In case of an adiabatic process,the correct relation in terms of pressure $p$ and density $\rho$ of a gas is:

$A$ fixed amount of a gas undergoes a thermodynamic process as shown in the figure,such that the heat interaction along path $B \to C \to A$ is equal to the work done by the gas along path $A \to B \to C$. Then the process $A \to B$ is:

$A$ spherical bubble inside water has radius $R$. Take the pressure inside the bubble and the water pressure to be $p_0$. The bubble now gets compressed radially in an adiabatic manner so that its radius becomes $(R-a)$. For $a \ll R$,the magnitude of the work done in the process is given by $(4 \pi p_0 R a^2) X$,where $X$ is a constant and $\gamma = C_p / C_V = 41 / 30$. The value of $X$ is:

For $1 \text{ mole}$ of an ideal gas,during an adiabatic process,the square of the pressure of a gas is found to be proportional to the cube of its absolute temperature. The specific heat of the gas at constant volume is ($R$ is universal gas constant)

At $N.T.P.$,one mole of a diatomic gas is compressed adiabatically to half of its volume. Given $\gamma = 1.41$,the work done on the gas will be approximately ....... $J$.

For an ideal gas,the internal energy is given by $U = 5pV/2 + C$,where $C$ is a constant. The equation of the adiabats in the $pV$-plane will be