Question

An ideal gas is undergoing a cyclic thermodynamic process in different ways as shown in the corresponding $P-V$ diagrams in column 3 of the table. Consider only the path from state 1 to state 2. 𝑊 denotes the corresponding work done on the system. The equations and plots in the table have standard notations as used in thermodynamic processes. Here $\gamma$ is the ratio of heat capacities at constant pressure and constant volume. The number of moles in the gas is $n$.

Column 1	Column 2	Column 3
(I) $W_{1\to 2}=\frac{1}{\gamma -1}\left(P_2{V}_2-P_I{V}_1\right)$	(i) Isothermal	(P)
(II) $W_{1\to 2}=-P{V}_2+P{V}_1$	(ii) Isochoric	(Q)
(III) $W_{1\to 2}=0$	(iii) Isobaric	(R)
(IV)$W_{l\to 2}=-nR{T}^{\prime }n\left(\frac{V_2}{V_1}\right)$	(iv) Adiabatic	(S)

Which one of the following options correctly represents a thermodynamic process that is used as a correction in the determination of the speed of sound in an ideal gas?

• A. (I) (ii) (Q)
• B. (IV) (ii) (R)
• C. (III) (iv) (R)
• D. (I) (iv) (Q)

Answer 1

Answer: (A)

The statement of the question is making reference to adiabatic process.
Now, in adiabatic process, work done by the gas,
$W_{gas}=-\frac{\Delta (PV)}{\gamma -1}$
$\therefore$ Work done on the gas $=+\frac{\Delta (PV)}{\gamma -1}=\frac{P_2{V}_2-P_1{V}_1}{\gamma -1}$
Also, note that $P-V$ graph for the process $1-2$ is steeper (adiabatic) in $Q$ (not in $R$ ).