A given sample of an ideal gas occupies a volume $V$ at a pressure $p$ and absolute temperature $T$. The mass

Question

A given sample of an ideal gas occupies a volume $V$ at a pressure $p$ and absolute temperature $T$. The mass of each molecule of the gas is $m$. Which of the following gives the density of the gas?

Accepted Answer

According to the Ideal Gas Equation in terms of the number of molecules ($N$):
$pV = NkT$, where $k$ is the Boltzmann constant .
Rearranging for number density ($N/V$):
$\frac{N}{V} = \frac{p}{kT}$.

The density ($ho$) of the gas is the total mass per unit volume. Since the mass of one molecule is $m$, the total mass is $N 	imes m$.
$ho = \frac{	ext{Total Mass}}{V} = \frac{N 	imes m}{V} = m \left( \frac{N}{V} ight)$.

Substituting the value of $N/V$ from the gas equation:
$ho = m \left( \frac{p}{kT} ight) = \frac{pm}{kT}$.

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