One mole of an ideal gas of volume $V$ and temperature $T$ is allowed to expand adiabatically to volume $2 V$ while doing no external work. The universal gas constant is $R$. What is the pressure of the gas after expansion?

Two identical boxes contain the same ideal gas. Let $\left(n_1, \lambda_1, T_1\right)$ and $\left(n_2, \lambda_2, T_2\right)$ be the number density, mean free path and temperature of the gas in the first and the second box, respectively. One of the boxes is empty into the other one. What will be the mean free path? $\lambda$ and temperature $T$ of the gas now?

57. Consider a Carnot heat engine operating between a heat reservoir at temperature 600 K , and an external atmosphere at temperature 300 K . In one cycle, 1000 kJ of heat is extracted from the heat reservoir, and the associated work is input to a reversible refrigerator that operates between 200 K and the same external atmosphere at 300 K . The refrigerator completes one cycle and releases heat into the atmosphere. How much heat is released into the atmosphere at the end of one cycle of the combined system?