Problem 6 - Advanced Problems in Physical Chemistry for Competitive Examinations

Question

The work involved \((w)\) in an isothermal expansion of \(n\) moles of an ideal gas from an initial pressure of ' \(P\) ' atm to final pressure of 1 atm in number of steps such that in every step the constant external pressure exactly 1 atm less than the initial pressure of gas is maintained, is given as (a) \(-n R T \sum_{i=1}^{i=P-1}\left(\frac{1}{P+1-i}\right)\) (b) \(-n R T \sum_{i=1}^{i=P}\left(\frac{1}{P+1-i}\right)\) (c) \(-n R T \sum_{i=1}^{i=P}\left(\frac{i}{P+1-i}\right)\) (d) \(-n R T \sum_{i=1}^{i=P-1}\left(\frac{i}{P+1-i}\right)\)

Step-by-Step Solution

Answer

The correct answer is (b) \-n R T \sum_{i=1}^{i=P}\left(\frac{1}{P+1-i}\right)\

Step 1: Understand the Process and Use of Formula in Isothermal Expansion

In an isothermal expansion of an ideal gas, the temperature remains constant. According to the first law of thermodynamics, the work done by the gas is the area under the pressure-volume curve for the process. Here, the expansion happens in such a way that the pressure decreases by 1 atm after every step until it reaches 1 atm. The work done at each step is equal to the external pressure times the change in volume. The total work done is the sum of the work done in each step.

Step 2: Analyze the Options

Each option represents a summation with varying indices and denominators. To determine the correct formula, we must consider that with each step, the pressure difference is 1 atm, and initially, the gas is at 'P' atm. In an isothermal process, work done is given by -nRTln(Vf/Vi), but since exact values of volume are not known, we seek a formula that correctly sums the incremental work done, having a decreasing pressure in the denominator.

Step 3: Identify the Correct Expression

In the given options, only option (b) is the correct expression for the work done during each step because it starts from i=1 to i=P. Since the external pressure is always 1 atm less than the initial pressure, it becomes P, P-1, P-2, ..., until the pressure becomes 1 atm. So, it matches our scenario. Option (b) correctly accounts for each step of the isothermal expansion against a decreasing external pressure.

Key Concepts

Thermodynamics

Thermodynamics is a branch of physics that deals with the relationships between heat and other forms of energy. In particular, it examines how energy transformations impact the physical properties of a system. Most thermodynamic processes are governed by the laws of thermodynamics, which are universal principles that apply to systems in thermal equilibrium.

In the context of gases and isothermal expansion, thermodynamics primarily focuses on the first law, which is a version of the law of conservation of energy. The first law states that the energy within an isolated system is constant. When a gas undergoes an isothermal expansion, the system's internal energy remains unchanged because the temperature is constant. Instead, any energy input or work done on the system results in work done by the system as the gas expands. This concept is central to understanding how work is calculated in isothermal processes for gases.

Ideal Gas Law

The ideal gas law is a fundamental equation in thermodynamics that describes the behavior of an ideal gas. It is expressed as

\( PV = nRT \),

where

• \( P \) represents the pressure of the gas,
• \( V \) is the volume,
• \( n \) is the number of moles,
• \( R \) is the ideal gas constant, and
• \( T \) is the absolute temperature.

In an isothermal process, the temperature (\( T \)) remains constant. Thus, if the volume (\( V \)) increases during the expansion, the pressure (\( P \)) of the gas must decrease to maintain the equation's balance. This relationship is used to calculate the work done by a gas when it expands isothermally, as described in the original exercise.

Work Done by Gas

During an isothermal expansion, a gas does work on its surroundings. The work (\( w \)) done by a gas in a thermodynamic process can be represented by the area under the pressure-volume curve. Work is expressed by the formula

\( w = -nRT \ln\left(\frac{V_f}{V_i}\right) \),

where \( V_i \) and \( V_f \) are the initial and final volumes, respectively. The negative sign indicates that the work is done by the system on the surroundings, leading to a loss in the internal energy of the system if additional heat is not supplied.

In our specific scenario, the work done is calculated incrementally, since the gas expands in steps, with each step featuring a pressure reduction of exactly 1 atm. The total work is the sum of work done at each step, which is determined by the constant external pressure and the resultant change in volume after each step.

Pressure-Volume Curve

A pressure-volume (PV) curve is a graphical representation of the relationship between the pressure and volume of a system during a thermodynamic process. For an isothermal expansion of an ideal gas, the PV curve is a hyperbola since the product of pressure and volume remains constant (if the temperature stays the same).

The work done by the gas during expansion can be visually understood by examining the area under the curve between two volumes on the PV diagram. The greater the area under the curve, the more work is done by the gas as it expands. In a step-wise isothermal expansion process, the shape of the PV curve changes slightly with each step due to the pressure reduction, and each segment under the step curve corresponds to the work done during that stage.