Question

Critical density of a gas having molecular weight 39 $\text{g mo}{{\text{l}}^{-1}}$ is $0.1\times {{10}^{3}}\text{gc}{{\text{m}}^{-3}}$. Its critical volume in $\text{L mo}{{\text{l}}^{-1}}$ is:
A. 0.390
B. 3.90
C. 0.039
D. 39.0
E. 390

Answer 1

Hint: There is no much difference between critical density and density while solving. So, use the relation between density, mass and volume of $\text{density=}\dfrac{\text{mass}}{\text{volume}}$. Use molar mass as critical mass of the gas. Convert the units of cubic centimetres to litres in the end.

Complete answer:
Let us solve this numerical question step by step;
Step (1)- We know that relation between density, mass and volume, which is $\text{density=}\dfrac{\text{mass}}{\text{volume}}$.
Step (2)- When we are talking about critical volume, critical density, then critical mass becomes molecular weight. As, critical means edging or nearing conditions.
The relation between the three will be critical density = $\dfrac{\text{critical mass}}{\text{critical volume}}\text{ or }\dfrac{\text{molecular mass}}{\text{critical volume}}$.
The molecular mass is given as 39$\text{g mo}{{\text{l}}^{-1}}$.
the critical density will be $\dfrac{39}{\text{critical volume}}$.
Step (3)- The critical density is given as $0.1\times {{10}^{3}}\text{gc}{{\text{m}}^{-3}}$ or $100\text{ gc}{{\text{m}}^{-3}}$.
The critical volume will be $\dfrac{\text{critical mass}}{\text{critical density}}$ or $\dfrac{39}{100}\text{ c}{{\text{m}}^{3}}\text{.mo}{{\text{l}}^{-1}}$.
The critical volume is $0.39\text{ c}{{\text{m}}^{3}}\text{.mo}{{\text{l}}^{-1}}$.
Step (4)- To convert cubic centimetres to litre, the conversion factor is $1\text{ c}{{\text{m}}^{3}}=0.001\text{ L}$.
The critical volume of the gas is $0.001\times 0.39$ or 0.00039 $\text{L}\text{.mo}{{\text{l}}^{-1}}$.

The correct option is ‘e’.

Note:
Critical density is an astronomical term also. If the density of the universe becomes less than the critical density, then, the universe will expand forever as there will not be enough matter to stop expansion. It is the value at which the universe is balanced. Don’t forget to convert units of critical volume from cubic centimetres to litres as we have to find the volume in litres.