MHT CET · Physics · Thermodynamics

An ideal gas has pressure $P$ , volume $V$ and absolute temperature $T$ . If $m$ is the mass of each molecule and $K$ is the Boltzmann constant then density of the gas is

A $\frac{P m}{K T}$
B $\frac{K T}{P m}$
C $\frac{K m}{P T}$
D $\frac{P K}{T m}$

Verified Solution

Answer & Solution

Correct Answer

(A) $\frac{P m}{K T}$

Step-by-step Solution

Detailed explanation

For ideal gas

P V = n R T

P V = \frac{m^{'}}{M} R T = \frac{m^{'}}{V} . \frac{R T}{M}

(where m' is the mass of the gas and M molecular weight)

∴ P = \frac{ρ R T}{M}

(Where

ρ

is the density of the gas)

∴ ρ = \frac{P M}{R T} = \frac{P M}{N K T}

(Where N is Avogadro number)

∴ ρ = \frac{P m}{K T}

(Where

m = \frac{M}{N} =

mass of each molecule)

Apply the relevant identity from the chapter and substitute the values established in the previous step, keeping the original constraint in view.

Use the simplified expression to isolate the unknown, then plug the result back into the question setup to confirm the working is internally consistent.

Cross-check against a boundary or limiting case. Both the dimensional balance and the qualitative behaviour at the extreme should agree with the candidate answer.

Combine the steps above to identify the correct option. The remaining choices each fail at least one of the consistency, dimensional, or boundary checks.

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An ideal gas has pressure P, volume V and absolute temperature T. If m is the mass of each molecule and K is the Boltzmann constant then density of the gas is

Step-by-step Solution

An ideal gas has pressure $P$ , volume $V$ and absolute temperature $T$ . If $m$ is the mass of each molecule and $K$ is the Boltzmann constant then density of the gas is