Question

Derive a relation between density and molar mass of the gas with the help of the ideal gas equation.

Answer 1

Hint: In order to answer this question, we need to have a basic idea about the ideal gas law. Even knowing the definition of density and molar mass of a gas, will be helpful in solving the question.

Step by step answer:
In terms of density we can derive the Ideal gas law as:
$\text{PV = nRT}$
Where,
P = pressure of the gas
V = Volume of the gas
T = Temperature of the gas
n = number of moles of the gas
R = gas constant
Number of moles $\text{= }\dfrac{\text{Given mass}}{\text{Molar mass}}$
$\text{PV = }\dfrac{\text{w}}{\text{M}}\text{ }\!\!\times\!\!\text{ RT}$
Where,
M = molar mass of the gas
w = mass of the gas
$\therefore \text{ P = }\dfrac{\text{wRT}}{\text{VM}}$
$\Rightarrow \text{P = }\dfrac{\text{ }\!\!\rho\!\!\text{ }\times \text{ RT}}{\text{M}}$
$\therefore \text{ }\!\!\rho\!\!\text{ = density = }\dfrac{\text{mass}}{\text{volume}}$
$\Rightarrow \text{ }\!\!\rho\!\!\text{ = }\dfrac{\text{PM}}{\text{RT}}$

Note: We should know that the ideal gas law, also called the general gas equation is the equation of state of a hypothetical ideal gas. It is a good approximation of the behaviour of many gases under many conditions although it has several limitations. Some of the limitations are mentioned below:
The ideal gas model tends to fail at lower temperatures or higher pressures when intermolecular forces and molecular size become important. It also fails for most heavy gases such as many refrigerants and for gases with strong intermolecular forces, notably water vapour.
We should also know that the ideal gas equation holds well as long as a density is kept low.