$A$ container is filled with $20$ moles of an ideal diatomic gas at absolute temperature $T$. When heat is supplied to the gas,the temperature remains constant,but $8$ moles dissociate into atoms. The heat energy given to the gas is ......... (in $R T$)

An ideal gas goes through a reversible cycle $a \to b \to c \to d$ and has the $V - T$ diagram shown below. Processes $d \to a$ and $b \to c$ are adiabatic. The corresponding $P - V$ diagram for the process is (all figures are schematic and not drawn to scale):

$A$ monoatomic gas at pressure $P$ having volume $V$ expands isothermally to a volume $2V$ and then adiabatically to a volume $16V$. The final pressure of the gas is $\left(\gamma = \frac{5}{3}\right)$.

$A$ thermally insulated vessel contains an ideal gas of molecular mass $M$ and ratio of specific heats $1.4$. The vessel is moving with speed $v$ and is suddenly brought to rest. Assuming no heat is lost to the surroundings,the temperature of the gas increases by ... ($R =$ universal gas constant)

For an ideal gas,which of the following statements is correct?

One mole of diatomic gas having rotational modes only is kept in a cylinder with a piston system. The cross-section area of the cylinder is $4 \text{ cm}^2$. The gas is heated slowly to raise the temperature by $1.2^\circ\text{C}$ during which the piston moves by $25 \text{ mm}$. The amount of heat supplied to the gas is . . . . . . $J$. (Atmospheric pressure = $100 \text{ kPa}$, $R = 8.3 \text{ J/mol}\cdot\text{K}$) (Neglect mass of the piston)