During a compression process involving an ideal gas at pressure \(P_{1}\), when the volume, \(V_{1}\) of the gas was halved, the temperature in Kelvin increases by half its initial value. The final pressure \(P_{2}\) is given by ____________
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Correct Answer: Option A
Explanation:
Given data
\begin{aligned}
&P_{1}=P_{1}, V_{1}=V_{1}, V_{2}=V_{1} / 2 . T_{1}=T_{1}, \\
&T_{2}=T_{1}+\frac{T_{1}}{2}=\frac{3 T_{1}}{2},
\end{aligned}
from the ideal gas equation,
\begin{aligned}
\frac{P_{1} V_{1}}{T_{1}} &=\frac{P_{2} V_{2}}{T_{2}} ; \frac{P_{1} V_{1} T_{2}}{V_{2} T_{1}} \\
&=\frac{P_{1} \times V_{1} \times \frac{3}{2} T_{1}}{\frac{V_{2}}{2} \times T_{1}}=P_{2}=3 P_{1}
\end{aligned}
Given data
\begin{aligned}
&P_{1}=P_{1}, V_{1}=V_{1}, V_{2}=V_{1} / 2 . T_{1}=T_{1}, \\
&T_{2}=T_{1}+\frac{T_{1}}{2}=\frac{3 T_{1}}{2},
\end{aligned}
from the ideal gas equation,
\begin{aligned}
\frac{P_{1} V_{1}}{T_{1}} &=\frac{P_{2} V_{2}}{T_{2}} ; \frac{P_{1} V_{1} T_{2}}{V_{2} T_{1}} \\
&=\frac{P_{1} \times V_{1} \times \frac{3}{2} T_{1}}{\frac{V_{2}}{2} \times T_{1}}=P_{2}=3 P_{1}
\end{aligned}