Question

Which of the following expressions is/are correct for an adiabatic process? (a) \(\frac{\mathrm{P}_{2}}{\mathrm{P}_{1}}=\left(\frac{\mathrm{T}_{1}}{\mathrm{~T}_{2}}\right)^{\gamma-1 / \gamma}\) (b) \(\frac{\mathrm{T}_{2}}{\mathrm{~T}_{1}}=\left(\frac{\mathrm{V}_{1}}{\mathrm{~V}_{2}}\right)^{\gamma-1}\) (c) \(\mathrm{P}_{1} \mathrm{~V}_{1}^{\gamma-1}=\mathrm{P}_{2} \mathrm{~V}_{2}^{\gamma-1}\) (d) \(\mathrm{P}_{2} \mathrm{~V}_{2}^{\gamma}=\mathrm{P}_{1} \mathrm{~V}_{1}^{\gamma}\)

Short answer

Expressions (b) and (d) are correct for an adiabatic process.

Step-by-step solution

Understanding Adiabatic Processes

In an adiabatic process, no heat is transferred into or out of the system. The process can be represented using the polytropic process equation, which in this case simplifies to \( PV^\gamma = \text{constant} \), where \( \gamma \) is the heat capacity ratio \( \frac{C_p}{C_v} \). This is the key equation to determine the correct expressions.

Analyzing Expression (a)

Expression \( (a) \) is \( \frac{\mathrm{P}_{2}}{\mathrm{P}_{1}}=\left(\frac{\mathrm{T}_{1}}{\mathrm{~T}_{2}}\right)^{\gamma-1 / \gamma} \). Use the adiabatic relation \( PV = nRT \), which can be rearranged using \( T = \) constant for ratio forms. After simplification, this expression doesn't match a known correct form for adiabatic processes, so it is incorrect.

Analyzing Expression (b)

Expression \( (b) \) is \( \frac{\mathrm{T}_{2}}{\mathrm{~T}_{1}}=\left(\frac{\mathrm{V}_{1}}{\mathrm{~V}_{2}}\right)^{\gamma-1} \). This matches the correct adiabatic relation derived from \( TV^{\gamma-1} = \text{constant} \). Thus, this expression correctly represents an adiabatic process.

Analyzing Expression (c)

Expression \( (c) \) is \( \mathrm{P}_{1} \mathrm{~V}_{1}^{\gamma-1}=\mathrm{P}_{2} \mathrm{~V}_{2}^{\gamma-1} \). This representation doesn't follow from \( PV^\gamma = \text{constant} \), suggesting that it is not a correct formulation of the adiabatic process.

Analyzing Expression (d)

Expression \( (d) \) is \( \mathrm{P}_{2} \mathrm{~V}_{2}^{\gamma}=\mathrm{P}_{1} \mathrm{~V}_{1}^{\gamma} \). This expression directly follows from the adiabatic process equation \( PV^\gamma = \text{constant} \), hence it is correct.

Key concepts

Polytropic Process Equation

In thermodynamics, the polytropic process equation provides a way to understand diverse processes which are not purely adiabatic or isothermal. For any polytropic process, the equation is given by: \[ PV^n = ext{constant} \] Here, *P* stands for the pressure, *V* for volume, and *n* is the polytropic index, which can vary depending on the nature of the process. This equation becomes very adjusted when it comes to an adiabatic process. In this special case, the value of the polytropic index becomes \( \)- the heat capacity ratio. In our study of adiabatic processes, we focus on the specific form of this equation: \[ PV^\gamma = ext{constant} \] This form signifies that during an adiabatic process, the product of pressure and volume raised to the power of the heat capacity ratio remains unchanged.

Heat Capacity Ratio

The heat capacity ratio, denoted as \( \gamma \), plays a crucial role in the study of thermodynamics, especially when dealing with adiabatic processes. The formula for calculating the heat capacity ratio is: \[ \gamma = \frac{C_p}{C_v} \] Here, \( C_p \) is the heat capacity at constant pressure, while \( C_v \) is the heat capacity at constant volume. The heat capacity ratio represents how compressible a material is under adiabatic conditions. For ideal gases: - A monatomic gas usually has a \( \gamma \) value of about 1.67. - Diatomic gases like nitrogen or oxygen tend to have around 1.40. This ratio essentially determines how the pressure, volume, and temperature of a gas change with each other during an adiabatic process.

Adiabatic Relation

In an adiabatic process, the system experiences no heat exchange with its surroundings. This key property results in unique behavior in terms of pressure, volume, and temperature changes, described by adiabatic relations. The main adiabatic relation is: \[ PV^\gamma = ext{constant} \] This equation describes the change in pressure and volume in such processes. When a gas undergoes compression or expansion without heat transfer, this relation holds. Another useful relation involves temperature and volume: \[ TV^{\gamma-1} = ext{constant} \] This highlights how temperature varies with changes in volume when no heat is added or removed. Together, these equations enable us to analyze the behavior of gases under adiabatic conditions effectively, allowing us to determine correctly formulated expressions, like in exercise (b) and (d), where one used temperature and the other used pressure-volume relationships.