MHT CET · Physics · Thermodynamics
A monoatomic ideal gas, initially at temperature \(T_1\) is enclosed in a cylinder fitted with a frictionless piston. The gas is allowed to expand adiabatically to a temperature \(T_2\) by releasing the piston suddenly. If \(L_1\) and \(L_2\) are the lengths of the gas column before and after expansion respectively then \(\frac{T_1}{T_2}\) is given by
- A \(\left(\frac{L_1}{L_2}\right)\)
- B \(\left(\frac{L_2}{L_1}\right)^{2 / 3}\)
- C \(\left(\frac{L_1}{L_2}\right)^{2 / 3}\)
- D \(\left(\frac{L_2}{L_1}\right)\)
Answer & Solution
Correct Answer
(B) \(\left(\frac{L_2}{L_1}\right)^{2 / 3}\)
Step-by-step Solution
Detailed explanation
For adiabatic process:
\(\begin{aligned} & T V Y-1=\text { constant } \\ & \Rightarrow T_1 V_2-1=T_2 V_2^{\gamma-1} \\ & \Rightarrow \frac{T_1}{T_2}=\left(\frac{A}{A} \frac{L_2}{L_1}\right)^{\frac{5}{3}-1} \\ & \Rightarrow \frac{T_1}{T_2}=\left(\frac{L_2}{L_1}\right)^{\frac{2}{3}}\end{aligned}\)
\(\begin{aligned} & T V Y-1=\text { constant } \\ & \Rightarrow T_1 V_2-1=T_2 V_2^{\gamma-1} \\ & \Rightarrow \frac{T_1}{T_2}=\left(\frac{A}{A} \frac{L_2}{L_1}\right)^{\frac{5}{3}-1} \\ & \Rightarrow \frac{T_1}{T_2}=\left(\frac{L_2}{L_1}\right)^{\frac{2}{3}}\end{aligned}\)
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