A cylinder with a movable piston contains $3$ moles of hydrogen at standard temperature and pressure. The walls of the cylinder are made of a heat insulator, and the piston is insulated by having a pile of sand on it. By what factor does the pressure of the gas increase if the gas is compressed to half its original volume ?

An ideal gas at a pressures of $1$ atmosphere and temperature of ${27^o}C$ is compressed adiabatically until its pressure becomes $8$ times the initial pressure, then the final temperature is ….. $^oC$ ($\gamma  = 3/2$)

Consider that an ideal gas ($n$ moles) is expanding in a process given by $P = f (V)$, which passes through a point $(V_0, \,p_0)$. Show that the gas is absorbing heat at $(p_0,\, V_0)$ if the slope of the curve $P = f (V)$ is larger than the slope of the adiabatic passing through $(p_0,\, V_0)$.

The equation of state for a gas is given by $PV = nRT + \alpha V$, where $n$ is the number of moles and $\alpha $ is a positive constant. The initial temperature and pressure of one mole of the gas contained in a cylinder are $T_o$ and $P_o$ respectively. The work done by the gas when its temperature doubles isobarically will be

The $PV$ diagram shows four different possible reversible processes performed on a monatomic ideal gas. Process $A$ is isobaric (constant pressure). Process $B$ is isothermal (constant temperature). Process $C$ is adiabatic. Process $D$ is isochoric (constant volume). For which process$(es)$ does the temperature of the gas decrease?