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Question
Two vessels A and B are filled with the same gas where the volume, temperature, and pressure in vessel A is twice the volume, temperature, and pressure in vessel B. Calculate the ratio of the number of molecules of the gas in vessel A to that in vessel B.
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Solution
Data: VA = 2VB, TA = 2TB, PA = 2PB
PV = nRT
or PV = NkBT
Find: `"N"_"A"/"N"_"B"` = ?
∴ The number of molecules, N = `"PV"/("k"_"B""T")`
∴ NA = `("P"_"A""V"_"A")/("k"_"B""T"_"A")` and NB = `("P"_"B""V"_"B")/("k"_"B""T"_"B")`
∴ `"N"_"A"/"N"_"B" = ("P"_"A"/"P"_"B")("V"_"A"/"V"_"B")("T"_"B"/"T"_"A")`
∴ `"N"_"A"/"N"_"B" = ((2"P"_"B")/"P"_"B")((2"V"_"B")/"V"_"B")("T"_"B"/(2"T"_"B"))`
∴ `"N"_"A"/"N"_"B" = cancel(2) xx 2 xx 1/cancel(2)`
∴ `"N"_"A"/"N"_"B" = 2/1`
∴ `"N"_"A"/"N"_"B"` = 2 : 1
This is the required ratio.
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