Question

An ideal gas at $27°\mathrm{C}$ has compressed adiabatically to $\frac{8}{27}$ of its original volume. If $\mathrm{\gamma }=\frac{5}{3}$ then the rise in temperature will be

• A. $475\mathrm{K}$
• B. $150\mathrm{K}$
• C. $275\mathrm{K}$
• D. $375\mathrm{K}$

Answer 1

The correct option is D

$375\mathrm{K}$

Explanation for the correct option

Option D: $375\mathrm{K}$

Adiabatic compression

• Adiabatic compression of a gas is defined as the compression in which no heat is added or subtracted from the gas, and the internal energy of the gas is increased, which is equal to the external work done on the gas.

Step 1: Given data

• The initial temperature of the gas $=27°\mathrm{C}$
• The volume of gas after compression $=\frac{8}{27}\mathrm{of}\mathrm{the}\mathrm{original}\mathrm{volume}$
• Degree of freedom,$\mathrm{\gamma }=\frac{5}{3}$

Step 2: Formula used

In the case of an adiabatic process, ${\mathrm{TV}}^{\mathrm{\gamma }-1}=\mathrm{constant}$ where

• $\mathrm{T}=\mathrm{temperature}$
• $\mathrm{V}=\mathrm{volume}$
• $\mathrm{\gamma }=\mathrm{degree}\mathrm{of}\mathrm{freedom}$

Step 3: Calculation of temperature

• Let the initial temperature be ${\mathrm{T}}_1$$=27°\mathrm{C}$
• Let the final temperature be ${\mathrm{T}}_2$
• Let the initial volume be ${\mathrm{V}}_1$ and the final volume be $V_2$.
• According to the question, $V_2\mathit{=}\frac{\mathit{8}}{\mathit{27}}{V}_{\mathit{1}}$
• Using the formula of the adiabatic process in the initial case, ${\mathrm{T}}_1{{\mathrm{V}}_1}^{\mathrm{\gamma }-1}=\mathrm{constant}$
• Using the formula of the adiabatic process in the final case, ${\mathrm{T}}_2{{\mathrm{V}}_2}^{\mathrm{\gamma }-1}=\mathrm{constant}$
• Hence, ${\mathrm{T}}_1{{\mathrm{V}}_1}^{\mathrm{\gamma }-1}={\mathrm{T}}_2{{\mathrm{V}}_2}^{\mathrm{\gamma }-1}$

$$\begin{gathered} \Rightarrow {\left(\frac{{\mathrm{V}}_1}{{\mathrm{V}}_2}\right)}^{\mathrm{\gamma }-1}=\frac{{\mathrm{T}}_2}{{\mathrm{T}}_1} \\ \Rightarrow {\left(\frac{{\mathrm{V}}_1}{{\displaystyle \frac{8}{27}}{\mathrm{V}}_1}\right)}^{\frac{5}{3}-1}=\frac{{\mathrm{T}}_2}{27} \\ \Rightarrow {\left(\frac{27}{{\displaystyle 8}}\right)}^{\frac{2}{3}}=\frac{{\mathrm{T}}_2}{27+273} \\ \Rightarrow {\left(\frac{27}{{\displaystyle 8}}\right)}^{\frac{2}{3}}=\frac{{\mathrm{T}}_2}{300} \\ \Rightarrow \frac{9}{{\displaystyle 4}}\times 300={\mathrm{T}}_2 \\ \Rightarrow {\mathrm{T}}_2=675\mathrm{K} \end{gathered}$$

• Rise in temperature = ${\mathrm{V}}_2-{\mathrm{V}}_1$=$675\mathrm{K}-300\mathrm{K}=375\mathrm{K}$

Hence this option is the correct answer.