In a thermodynamic system,$\Delta U$ represents the increase in its internal energy and $dW$ is the work done by the system. Then,which of the following statements is correct?

$A$ gas for which $\gamma = 1.5$ is suddenly compressed to $\frac{1}{4}$ of its initial volume. What is the ratio of the final pressure to the initial pressure?

An ideal gas at atmospheric pressure is adiabatically compressed so that its density becomes $32$ times its initial value. If the final pressure of the gas is $128$ atmospheres,the value of $\gamma$ for the gas is:

In an adiabatic process,the initial and final temperatures of $1 \, mol$ of an ideal gas are $T_1$ and $T_2$ respectively. The change in internal energy of the gas is .......

The following figure shows an adiabatic cylindrical container of volume $V_0$ divided by an adiabatic smooth piston (area of cross-section = $A$) into two equal parts. An ideal gas $(C_P/C_V = \gamma)$ is at pressure $P_1$ and temperature $T_1$ in the left part,and a gas at pressure $P_2$ and temperature $T_2$ is in the right part. The piston is slowly displaced and released at a position where it can stay in equilibrium. The final pressure of the two parts will be (Suppose $x$ = displacement of the piston):

For an adiabatic process,the wrong statement is: