Question

The van der Waal's equation for \((1 / 2)\) mole of a gas (a) \(\left(P+\frac{a}{V^{2}}\right)(V-b)=R T\) (b) \(\left(P+\frac{a}{4 V^{2}}\right)\left(V-\frac{b}{2}\right)=\frac{R T}{2}\) (c) \(\left(P+\frac{a}{4 V^{2}}\right)\left(\frac{V-b}{2}\right)=R T\) (d) \(\left(P+\frac{a}{4 V^{2}}\right)\left(\frac{V-b}{2}\right)=2 R T\)

Short answer

The correct Van der Waals equation for (1 / 2) mole of a gas is option (b) \(\left(P + \frac{a}{4 V^{2}}\right)\left(V - \frac{b}{2}\right) = \frac{R T}{2}\).

Step-by-step solution

Understanding Van der Waals Equation for One Mole of Gas

The general form of the Van der Waals equation for one mole of gas is given by \((P + \frac{a}{V^2})(V - b) = RT\), where \(P\) is the pressure, \(V\) is the volume, \(a\) and \(b\) are Van der Waals constants for the gas, \(R\) is the universal gas constant, and \(T\) is the temperature.

Adjusting Van der Waals Equation for Half a Mole of Gas

For half a mole, the volume term would remain the same since it's an extensive property, but the other terms need to be adjusted to account for the change in the number of moles (from 1 mole to 0.5 mole).

Applying Adjustments to the Pressure Term

The pressure correction term \(\frac{a}{V^2}\) depends on the number of moles squared. Since we have half the number of moles, the correction term becomes \(\frac{a}{(2V)^2} = \frac{a}{4V^2}\).

Applying Adjustments to the Volume Term

The volume correction term \(V - b\) is a linear term and should be halved since we have half the number of moles. Thus, it becomes \(\frac{V - b}{2}\).

Applying Adjustments to the Right Side of the Equation

The right side of the equation embodies the ideal gas law \(PV = nRT\) where \(n\) is the number of moles. For half a mole, this term becomes \(\frac{1}{2}RT\).

Combine the Adjusted Terms

After adjusting the terms for half a mole of gas, the Van der Waals equation becomes \(\left(P + \frac{a}{4V^2}\right)\left(\frac{V - b}{2}\right) = \frac{RT}{2}\), which matches option (b).

Key concepts

Physical Chemistry

Physical chemistry is the study of how matter behaves on a molecular and atomic level and how chemical reactions occur. It combines principles of physics and chemistry to understand the physical properties of molecules, the forces that act upon them, and their reactions.
Within this field, the behavior of gases is a fundamental topic, often involving the investigation of how temperature, pressure, and volume affect a gas. Problems like the one in our exercise often appear in physical chemistry, as they involve understanding the interactions between gas particles, which can deviate from the ideal case, often described by the ideal gas law. These deviations are particularly significant in real gases, where intermolecular forces and the finite size of gas molecules are taken into account.

Gas Laws

Gas laws are the scientific laws that describe the behavior of gases, and they include Boyle's Law, Charles's Law, Avogadro's Law, and the Ideal Gas Law. The Ideal Gas Law is a fundamental equation in the study of gases, combining the other three laws and represented as PV=nRT. However, real gases do not always follow the Ideal Gas Law, especially under high pressure and low temperature.
These deviations are accounted for by the Van der Waals equation, by introducing correction factors for gas pressure and volume. Understanding these laws not merely conceptually but also mathematically is key for solving textbook exercises and excelling in competitive examinations that test these principles.

Competitive Examinations Chemistry

Competitive examinations often test a student's grasp of chemistry principles through various question formats, ranging from multiple-choice questions to problem-solving exercises. They require not only an understanding of core concepts but also the ability to apply these concepts to solve complex problems.
Questions on gas laws, including the Van der Waals equation, are common in these examinations. They evaluate a student's proficiency in physical chemistry and often require a thorough comprehension of the equations and the ability to adjust them according to the conditions of the problem, like the mole adjustment shown in the step-by-step solution of our exercise.

Van der Waals Constants

Van der Waals constants 'a' and 'b' are empirically determined values specific to individual gases. The constant 'a' corrects for the attractive forces between gas molecules, and its value varies depending on the strength of these intermolecular forces. The higher the value of 'a', the stronger the attraction between particles.
The constant 'b' accounts for the volume occupied by the gas molecules themselves, effectively reducing the space available for movement. It is related to the size of the molecules; larger molecules have larger 'b' values.
Both constants are crucial when using the Van der Waals equation to predict the behavior of real gases, as they allow us to adjust the Ideal Gas Law to more accurately reflect the non-ideal behavior of real gases, demonstrated by the adjustments in the step-by-step solution.