Question

Select the correct formula:
(where k=Boltzmann’s constant, R=gas constant, n=moles, r=density, M=molecular weight, p=pressure, T=kelvin temperature, V=volume)
a). \[k=R{{N}_{av}}\]
b). \[r=\dfrac{nM}{V}\]
c). \[\dfrac{p}{r}=\dfrac{RT}{M}\]
d). \[R=k{{N}_{av}}\]

A). a, b, c
B). a, b, d
C). b, c, d
D). a, c, d

Answer 1

Hint: To find the correct answer we need to know the correct formula involving each physical quantity. Here we require the relation between Boltzmann constant and the universal gas constant, formula for density in terms of mass and volume and the ideal gas equation.

Formula used: Relation between Boltzmann constant and the universal gas constant (ideal gas constant) is:
\[R=k{{N}_{av}}\]
The formula for density is:
\[\text{Density=}\dfrac{\text{Mass}}{\text{Volume}}\]
And the ideal gas equation is:
 \[PV=nRT\]

Complete step by step answer:
We will check the validity for each equation given in the question-

For a). \[k=R{{N}_{av}}\] we already know that the relation between Boltzmann constant and the universal gas constant (ideal gas constant) is:
\[R=k{{N}_{av}}\] therefore, the option a). is incorrect but option d). is correct. Here, \[{{N}_{av}}\] stands for Avogadro’s constant.

For b). \[r=\dfrac{nM}{V}\] here r is density, n is moles, M is the molecular weight and V is the volume. From the relation of density with mass and volume,
 \[\text{Density=}\dfrac{\text{Mass}}{\text{Volume}}\]
It satisfies the above equation for n moles. Hence, option b). is correct.
For c). \[\dfrac{p}{r}=\dfrac{RT}{M}\]
We know that the ideal gas equation is:
\[PV=nRT\]
And \[r=\dfrac{M}{V}\]
\[\Rightarrow V=\dfrac{M}{r}\]
On substituting,
\[\dfrac{PM}{r}=nRT\]
\[\dfrac{P}{r}=\dfrac{nRT}{M}\]
For \[n=1\] it satisfies the above equation so option c). is correct.
Hence, the correct answer is option C. b, c, d

Note: Students must note that Boltzmann constant is \[k=1.38\times {{10}^{-23}}\] and it relates the average kinetic energy of a particle of a gas with the temperature of the gas. It has the same SI unit as Entropy i.e. \[J{{K}^{-1}}\].