$V$ cc volume of a gas having $\gamma = \frac{5}{2}$ is suddenly compressed to $\frac{V}{4}$ cc. The initial pressure of the gas is $P$. The final pressure of the gas will be

In a thermodynamic system,$\Delta U$ represents the increase in internal energy and $W$ represents the work done by the system. Which of the following statements is true?

An ideal gas at $27^{\circ}\text{C}$ is compressed adiabatically to $(8/27)$ of its original volume. If $\gamma = 5/3$, then the rise in temperature of a gas is (in $\text{ K}$)

$A$ monoatomic ideal gas, initially at temperature $T_{1}$, is enclosed in a cylinder fitted with a frictionless piston. The gas is allowed to expand adiabatically to a temperature $T_{2}$ by releasing the piston suddenly. If $l_{1}$ and $l_{2}$ are the lengths of the gas column before and after the expansion respectively, then the value of $\frac{T_{1}}{T_{2}}$ will be

During an adiabatic process,the pressure of a gas is found to be proportional to the cube of its temperature. The ratio of $\frac{C_P}{C_V}$ for the gas is

The temperature of a hypothetical gas increases to $\sqrt{2}$ times when compressed adiabatically to half the volume. Its equation can be written as