MHT CET · Physics · Thermodynamics
An ideal gas at \(27^{\circ} \mathrm{C}\) is compressed adiabatically to (8/27) of its original volume. If ratio of specific heats, \(\gamma=5 / 3\) then the rise in temperature of the gas is
- A \(500 \mathrm{~K}\)
- B \(125 \mathrm{~K}\)
- C \(250 \mathrm{~K}\)
- D \(375 \mathrm{~K}\)
Answer & Solution
Correct Answer
(D) \(375 \mathrm{~K}\)
Step-by-step Solution
Detailed explanation
For an adiabatic process \(\mathrm{TV}^{\gamma-1}=\) constant
\(\therefore \frac{\mathrm{T}_2}{\mathrm{~T}_1}=\left(\frac{\mathrm{V}_1}{\mathrm{~V}_2}\right)^{\gamma-1}=\left(\frac{27}{8}\right)^{\frac{5}{3}-1}=\left(\frac{27}{8}\right)^{\frac{2}{3}}=\) \(\frac{9}{4} \)
\( \therefore \mathrm{T}_2=\frac{9}{4} \cdot \mathrm{T}_1=\frac{9}{4} \times 300=675 \mathrm{~K} \)
\( \therefore \mathrm{T}_2-\mathrm{T}_1=675-300=375 \mathrm{~K}\)
\(\therefore \frac{\mathrm{T}_2}{\mathrm{~T}_1}=\left(\frac{\mathrm{V}_1}{\mathrm{~V}_2}\right)^{\gamma-1}=\left(\frac{27}{8}\right)^{\frac{5}{3}-1}=\left(\frac{27}{8}\right)^{\frac{2}{3}}=\) \(\frac{9}{4} \)
\( \therefore \mathrm{T}_2=\frac{9}{4} \cdot \mathrm{T}_1=\frac{9}{4} \times 300=675 \mathrm{~K} \)
\( \therefore \mathrm{T}_2-\mathrm{T}_1=675-300=375 \mathrm{~K}\)
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