An ideal gas undergoes a quasi-static, reversible process in which its molar heat capacity \(C\) remains constant. If during this process the relation of pressure \(P\) and volume \(V\) is given by \(PV^n\) = constant, then \(n\) is given by: (here \(C_P\) and \(C_V\) are molar specific heat at constant pressure and constant volume, respectively) 
1. \( n =\dfrac{C_P}{C_V} \)
2. \(n =\dfrac{C-C_P}{C-C_V} \)
3. \(n =\dfrac{C_P-C}{C-C_V} \)
4. \(n =\dfrac{C-C_V}{C-C_P}\)

\(C_P\) and \(C_V\) are specific heats at constant pressure and constant volume respectively. It is observed that
\(C_P-C_V=a\) for hydrogen gas
\(C_P-C_V=b\) for nitrogen gas
The correct relation between \(a\) and \(b\) is:
1. \(a=\frac{1}{14} b\)
2. \(a= b\)
3. \(a=14b\)
4. \(a=28b\)

A cylinder with fixed capacity of \(67.2\) litres contains helium gas at STP. The amount of heat needed to raise the temperature of the gas by \(20^\circ \text{C}\) is: [Given that \(R = 8.31\) J mol^–1K^–1]
1. \(748~\text{J}\)
2. \(350~\text{J}\)
3. \(374~\text{J}\)
4. \(700~\text{J}\)

When heat \(Q\) is supplied to a diatomic gas of rigid molecules, at constant volume its temperature increases by \(\Delta T\). the heat required to produce the same change in temperature, at a constant pressure is:
1. \( \frac{7}{5} Q \)
2. \(\frac{3}{2} Q \)
3. \( \frac{2}{3} Q \)
4. \( \frac{5}{3} Q\)
 

Two moles of helium gas are mixed with three moles of hydrogen molecules (taken to be rigid). What is the molar specific heat of the mixture at constant volume? (\(R = 8.3~\text{J/mol K}\))
1. \(21.6~\text{J/mol K}\)
2. \(19.7~\text{J/mol K}\)
3. \(15.7~\text{J/mol K}\)
4. \(17.4~\text{J/mol K}\)

A diatomic gas, having \(C_P=\frac{7}{2}R\) and \(C_V=\frac{5}{2}R\) is heated at constant pressure. The ratio of \(dU:dQ:dW\) is:
1. \(5:7:3\)
2. \(5:7:2\)
3. \(3:7:2\)
4. \(3:5:2\)

One mole of a rigid diatomic gas performs a work of \(\dfrac{Q}{5}\) when heat \(Q\) is supplied to it. The molar heat capacity of the gas during this transformation is:
1. \(\dfrac{15}{8} R\)

2. \(\dfrac{5}{8} R\)

3. \(\dfrac{25}{8} R\)

4. \(\dfrac{5}{7} R\)