Problem 920 - JEE Physics

Question

The specific heat at constant pressure and at constant volume for an ideal gas are \(\mathrm{C}_{\mathrm{p}}\) and \(\mathrm{C}_{\mathrm{v}}\) and adiabetic \(\&\) isothermal elasticities are \(E_{\Phi}\) and \(E_{\theta}\) respectively. What is the ratio of \(\mathrm{E}_{\Phi}\) and \(\mathrm{E}_{\theta}\) (A) \(\left(\mathrm{C}_{\mathrm{v}} / \mathrm{C}_{\mathrm{p}}\right)\) (B) \(\left(\mathrm{C}_{\mathrm{p}} / \mathrm{C}_{\mathrm{v}}\right)\) (C) \(\mathrm{C}_{\mathrm{p}} \mathrm{C}_{\mathrm{y}}\) (D) \(\left[1 / C_{p} C_{y}\right]\)

Step-by-Step Solution

Answer

The ratio of adiabatic elasticity (E_F) and isothermal elasticity (E_θ) is given by: \(\frac{E_F}{E_θ} = \frac{C_p}{C_v}\) Hence, the correct answer is (B) \(\left(\mathrm{C}_{\mathrm{p}} / \mathrm{C}_{\mathrm{v}}\right)\).

Step 1: 1. Recall the relationship between Cp, Cv, and the adiabatic index

We know that the adiabatic index, denoted by γ (gamma), is given by the ratio of specific heat capacities: \(γ = \frac{C_p}{C_v}\)

Step 2: 2. Recall the expression for adiabatic elasticity (E_F)

The adiabatic elasticity, E_F, is related to the adiabatic index as follows: \(E_F = γ\)

Step 3: 3. Recall the expression for isothermal elasticity (E_θ)

Isothermal elasticity, E_θ, is given by: \(E_θ = 1\)

Step 4: 4. Find the ratio E_F / E_θ

Now, we can find the ratio of adiabatic elasticity to isothermal elasticity: \(\frac{E_F}{E_θ} = \frac{γ}{1} = \frac{C_p}{C_v}\) So, the correct answer is (B) \(\left(\mathrm{C}_{\mathrm{p}} / \mathrm{C}_{\mathrm{v}}\right)\).

Key Concepts

Specific heat capacities

Specific heat capacities are key concepts in thermodynamics, especially when dealing with gases. When we talk about specific heat, we're referring to the amount of heat energy required to raise the temperature of a unit mass of a substance by one degree Celsius. The specific heat capacity can be measured at constant volume, denoted as \(C_v\), or at constant pressure, represented by \(C_p\).
For gases, \(C_p\) is typically higher than \(C_v\). This is because when heating at constant pressure, the gas has to do extra work to expand against the surrounding pressure, thus requiring more energy.

• \(C_v\): The heat capacity when the volume remains constant. No work is done by the system.
• \(C_p\): The heat capacity when the pressure is constant. The system may do work as it expands.

Understanding these concepts is crucial for solving problems related to energy transfer in gases.

Adiabatic processes

An adiabatic process is a type of thermodynamic process where no heat is exchanged with the surroundings. This means that all the energy in the system is used to do work or increase internal energy. These processes are common in rapid compressions and expansions, where the changes occur so quickly that there's no time for heat exchange.
The key parameter in adiabatic processes is the adiabatic index, \( \gamma \), defined as the ratio \( \gamma = \frac{C_p}{C_v} \). This index is crucial in determining how a gas behaves when it undergoes an adiabatic process. Adiabatic elasticity, \( E_\Phi \), relates to how compressible a gas is under adiabatic conditions.
Understanding adiabatic processes helps in predicting the behavior of gases in various applications, such as engines and atmospheric processes.

Isothermal processes

Isothermal processes are thermodynamic processes where the temperature of the system remains constant throughout. In such processes, the system exchanges heat with its surroundings to ensure the temperature does not change, even though the internal energy may change due to work done on or by the system.
For an ideal gas undergoing an isothermal process, the pressure and volume change in such a manner that the product \( PV \) remains constant. Here's what's important:

• Includes slow compressions and expansions, allowing heat exchange.
• The isothermal elasticity, \( E_\theta \), is 1, which is different from adiabatic elasticity, making the concept of elasticity simpler in isothermal conditions.

Understanding isothermal processes is vital for systems like refrigerators and heat engines, where maintaining a constant temperature is essential.

Elasticity in gases

Elasticity in gases refers to how a gas responds to changes in pressure and volume. It's like the flexibility of gases when they're compressed or expanded. There are two main types relevant to thermodynamic processes:

• Adiabatic elasticity, \( E_\Phi \), measures the gas's elasticity during adiabatic changes and is equal to the adiabatic index \( \gamma \).
• Isothermal elasticity, \( E_\theta \), measures the elasticity during isothermal processes and is always equal to 1.

The difference in elasticities is important for understanding and calculating how gases behave under different conditions. In the context of the exercise, it explains why the ratio \( \frac{E_\Phi}{E_\theta} \) is \( \frac{C_p}{C_v} \). It's crucial for solving such problems, especially in thermodynamics or chemical engineering.