Question

Which of the following gas properties is needed to calculate the work done by an expanding gas? L. The initial and final pressures II. The initial and final volumes III. The path followed during the expansion. A. I only B. II only C. I and II only D. I, II, and III

Short answer

D. I, II, and III

Step-by-step solution

Understand the Concept of Work Done by an Expanding Gas

The work done by an expanding gas in a thermodynamic process is given by the integral \( W = \int_{V_i}^{V_f} P dV \), where \( P \) is the pressure and \( V \) is the volume. This means the work done depends on the pressure, volume, and the specific path taken during the expansion process.

Evaluate Each Option

1. Initial and final pressures (I): These are necessary to understand the changes in the gas state. 2. Initial and final volumes (II): These determine the extent of expansion or compression. 3. Path followed during the expansion (III): This is important because the work done can vary based on the specific process (e.g., isothermal, adiabatic).

Determine the Correct Combination

Given that the work done depends on the initial and final pressures, the initial and final volumes, and the path followed, all three properties are needed for an accurate calculation.

Select the Correct Answer

The combination of initial and final pressures, initial and final volumes, and the path followed during the expansion is required to calculate the work done by the gas.

Key concepts

thermodynamics

Thermodynamics is the branch of physics that deals with heat and temperature, and their relation to energy and work. It encompasses a set of principles explaining how different forms of energy are interconverted. When we talk about the work done by an expanding gas, we are essentially diving into the first law of thermodynamics, which states that the energy of an isolated system is constant. So, understanding thermodynamics helps us figure out how energy is transferred within a system, such as how and why gases expand and do work.

pressure-volume relationship

The relationship between pressure and volume is fundamental in comprehending how gases behave under different conditions. As per Boyle's Law, pressure and volume are inversely related when the temperature is constant. This relationship can be represented mathematically as \( P_i V_i = P_f V_f \). In the context of an expanding gas, knowing initial and final volumes along with pressures is crucial. These parameters tell us how much the gas has expanded and at what pressures, helping calculate the work done: \[ W = \int_{V_i}^{V_f} P dV \].

gas laws

Gas laws are a collection of rules that describe how gases behave under various conditions of pressure, volume, and temperature. These include laws such as Boyle's Law, Charles's Law, and the Ideal Gas Law. Each law provides a different perspective on the behavior of gases, but they all intersect with the concept of work done by an expanding gas. For example, the Ideal Gas Law \( PV = nRT \) combines pressure, volume, and temperature, offering a comprehensive method for calculating changes in a gas when it expands.

thermodynamic paths

A thermodynamic path refers to the specific route taken from an initial to a final state within a process. This path can be isothermal (constant temperature), adiabatic (no heat exchange), or any other type. The significance of the path comes into play while calculating the work done by an expanding gas. That's because the work done by the gas during expansion depends on the specific process or path followed. For instance, in an isothermal process, the work done can be calculated using the formula \[ W = nRT \ln\left(\frac{V_f}{V_i}\right) \]. So, the path determines the method and the result of the work done calculation.