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Kinetic Theory — Physics STD 11 Science — Question

Gujarat BoardEnglish MediumSTD 11 SciencePhysicsKinetic Theory5 Marks
Question
Figure. shows two vessels A and B with rigid walls containing ideal gases. The pressure, temperature and the volume are $p_A, T_A, V$ in the vessel A and $p_B, T_B$, V in the vessel B. The vessels are now connected through a small tube. Show that the pressure p and the temperature T satisfy $\frac{\text{p}}{\text{T}}=\frac{1}{2}\Big(\frac{\text{P}_\text{A}}{\text{T}_\text{A}}+\frac{\text{p}_\text{B}}{\text{T}_\text{B}}\Big)$ when equilibrium is achieved.
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Answer



Now, Let the final pressure; Volume & Temp be
After connection = $P_A' \rightarrow$ Partial pressure of A
$P_B' \rightarrow$ Partial pressure of B
Now, $\frac{\text{P}_\text{A}'\times2\text{V}}{\text{T}}=\frac{\text{P}_\text{A}\times\text{V}}{\text{T}_\text{A}}$
Or $\frac{\text{P}_\text{A}'}{\text{T}}=\frac{\text{P}_\text{A}}{2\text{T}_\text{A}}\ ...(1)$
Similarly, $\frac{\text{P}_\text{B}'}{\text{T}}=\frac{\text{P}_\text{B}}{2\text{T}_\text{B}}\ ...(2)$
Adding (1) & (2)
$\frac{\text{P}_\text{A}'}{\text{T}}+\frac{\text{P}_\text{B}'}{\text{T}}=\frac{\text{P}_\text{A}}{2\text{T}_\text{A}}+\frac{\text{P}_\text{B}}{2\text{T}_\text{B}}=\frac{1}{2}\Big(\frac{\text{P}_\text{A}}{\text{T}_\text{A}}+\frac{\text{P}_\text{B}}{\text{T}_\text{B}}\Big)$
$\Rightarrow\frac{\text{P}}{\text{T}}=\frac{1}{2}\Big(\frac{\text{P}_\text{A}}{\text{T}_\text{A}}+\frac{\text{P}_\text{B}}{\text{T}_\text{B}}\Big)$ $\big[\therefore\text{P}_\text{A}'+\text{P}_\text{B}'=\text{P}\big]$

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