MHT CET · Physics · Thermodynamics
A jar ' \(P\) ' is filled with gas having pressure, volume and temperature \(\mathrm{P}, \mathrm{V}, \mathrm{T}\) respectively. Another gas jar Q filled with a gas having pressure \(2 \mathrm{P}\), volume \(\frac{\mathrm{V}}{4}\) and temperature \(2 \mathrm{~T}\). The ratio of the number of molecules in jar \(\mathrm{P}\) to those in jar Q is
- A \(1:1\)
- B \(1:2\)
- C \(2:1\)
- D \(4:1\)
Answer & Solution
Correct Answer
(D) \(4:1\)
Step-by-step Solution
Detailed explanation
According to the ideal gas equation \(\mathrm{PV}=\mathrm{Nk}_{\mathrm{B}} \mathrm{T}\)
For jar \(\mathrm{P}\), we have
\(\mathrm{PV}=\mathrm{N}_1 \mathrm{k}_{\mathrm{B}} \mathrm{T}\)... \((i)\)
For jar Q, we have,
(2P) \(\left(\frac{\mathrm{V}}{4}\right)=\mathrm{N}_2 \mathrm{k}_{\mathrm{B}}(2 \mathrm{~T})\)
\(\Rightarrow \mathrm{PV}=4 \mathrm{~N}_2 \mathrm{k}_{\mathrm{B}} \mathrm{T}\)... \((ii)\)
From equations (i) and (ii)
\(\begin{array}{ll}
& \mathrm{N}_1=4 \mathrm{~N}_2 \quad \Rightarrow \frac{\mathrm{N}_1}{\mathrm{~N}_2}=4 \\
\therefore \quad & \mathrm{N}_1: \mathrm{N}_2=4: 1
\end{array}\)
For jar \(\mathrm{P}\), we have
\(\mathrm{PV}=\mathrm{N}_1 \mathrm{k}_{\mathrm{B}} \mathrm{T}\)... \((i)\)
For jar Q, we have,
(2P) \(\left(\frac{\mathrm{V}}{4}\right)=\mathrm{N}_2 \mathrm{k}_{\mathrm{B}}(2 \mathrm{~T})\)
\(\Rightarrow \mathrm{PV}=4 \mathrm{~N}_2 \mathrm{k}_{\mathrm{B}} \mathrm{T}\)... \((ii)\)
From equations (i) and (ii)
\(\begin{array}{ll}
& \mathrm{N}_1=4 \mathrm{~N}_2 \quad \Rightarrow \frac{\mathrm{N}_1}{\mathrm{~N}_2}=4 \\
\therefore \quad & \mathrm{N}_1: \mathrm{N}_2=4: 1
\end{array}\)
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