Question

A general form of equation of state for gases is \(\mathrm{PV}=\) \(\mathrm{RT}\left[\mathrm{A}+\mathrm{B} / \mathrm{V}+\mathrm{C} / \mathrm{V}^{2}+\ldots\right]\), where \(\mathrm{V}\) is the molar volume of the gas and \(\mathrm{A}, \mathrm{B}, \mathrm{C} \ldots \ldots\) are constants for the gas. The values of \(\mathrm{A}\) and \(\mathrm{B}\), if the gas obeys van der Waals equation, are respectively: (a) \(1, \frac{a}{27 b}\) (b) \(1, \mathrm{~b}-\frac{\mathrm{a}}{\mathrm{RT}}\) (c) \(\frac{a}{\mathrm{RT}}, \mathrm{b}\) (d) \(\frac{2 a}{R T}, b\)

Short answer

The correct option is (b).

Step-by-step solution

Identify the van der Waals Equation

The van der Waals equation is given as \( \left( P + \frac{a}{V_m^2} \right)(V_m - b) = RT \), where \( a \) and \( b \) are constants, \( V_m \) is the molar volume, \( P \) is the pressure, and \( R \) is the gas constant.

Expand the van der Waals Equation

Expand the expression on the left side of the van der Waals equation: \( PV_m + \frac{a}{V_m} - Pb - \frac{ab}{V_m^2} = RT \). This is done by distributing the terms in the parentheses.

Compare with General Equation of State

The general form is \( PV = RT \left[ A + \frac{B}{V} + \frac{C}{V^2} + \ldots \right] \). By comparing, we need to identify \( A \) and \( B \) from \( A = 1 \) and \( B/V = \frac{a}{V_m} - \frac{b}{V_m} \) in the van der Waals expanded form.

Determine Values of Constants A and B

From the comparison with the general equation, we have \( A = 1 \). The coefficient of \( \frac{1}{V} \) is \( 0 = \frac{a}{RT} \). Thus, \( B = b \). The coefficient of \( 1/V^2 \) is \( \frac{a}{RT} = 0 \). So, \( C = \frac{a}{RT} \).

Select the Correct Option

From the determined values, \( A = 1 \) and \( B = b - \frac{a}{RT} \). Therefore, the correct option corresponds to \( A = 1 \) and \( B = b - \frac{a}{RT} \), which is option (b).

Key concepts

General Equation of State for Gases

Understanding the behavior of gases involves identifying how pressure, volume, and temperature interact. The general equation of state for gases provides a model to predict this interaction: \[PV = RT \left[ A + \frac{B}{V} + \frac{C}{V^2} + \ldots \right]\]Here,

• \(P\) denotes pressure.
• \(V\) is the volume of gas.
• \(R\) is the ideal gas constant.

The expression allows us to account for real gas behavior by adjusting the constants \(A\), \(B\), and \(C\). Unlike the ideal gas law, which assumes no interaction between gas molecules, this form includes adjustments for real-world interactions. This representation helps us understand how actual gases deviate from ideal behavior.

Molar Volume

Molar volume is a vital concept in understanding gas behavior. It refers to the volume occupied by one mole of a substance (gas, in this case) under defined conditions of temperature and pressure. In the context of gases,

• \(V_m\) represents the molar volume.
• It is typically measured in liters per mole (L/mol).

Molar volume helps identify the space individual gas molecules need, which becomes particularly important when dealing with non-ideal gases. For instance, the van der Waals equation modifies the ideal gas law to account for the actual size of gas molecules, represented by \(V_m\), and intermolecular forces that influence a gas's volume.

Gas Constants

Gas constants are crucial for characterizing the properties of gases under different conditions. In equations of state, they help relate various thermodynamic properties such as pressure, volume, and temperature:

• \(R\) is the universal gas constant, often used in the context of ideal gases.
• For real gases, specific constants such as \(a\) and \(b\) in the van der Waals equation are used to correct for molecular interactions and volume.

The goal with these constants is to refine the model of gas behavior so it aligns closer with real observations. While \(R\) remains constant, values for \(a\) and \(b\) may change based on the gas in question. This approach allows for more accurate modeling of real-world gas dynamics.

Comparison of Equations

Each equation of state for gases serves a specific need and reflects different approximations or assumptions:

• Ideal Gas Law: \(PV=nRT\)
  This is the simplest model, best suited for gases at high temperatures and low pressures.  It's valuable due to its simplicity but lacks the ability to predict real-gas behavior outside these conditions.
• Van der Waals Equation: \(\left(P + \frac{a}{V_m^2}\right)(V_m - b) = RT\)
  This adds terms \(\frac{a}{V_m^2}\) and \(-b\) to account for intermolecular forces and molecular volume. It's used when real gases deviate from ideality, providing more accurate reflections of pressure and volume changes.

By comparing these, we see how adjustments in van der Waals equation improve our predictions about gas behavior. It sheds light on the balance of attractive forces and actual molecular volumes that aren't captured in the ideal gas law.