Problem 52 - A Complete Resource Book in Chemistry for JEE Main

Question

1 mole of an ideal gas at \(300 \mathrm{~K}\) is expanded isothermally and reversible from an initial volume of 1 litre to 10 litre. The work in this process is \(\left(\mathrm{R}=2 \mathrm{cal} \mathrm{mol}^{-1} \mathrm{~K}^{-1}\right)\) (a) \(163.7 \mathrm{cal}\) (b) zero (c) \(-1381.8 \mathrm{cal}\) (d) 9 litreatm

Step-by-Step Solution

Answer

The work done is \(-1381.8 \text{ cal}\), matching option (c).

Step 1: Identify the formula for isothermal work in an ideal gas

For an isothermal and reversible process, the work done by an ideal gas is given by the formula \( W = -nRT \ln \left( \frac{V_f}{V_i} \right) \), where \( n \) is the number of moles, \( R \) is the gas constant, \( T \) is the temperature, \( V_f \) is the final volume, and \( V_i \) is the initial volume.

Step 2: Insert known quantities

In this exercise, we are given \( n = 1 \) mole, \( R = 2 \text{ cal mol}^{-1} \text{K}^{-1} \), \( T = 300 \text{ K} \), \( V_i = 1 \text{ litre} \), and \( V_f = 10 \text{ litre} \). Substitute these values into the formula for work.

Step 3: Calculate the natural logarithm term

Compute \( \ln \left( \frac{10}{1} \right) = \ln(10) \). The value of \( \ln(10) \) is approximately 2.302.

Step 4: Compute the work done during the expansion

Plug the computed logarithm back into the formula: \[ W = -1 \times 2 \times 300 \times 2.302 \]. This simplifies to \[ W = -1381.2 \text{ cal} \].

Step 5: Compare your result with the given options

The calculated work is approximately \(-1381.2 \text{ cal}\). The option closest to this value is \( -1381.8 \text{ cal} \).

Key Concepts

Ideal Gas

An ideal gas is a theoretical gas composed of many randomly moving point particles that interact only through elastic collisions. In real-world conditions, gases do not exactly follow these assumptions, but the ideal gas model is a good approximation for many gases under a variety of conditions. The idea is based on the "Ideal Gas Law," which combines various properties of gases into a single equation:

• **Pressure (P):** The force exerted by gas particles hitting the walls of the container.
• **Volume (V):** The space occupied by the gas.
• **Temperature (T):** A measure of the average kinetic energy of the gas particles. Always in Kelvin for the equation.
• **Amount of gas (n):** Often measured in moles.

The equation is given by: \[ PV = nRT \] where \( R \) is the gas constant. This formula provides the relationship among pressure, volume, temperature, and moles, making it easier to analyze processes like isothermal expansion.

Work Calculation

In thermodynamics, calculating the work done by or on a gas is key to understanding energy transfer. During an isothermal expansion, where temperature remains constant, the work done by the gas can be calculated using the formula: \[ W = -nRT \ln \left( \frac{V_f}{V_i} \right) \]

• **\( n \):** Number of moles of the gas. In our exercise, it is 1 mole.
• **\( R \):** Gas constant, given as \( 2 \text{ cal mol}^{-1} \text{K}^{-1} \).
• **\( T \):** Temperature in Kelvin, constant at 300 K in the exercise.
• **\( V_i \)** and **\( V_f \):** Initial and final volumes, 1 litre and 10 litres respectively.

The negative sign in the formula indicates that work is done by the gas during expansion. Plugging in the numbers from the exercise showed a calculated work of \(-1381.2 \text{ cal}\), reflecting energy given off by the gas as it expands.

Reversible Process

A reversible process in thermodynamics is an idealization where the process can be reversed without leaving any trace or effect on the surrounding environment. It maximizes work done in thermodynamic processes, as opposed to irreversible processes which lose some energy to characteristics like friction or turbulence.

• **Key Characteristics:** Very slow changes ensuring equilibrium at all stages, allowing systems to be turned back without any net flow of energy or entropy changes.
• **Practicality:** Real processes are never truly reversible, but this concept is useful for understanding the limits of efficiency and work output.
• **In Our Exercise:** The gas expansion was isothermal and considered reversible. This allowed for the precise application of the work formula and demonstrated maximum efficiency.

Reversible processes are theoretical, but they provide crucial insights into the optimal conditions for energy transformations.

Natural Logarithm

A natural logarithm is a logarithm to the base \(e\), where \(e\) is an irrational and transcendental constant approximately equal to 2.71828. It is denoted as \(\ln\) and is a fundamental concept in mathematics, used extensively for calculations in physics and engineering.

• **Usage:** In the context of the ideal gas work formula, \( \ln \left( \frac{V_f}{V_i} \right) \) calculates the change in volume for isothermal processes.
• **Properties:** The natural logarithm has unique properties such as \( \ln(1) = 0 \) and \( \ln(xy) = \ln(x) + \ln(y) \).
• **In Our Exercise:** We used \( \ln(10) \approx 2.302 \) to determine the work done. This value reflects the ratio of final to initial volume (\( \frac{10}{1} \)).

Understanding how to apply natural logarithms can simplify calculations and provide a clear path to solving problems in thermodynamics effectively.