Gas Law and Polytropic Processes

The question describes a fundamental gas law, $\text{pV}^\text{n} = \text{C}$, which is a generalized expression for various thermodynamic processes experienced by gases. This equation is known as the polytropic process equation, where:

• $\text{p}$ represents the pressure of the gas.
• $\text{V}$ represents the volume of the gas.
• $\text{n}$ is the polytropic index, a constant that varies for different processes.
• $\text{C}$ is a constant.

The value of $\text{n}$ determines the specific type of thermodynamic process the gas undergoes. Let's explore what happens when the value of $\text{n}$ is equal to 1.

Process Identification when n = 1

When the polytropic index $\text{n}$ is equal to 1, the general gas law $\text{pV}^\text{n} = \text{C}$ simplifies significantly. Substituting $\text{n} = 1$ into the equation, we get:

$\text{pV}^1 = \text{C}$

Which further simplifies to:

$\text{pV} = \text{C}$

This simplified equation, $\text{pV} = \text{C}$ (where C is a constant), is the mathematical representation of Boyle's Law. Boyle's Law states that for a fixed amount of gas at constant temperature, the pressure and volume are inversely proportional. Therefore, a process where $\text{pV}$ remains constant implies that the temperature of the gas also remains constant.

Isothermal Process Explanation

A thermodynamic process during which the temperature of the system remains constant is called an isothermal process. In such a process, any heat added to or removed from the system is precisely balanced by the work done by or on the system, ensuring that the internal energy and thus the temperature do not change.

Let's consider why the other options are not correct for $\text{n} = 1$:

• Adiabatic Process: An adiabatic process is one where no heat transfer occurs between the system and its surroundings. For an adiabatic process, the polytropic index $\text{n}$ is equal to $\gamma$ (gamma), which is the heat capacity ratio ($\text{C}_\text{p}/\text{C}_\text{v}$) for the gas. The value of $\gamma$ is typically around 1.4 for diatomic gases like air, not 1. So, $\text{pV}^\gamma = \text{C}$ represents an adiabatic process.
• Isentropic Process: An isentropic process is a reversible adiabatic process. Since it's also adiabatic, the condition $\text{n} = \gamma$ applies. Therefore, $\text{n} = 1$ does not correspond to an isentropic process.
• Polytropic Process: The equation $\text{pV}^\text{n} = \text{C}$ itself defines a polytropic process. However, the question asks for the specific name of the process when $\text{n} = 1$. While it is a type of polytropic process, "isothermal" is the more specific and accurate name for the case where $\text{n} = 1$. The polytropic process is a general category that includes isothermal, adiabatic, isobaric ($\text{n}=0$), and isochoric ($\text{n}=\infty$) processes as special cases.

Here's a summary table of common thermodynamic processes and their corresponding 'n' values:

Based on the analysis, when $\text{n} = 1$ in the gas law $\text{pV}^\text{n} = \text{C}$, the process is specifically known as an isothermal process because the temperature remains constant.