Question

2 moles of an ideal gas is expanded isothermally and reversibly from 1 litre of 10 litre at \(300 \mathrm{~K}\). The enthalpy change (in \(\mathrm{kJ}\) ) for the process is (a) \(11.4 \mathrm{~kJ}\) (b) \(-11.4 \mathrm{~kJ}\) (c) \(0 \mathrm{~kJ}\) (d) \(4.8 \mathrm{~kJ}\).

Short answer

(c) \(0 \mathrm{~kJ}\)

Step-by-step solution

Understand Key Concepts

An isothermal process occurs at a constant temperature. For an ideal gas undergoing an isothermal reversible expansion, the change in internal energy (\( \Delta U \)) is zero because temperature remains constant.

Apply PV=nRT

For an ideal gas, we use the equation \( PV = nRT \). Since \( T \) and \( n \) are constant, any change in pressure or volume does not affect ideal gas enthalpy.

Calculate Enthalpy Change in Isothermal Process

Enthalpy change (\( \Delta H \)) in an isothermal process of an ideal gas is zero because enthalpy depends on internal energy and work done at constant temperature is countered by heat absorbed.

Conclude Based on Options Provided

Since \( \Delta H = 0 \) for the isothermal expansion of an ideal gas, the correct answer is when the enthalpy change is negligible or zero.

Key concepts

Ideal Gas Laws

The Ideal Gas Laws are fundamental principles used to describe the behavior of ideal gases, which are hypothetical gases that perfectly follow the laws over a wide range of conditions. They allow us to predict how gases will behave under various conditions of pressure, temperature, and volume. The principal equation, known as the Ideal Gas Law, is expressed as:\[ PV = nRT \]Where:

• \( P \) is the pressure of the gas
• \( V \) is the volume
• \( n \) is the number of moles
• \( R \) is the universal gas constant
• \( T \) is the temperature in Kelvin

This equation links together the properties of a gas and allows us to delve into deeper analyses of various processes, such as isothermal expansions where the temperature remains constant. It is vital to understand how they interconnect to predict how a given change in one parameter will affect the others.

Internal Energy

In the context of ideal gases, the internal energy is a crucial concept that refers to the total energy contained within a system due to the kinetic energy of the gas molecules. For an ideal gas, the internal energy (\( U \)) depends solely on the temperature. Therefore, if the temperature remains constant, as in an isothermal process, the internal energy does not change.During an isothermal process, the equation \[\Delta U = 0\]applies. This implies that regardless of any other changes occurring in the system, such as pressure or volume, the internal energy remains unchanged if the temperature is maintained constant. This fundamental truth helps set the foundation for understanding the energy dynamics during processes like isothermal expansions.

Isothermal Expansion

Isothermal expansion is a type of thermodynamic process where a gas expands at a constant temperature. This means that even as the gas expands and its volume increases, the temperature does not change. In an isothermal process for an ideal gas, the heat added to the system is used completely to do the work of expansion. Because of this balance, even though there might be external interventions like heating, the internal energy of the system remains constant.The work done by the gas during isothermal expansion is given by the formula:\[ W = nRT \ln\left(\frac{V_f}{V_i}\right) \]where

• \( V_f \) is the final volume
• \( V_i \) is the initial volume

Understanding this helps us realize how gas systems perform work and transfer energy without changes in temperature, facilitating many practical processes seen in engineering and physics.

Reversible Process

A reversible process in thermodynamics is an idealized process that occurs in such a manner that the system involved and its surrounding can be returned to their original states through infinitesimal changes in a variable. In the context of an isothermal expansion, a reversible process implies that the expansion occurs slowly enough that the system remains in thermal equilibrium through every stage of the process. In practical terms, reversible processes allow us to explore maximum efficiency scenarios because they – theoretically – allow the system to recover all the energy invested into it without losses. For an isothermal and reversible process, the external pressure is adapted incrementally to the system's changes, ensuring that no extra energy is dissipated, and every step can be precisely accounted for in calculations of work and heat transfer. This principle is crucial for understanding thermodynamics in ideal conditions and provides a foundational concept for more complex processes and scientific exploration.