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(A) The ideal gas equation is given by $PV = nRT$,where $n$ is the number of moles.Since $n = N/N_A$ (where $N$ is the number of molecules and $N_A$ i

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$4:1$

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(A) The ideal gas equation is given by $PV = nRT$,where $n$ is the number of moles.Since $n = N/N_A$ (where $N$ is the number of molecules and $N_A$ is Avogadro's number),the equation becomes $PV = (N/N_A)RT$.For container $A$: $P_A V_A = (N_A / N_{Avogadro}) R T_A \Rightarrow PV = (N_A / N_{Avogadro}) R T$.For container $B$: $P_B V_B = (N_B / N_{Avogadro}) R T_B \Rightarrow (2P)(V/4) = (N_B / N_{Avogadro}) R (2T) \Rightarrow PV/2 = (N_B / N_{Avogadro}) R (2T)$.Dividing the two equations:$\frac{PV}{PV/2} = \frac{N_A}{N_B} 	imes \frac{T}{2T} \Rightarrow 2 = \frac{N_A}{N_B} 	imes \frac{1}{2}$.Therefore,$\frac{N_A}{N_B} = 4$,which is $4:1$.

$A$ container $A$ contains a gas at thermodynamic coordinates $P, V,$ and $T$. Another container $B$ contains a different gas at $2P, V/4,$ and $2T$. The ratio of the number of molecules in container $A$ to the number of molecules in container $B$ is:

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