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(B) The ideal gas equation is given by $PV = nRT$,where $n$ is the number of moles. Since the number of molecules $N$ is proportional to the number of

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$4$

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(B) The ideal gas equation is given by $PV = nRT$,where $n$ is the number of moles. Since the number of molecules $N$ is proportional to the number of moles $(N = nN_A)$,the ratio of the number of molecules is equal to the ratio of the number of moles.For the first sample: $P_1 = P, V_1 = V, T_1 = T$. Thus,$n_1 = \frac{PV}{RT}$.For the second sample: $P_2 = 2P, V_2 = V/4, T_2 = 2T$. Thus,$n_2 = \frac{(2P)(V/4)}{R(2T)} = \frac{PV/2}{2RT} = \frac{PV}{4RT}$.The ratio of the number of molecules is $\frac{N_1}{N_2} = \frac{n_1}{n_2} = \frac{PV/RT}{PV/4RT} = \frac{1}{1/4} = 4:1$.

We have a sample of gas characterized by $P, V$ and $T$ and another sample of gas characterized by $2P, V/4$ and $2T$. What is the ratio of the number of molecules in the first and second samples (in $: 1$)?

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