The amount of heat needed to raise the temperature of 4 moles of rigid diatomic gas from 0$$^\circ$$ C to 50$$^\circ$$ C when no work is done is ___________. (R is the universal gas constant).

The entropy of any system is given by
$$S = {\alpha ^2}\beta \ln \left[ {{{\mu kR} \over {J{\beta ^2}}} + 3} \right]$$ where $$\alpha$$ and $$\beta$$ are the constants. $$\mu$$, J, k and R are no. of moles, mechanical equivalent of heat, Boltzmann constant and gas constant respectively.
[Take $$S = {{dQ} \over T}$$]
Choose the incorrect option from the following :

Consider a sample of oxygen behaving like an ideal gas. At 300 K, the ratio of root mean square (rms) velocity to the average velocity of gas molecule would be :

(Molecular weight of oxygen is 32g/mol; R = 8.3 J K^$$-$$1 mol^$$-$$1)

For an adiabatic expansion of an ideal gas, the fractional change in its pressure is equal to (where $$\gamma$$ is the ratio of specific heats) :