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:

Question

Given $C_p = a + bT$
$a = 19.5$, $b = 0.042$, $n = 1$ mole
Temperature changes from $27^\circ C$ to $327^\circ C$.
Find $\Delta H$.

Answer 1

Ideal Gas Law and Density Relationship:

The ideal gas equation is stated as:

$ PV = nRT $

This can be rewritten as:

$ P = \frac{nRT}{V} $

Here, $ P $ represents pressure, $ T $ is temperature, $ R $ is the gas constant, $ n $ denotes the number of moles, and $ V $ signifies volume.

To express $ P $ in terms of density $ \rho $, we substitute $ \rho = \frac{m}{V} $, where $ m $ is the mass of the gas:

$ P = \frac{\rho RT}{M} $

In this equation, $ M $ is the molar mass of the gas. Rearranging this formula yields:

$ \rho = \frac{PM}{RT} $

Analysis of PT Graph for Varying Densities:

Given the relationship $ \rho = \frac{PM}{RT} $, at a constant temperature $ T $, the density $ \rho $ is directly proportional to the pressure $ P $:

$ \rho \propto P $

Consequently, for a fixed temperature, higher pressure implies higher density.

Interpretation of the PT Diagram:

Observing the provided PT diagram, we note:

$ P_1 > P_2 > P_3 $ at the same temperature $ T $

Based on the direct proportionality $ \rho \propto P $ at constant temperature, we deduce:

$ \rho_1 > \rho_2 > \rho_3 $

Conclusion:

The accurate statement is $ \rho_1 > \rho_2 $, which aligns with Option (2).

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:

The Correct Option is B

Solution and Explanation

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