Problem 75 - A Complete Resource Book in Chemistry for JEE Main

Question

A monoatomic ideal gas undergoes a process in which the ratio of \(\mathrm{P}\) to \(\mathrm{V}\) at any instant is constant and equals to 1 . What is the molar heat capacity of the gas? (a) \(4 \mathrm{R} / 2\) (b) \(3 \mathrm{R} / 2\) (c) \(5 \mathrm{R} / 2\) (d) 0

Step-by-Step Solution

Answer

The molar heat capacity of the gas is 0, matching option (d).

Step 1: Understanding the Process

The given process has the condition \( \frac{P}{V} = 1 \), meaning pressure \(P\) is proportional to volume \(V\), indicating an isothermal process where the temperature \(T\) remains constant.

Step 2: Apply Equation of State for Ideal Gas

For an ideal gas, using the ideal gas law \( PV = nRT \), with \( \frac{P}{V} = 1 \), we can deduce that \( T \) is constant and hence \( \Delta T = 0 \) over the process.

Step 3: Recall Definition of Molar Heat Capacity

The molar heat capacity at constant temperature, \( C_T \), generally is given by: \[ C = \frac{Q}{n \Delta T} \] where \(Q\) is heat added, \(n\) is number of moles, and \(\Delta T\) is the temperature change. Since \(\Delta T = 0\), the molar heat capacity in an isothermal process is \( C = 0 \).

Key Concepts

Ideal Gas Law

The ideal gas law is a fundamental principle in physics that describes how gases behave under various conditions. It's represented by the equation \( PV = nRT \), where \(P\) stands for pressure, \(V\) is volume, \(n\) is the number of moles, \(R\) is the ideal gas constant, and \(T\) is temperature. This equation helps us understand the relationship between pressure, volume, and temperature in a gas. By holding one of these variables constant, you can predict how the others will change.

• If you increase the pressure on a gas while keeping the volume constant, the temperature will increase.
• If you increase the volume, keeping the pressure constant, the temperature must also increase.
• If the temperature is held constant, as in an isothermal process, changes in volume inversely affect pressure.

Knowing these relationships allows scientists and engineers to predict how gases will behave in different environments, making the ideal gas law a crucial tool in thermodynamics.

Isothermal Process

An isothermal process is a thermodynamic process in which the temperature of the system remains constant. This means that any added heat gets fully utilized to do work or is absorbed without changing the temperature. When a process is isothermal, \( \Delta T = 0 \), meaning there is no change in temperature.
During an isothermal process involving an ideal gas, pressure and volume change inversely. When volume increases, pressure decreases and vice versa, maintaining a steady temperature. This property can be observed through Boyle's law, which is a specific case of the ideal gas law for isothermal conditions.
Some key characteristics:

• Typically requires slow enough changes that heat transfer can keep the temperature stable.
• Commonly occurs in systems coupled with an external reservoir for thermal energy exchange.
• Featured in the workings of devices like refrigerators and heat engines.

Due to the constant temperature, the internal energy change in an isothermal process of an ideal gas is zero, which leads to a molar heat capacity of zero for such processes.

Monoatomic Ideal Gas

A monoatomic ideal gas consists of single atoms, like helium or neon. These types of gases have distinct properties different from diatomic or polyatomic gases due to their simpler structure. For monoatomic gases, the internal energy relates directly to their temperature since they don't possess rotational or vibrational energy levels like more complex molecules.
Key features of monoatomic gases include:

• Their molar heat capacity at constant volume is \( \frac{3R}{2} \).
• Since they are monoatomic, they hold only translational kinetic energy.
• They follow the same laws as other ideal gases, but calculation and behaviors are simpler due to their lack of rotational and vibrational modes.

This makes calculations involving monoatomic ideal gases straightforward, which is why they often serve as good approximations for more complex systems in physics education and research.

Thermodynamics

Thermodynamics is the study of heat, energy, and work, and how they interrelate within physical systems. It involves understanding how energy transfers from one form to another and how systems change when achieving different energy states. Laws of thermodynamics provide foundational principles to describe these changes.
Thermodynamics can be broken down into several key laws:

• First Law: Energy cannot be created or destroyed, only transformed (the principle of energy conservation).
• Second Law: Entropy, a measure of disorder, increases over time in an isolated system.
• Third Law: As temperature approaches absolute zero, the entropy of a perfect crystal approaches zero.
• Zeroeth Law: If two systems are each in thermal equilibrium with a third, they are in equilibrium with each other.

Thermodynamics is crucial for designing engines, refrigerators, and understanding natural processes. By studying how energy flows and transforms, it helps us appreciate the efficient utilization of energy resources in various applications.