Question

In an adiabatic process on a gas with γ = 1.4, the pressure is increased by 0.5%. The volume decreases by about
(a) 0.36%
(b) 0.5%
(c) 0.7%
(d) 1%

Answer 1

(a) 0.36 %

Let p and p^, be the initial and final pressures of the system and V and V^, be the initial and final volumes of the system. p^, is 0.5% more than p and the process is adiabatic. So,

$$\begin{gathered} p{V}^{\mathit{\gamma }}=p\text{'}V{\text{'}}^{\gamma },\mathit{}\gamma =1.4=\frac{7}{5} \\ \Rightarrow \frac{V{\text{'}}^{\gamma }}{V^{\gamma }}=\frac{p}{p\text{'}} \\ \Rightarrow {\left(\frac{V\text{'}}{V}\right)}^{\gamma }=\frac{p}{p+{\displaystyle \frac{0.5p}{100}}} \\ \Rightarrow {\left(\frac{V\text{'}}{V}\right)}^{\raisebox{1ex}{$7$}\!\left/ \!\raisebox{-1ex}{$5$}\right.}=\frac{100}{100.5} \\ \Rightarrow \frac{V\text{'}}{V}={\left(\frac{100}{100.5}\right)}^{\raisebox{1ex}{$5$}\!\left/ \!\raisebox{-1ex}{$7$}\right.}=(\frac{200}{201}{)}^{\frac{5}{7}} \\ \Rightarrow \frac{V\text{'}}{V}=0.99644 \\ \Rightarrow V\text{'}=0.99644V \\ \Rightarrow V\text{'}=V-0.00356V \end{gathered}$$
Therefore, V^, is 0.36 % less than V.