Thermodynamics: closed system problem

[jofree87]

As shown in the figure attached, an insulated box is initially divided into halves by a frictionless, thermally conducting piston. On one side of the piston is 1.5 m^3 of air at 400 K, 4 bar. On the other side is 1.5 m^3 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.

The answer provided by my professor for temperature is 400 K, and the final pressure, 3 bar. I would like to know how is this problem solved?

 

[jbsteele]

Thank you. Solution:a) The final temperature in this case remains at 400 K since the piston is frictionless and thermally conducting, and therefore the heat produced due to work done by the internal forces in the system (due to the pressure difference) will be uniformly distributed to both sides of the piston. Thus, the temperatures of both sides remain the same.b) To determine the final pressure, we use the ideal gas law:PV = nRTWhere P is the pressure, V is the volume, n is the number of moles of gas, R is the universal gas constant, and T is the temperature.Since both sides of the piston are initially at the same temperature and volume, we can rewrite the equation as follows:P1V1 = P2V2where P1 and V1 are the pressure and volume on one side of the piston, and P2 and V2 are the pressure and volume on the other side.We know that V1 = V2 = 1.5 m^3. We also know that P1 = 4 bar and P2 = 2 bar. Substituting these values into the equation above, we get:4(1.5 m^3) = 2(1.5 m^3)Solving for P2 gives us a final pressure of 3 bar.