Problem 36 - 43 Years JEE ADVANCED (1978-2020) + JEE MAIN Chapterwise & Topicwise Solved Papers Chemistry

Question

In thermodynamics, the \(P-V\) work done is given by $$ w=-\int d V P_{e x t} $$ For a system undergoing a particular process, the work done is, $$ w=-\int d V\left(\frac{R T}{V-b}-\frac{a}{V^{2}}\right) $$ This equation is applicable to a [Adv. 2020] (a) system that satisfies the van der Waals equation of state. (b) process that is reversible and isothermal. (c) process that is reversible and adiabatic. (d) process that is irreversible and at constant pressure.

Step-by-Step Solution

Answer

(b) process that is reversible and isothermal.

Step 1: Understand the Given Equation

The provided equation for work done is \(w=-\int d V\left(\frac{RT}{V-b}-\frac{a}{V^{2}}\right)\). This equation incorporates two correction terms: one for the volume \((V-b)\) and one for pressure \((a/V^2)\), suggesting that it modifies the ideal gas law.

Step 2: Recognize the Equation Type

The terms \(\frac{RT}{V-b}\) and \(-\frac{a}{V^2}\) in the integral suggest modifications similar to those in the van der Waals equation, which corrects the ideal gas law for intermolecular forces and finite molecular sizes.

Step 3: Identify the Associated Process

This specific equation form with van der Waals corrections is used to describe a reversible process, following the characteristics of the van der Waals equation, which is an approximation for real gas behaviors, typically at constant temperature (isothermal).

Step 4: Choose the Correct Option

Considering the involvement of the van der Waals equation-like terms in the work done expression and typical applications, it is linked to process conditions that are reversible and isothermal.

Key Concepts

Van der Waals equation

The Van der Waals equation is a pivotal concept in thermodynamics, especially when dealing with real gases. This equation modifies the ideal gas law, which assumes no interactions between gas molecules and that their volume is negligible. However, real gases do not strictly follow these assumptions. The Van der Waals equation introduces two correction factors:

• Pressure correction: Accounts for intermolecular forces, represented by the term \(- \frac{a}{V^{2}}\). The constant \(a\) reflects the magnitude of these forces.
• Volume correction: Considers the finite size of molecules, represented by \(V-b\). The constant \(b\) is related to the volume occupied by the gas molecules.

The complete formula is expressed as:\[(P + \frac{a}{V^2})(V - b) = RT\]Here, \(R\) is the universal gas constant and \(T\) is the temperature. This equation effectively models real gas behavior by considering volume exclusion and attractive forces, providing better predictions than the ideal gas law at higher pressures and lower temperatures.

Reversible process

In thermodynamics, a reversible process is one where the system changes state in such a way that the process could reverse itself without leaving any net change in the system and surroundings. This means that each infinitesimal step in the process is in perfect equilibrium. Here are important aspects of a reversible process:

• Reversibility ensures maximum efficiency and work output, as the process proceeds through a sequence of equilibrium states.
• Endless slowness and meticulous control over the process parameters, like pressure and volume changes, are required.
• If reversed, every step would retrace and remove the effects of changes made initially.

In practice, truly reversible processes do not exist because some energy dissipation inevitably occurs. Nonetheless, reversible processes are crucial theoretical models. They serve as benchmarks for engineering systems, providing an upper limit on the performance of real processes.

Isothermal process

An isothermal process is a thermodynamic process where the temperature of the system remains constant throughout. This constant temperature is maintained by allowing heat exchange with the surroundings, balancing any heat absorption or release due to work done by or on the system. Key characteristics of isothermal processes include:

• For an ideal gas, the internal energy change \(\Delta U\) is zero because it only depends on temperature.
• The work done \(w\) in an isothermal process is calculated as \(w = -\int PdV\), integrating over the path on a \(PV\)-diagram, under constant temperature conditions.
• Utilization of isothermal processes highlights real-world processes like melting or boiling, where temperature remains unchanged.

In the context of real gases, as seen with the Van der Waals equation, isothermal processes still assume temperature constancy but must account for intermolecular forces and molecular sizes. Such realistic corrections are necessary for accurate predictions of behavior and work calculations in real gas systems.