Question

A sample of an ideal gas occupies a volume V at a pressure P and absolute temperature T. The mass of each molecule is m. The equation for density is
A. $mKT$
B. $\dfrac{P}{{KT}}$
C. $\dfrac{P}{{KTV}}$
D. $\dfrac{{Pm}}{{KT}}$

Answer 1

Hint: We can use the ideal gas equation to find the solution. The universal gas constant is the product of Boltzmann constant and Avogadro number. The volume is calculated as the ratio of total mass by density. By substituting these in the ideal gas equation, we can find the final answer.

Complete step by step answer:
It is given that a sample of ideal gas has a volume $V$ .
The pressure of the ideal gas is given as $P$ .
Absolute temperature of ideal gas is given as $T$ .
Mass of each molecule of the ideal gas is $m$ .
We need to find an equation for density.
Let us use the ideal gas equation to find the answer to this question.
The ideal gas equation is given as
$PV = nRT$
Where, P is the pressure, V is the volume, n is the number of moles, R is the universal gas constant, T is temperature of the gas.
Let us take the number of molecules to be one.
So, the ideal gas equation becomes
$PV = RT$ (1)
We know that the density is given as mass divided by volume.
$\rho = \dfrac{M}{V}$
Where, $\rho $ is the density, M is the mass and V is the volume.
From this we get volume as
$ \Rightarrow V = \dfrac{M}{\rho }$
Mass of each molecule is given as m. Thus, one mole will contain Avogadro number of molecules so we can write the total mass as
$M = Nm$
Where, $N$ denotes the Avogadro number.
So, the volume will be
$ \Rightarrow V = \dfrac{{Nm}}{\rho }$
On substituting for volume in the equation 1, we get
$ \Rightarrow P\dfrac{{Nm}}{\rho } = RT$ (2)
We know that universal gas constant is the product of Boltzmann constant, K and the Avogadro number N.
$ \Rightarrow R = KN$
On substituting this in equation 2, we get
$ \Rightarrow P\dfrac{{Nm}}{\rho } = KNT$
$ \Rightarrow P\dfrac{m}{\rho } = KT$
$ \Rightarrow \rho = \dfrac{{Pm}}{{KT}}$
This is the value of density.
So, the correct answer is option D.

Note: The ideal gas equation is valid only under low density conditions. While using the ideal gas equation we assume that the intermolecular forces acting between the gas molecules is negligible and the size of the molecule is also considered as negligible. So, in the case of heavy gases and in the case of gases having strong intermolecular forces we cannot use ideal gas law.