P-T diagram of an ideal gas having three different densities $\rho_{1}, \rho_{2}, \rho_{3}$ (in three different cases) is shown in the figure. Which of the following is correct :

(D) $\rho_{1}=\rho_{2}=\rho_{3}$

JEE Main · 2024 · 04 Apr 2024 · Shift 1 · Heat & Thermodynamics

P-T diagram of an ideal gas having three different densities

ρ_{1}, ρ_{2}, ρ_{3}

(in three different cases) is shown in the figure. Which of the following is correct :

(A) $ρ_{2} < ρ_{3}$
(B) $ρ_{1} > ρ_{2}$ ✓ Correct
(C) $ρ_{1} < ρ_{2}$
(D) $ρ_{1} = ρ_{2} = ρ_{3}$

Answer: (B)

Solution

For an ideal gas,

P V = \frac{m}{M} R T

, so

P = \frac{ρR}{M} T .

On a

P

–

T

diagram each density is a straight line through the origin with slope

\frac{ρR}{M}

; a steeper line means larger density. At any fixed temperature, the curve with the greater pressure has the greater density, so from the figure

ρ_{1} > ρ_{2} > ρ_{3} .

In particular

ρ_{1} > ρ_{2}

\Rightarrow

option B.

JEE Main · 2024 · 04 Apr 2024 · Shift 1 · Single correct option

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