Problem 124 - A Complete Resource Book in Chemistry for JEE Main

Question

If a real gas follows equation \(\mathrm{P}(\mathrm{V}-\mathrm{nb})=\mathrm{RT}\) at low pressure, then for a graph between \(\mathrm{d} / \mathrm{P}\) vs. \(\mathrm{P}\), (where \(\mathrm{d}\) is the density of gas) (a) Intercept is \(\frac{\mathrm{MR}}{\mathrm{T}}\) (b) Intercept is \(\frac{\mathrm{M}}{\mathrm{RT}}\) (c) Slope is \(-\frac{b}{M(R T)^{2}}\) (d) Slope is \(-\frac{\mathrm{Mb}}{(\mathrm{RT})^{2}}\)

Step-by-Step Solution

Answer

The answer is (b) Intercept is \(\frac{M}{RT}\).

Step 1: Write the given equation

The given equation is for a real gas:\[ \mathrm{P} (\mathrm{V} - \mathrm{nb}) = \mathrm{RT} \]where \( \mathrm{P} \) is the pressure, \( \mathrm{V} \) is the volume, \( \mathrm{n} \) is the number of moles, \( \mathrm{b} \) is a constant, \( \mathrm{R} \) is the gas constant, and \( \mathrm{T} \) is the temperature.

Step 2: Express Pressure in terms of volume and density

First, recall that the density \( \mathrm{d} \) is defined as mass \( m \) per unit volume \( \mathrm{V} \), so \( \mathrm{V} = \frac{M}{\mathrm{d}} \), where \( M \) is the molar mass: \[ V = n \cdot \frac{M}{d} \]Substitute the expression for \( V \) from density:\[ \mathrm{P} \left( \frac{nM}{\mathrm{d}} - \mathrm{nb} \right) = \mathrm{RT} \]

Step 3: Solve for Pressure equation

Rearrange the equation to express pressure \( \mathrm{P} \):\[ \mathrm{P} \left( \frac{Mn}{\mathrm{d}} - \mathrm{nb} \right) = \mathrm{RT} \]which gives:\[ \mathrm{P} \cdot \frac{Mn}{\mathrm{d}} = \mathrm{RT} + \mathrm{P} \cdot nb \]

Step 4: Complete the derivation for density-pressure relationship

We rearrange the equation further:\[ \mathrm{P} \cdot \frac{Mn}{\mathrm{d}} + \mathrm{P} \cdot nb = \mathrm{RT} \]and solve for density \( \mathrm{d} \):\[ \frac{1}{\mathrm{P}} = \frac{Mn}{\mathrm{RT}} - \frac{nMb}{RT} \]

Step 5: Interpretation for graph (1/d) vs. P

An equation of a linear graph is generally \( y = mx + c \). Here, we rearrange:\[ \frac{1}{\mathrm{d}} = \frac{Mn}{\mathrm{RT}}\frac{1}{\mathrm{P}} - \frac{bMn}{RT} \]\[ \frac{1}{\mathrm{d}} = \left( \frac{-bMn}{RT} \right)\mathrm{P} + \left( \frac{Mn}{RT} \right) \]Comparing with \( y = mx + c \), \( m = -\frac{bM}{(RT)^2} \), \( c = \frac{M}{RT} \).

Key Concepts

Density-Pressure Relationship

Understanding the relationship between density and pressure in gases is crucial, especially for real gases. This relationship can vary based on how closely a gas follows the ideal gas law versus a real gas behavior. For gases, density \(d\) is defined as the mass \(m\) per unit volume \(V\), i.e., \(d = \frac{m}{V}\). In the context of gases, where mass can also be expressed in terms of molar mass \(M\) and number of moles \(n\), the equation for density can be adjusted based on the pressure the gas is under.
To analyze the relationship between density and pressure graphically, it's enlightening to plot them against each other, such as a 1/d vs. P plot as seen in this context. Graphs of density versus pressure for gases primarily reflect changes in volume at given pressure and temperature conditions, illustrating how density inversely correlates with pressure in an ideal scenario. However, the slope and intercept in the real gas equation graphically help to assess the deviation of real gases from ideal behavior at varying pressures.

Density Equation for Gases

The density equation for gases connects the density (\(d\)), pressure (\(P\)), and other parameters like molar mass (\(M\)), temperature (\(T\)), and the ideal gas constant (\(R\)). For real gases, the standard formula \(PV = nRT\) has to be modified to better describe the gas behavior.
A hallmark adjustment includes factors such as the volume correction term \(nb\). For a real gas, the equation becomes \(P(V-nb) = RT\). Substituting \(V\) with its expression in terms of density (\( \frac{Mn}{d}\)) allows one to explore how density impacts the gas equation. Transforming this into a more useful form for plotting, 1/d vs. P yields insight into the behavior under different conditions.
Additionally, this transformed equation's intercept and slope provide intrinsic properties of the gas, such as molar volume and deviations indicating molecular interactions, respectively. Hence, understanding these relationships is key to interpreting physical characteristics of gases under various conditions.

Real Gas Behavior

Real gases deviate from the ideal gas law primarily due to interactions between particles and the volume occupied by gas molecules themselves. While the ideal gas law assumes no intermolecular interactions and point particles, real gases have slight attractions and volume occupation that must be accounted for. Concepts like the Van der Waals equation are included for these adjustments.
In the problem at hand, the equation \(P(V-nb) = RT\) reflects these deviations by incorporating a constant \(b\) that corrects the volume. This adjustment helps compensate for the non-ideal behavior by accounting for the actual volume occupied by gas molecules.
Exploring how \(b\) contributes to changes in slope \(-\frac{bM}{(RT)^2}\) in a density-pressure plot gives insight into the extent of deviation from the ideal gas model. It helps quantify the real-world complexity by which gases behave, providing a more accurate understanding for practical applications such as in engineering calculations.