different pressures but the same temperature. When the piston is released, the air will tend to equalize the pressure on both sides. According to the ideal gas law, PV = nRT, where P is pressure, V is volume, n is the number of moles, R is the gas constant, and T is temperature. Since the piston is frictionless and thermally conducting, it allows for the exchange of energy without work, and the temperatures will equalize.

Given that the temperatures and volumes are the same initially and there's no change in volume after equilibrium, the final temperature remains at 400 K. To find the final pressure, we use the relationship between pressures and temperatures for an ideal gas
`\(P_1 / P_2 = T_1 / T_2\)`. Solving for the final pressure, we get
`\(4 \, \text{bar} / 2 \, \text{bar} = 400 \, \text{K} / T_2\)`, yielding a final pressure of 3 bar.

Regarding entropy production, since the process occurs at constant temperature and there's no irreversibility or change in the system's disorder, the entropy produced is zero. This is because there's no net change in entropy when the system goes from an initial state to a final equilibrium state at constant temperature.