Problem 18 - 43 Years JEE ADVANCED (1978-2020) + JEE MAIN Chapterwise & Topicwise Solved Papers Chemistry

Question

A mono-atomic ideal gas undergoes a process in which the ratio of \(P\) to \(V\) at any instant is constant and equals to 1 . What is the molar heat capacity of the gas \([\mathbf{2 0 0 6}-\mathbf{3 M} ;-1]\) (a) \(\frac{3 R}{2}\) (b) \(2 R\) (c) 0 (d) \(\frac{5 R}{2}\)

Step-by-Step Solution

Answer

The molar heat capacity of the gas is 0.

Step 1: Understanding the Process

The problem describes a process in which the ratio \( P/V \) is constant, specifically equal to 1. This implies that the pressure and volume are related by the equation \( P = V \). Since \( P \) is pressure and \( V \) is volume, this describes a special process where, at each point, \( P = V \).

Step 2: Recall the Ideal Gas Law

The ideal gas law is given by \( PV = nRT \). Given that the ratio \( P/V \) is 1 at all times, we can express \( P = V = nRT/V \) for simplicity in our terms \( PV = nRT \). This implies the functional relation of variables in this state is identical such that pressure and volume might change, but their ratio stays constant.

Step 3: Analyzing the Process Dynamics

Since the problem hints at an ideal characteristic that \( P \) and \( V \) follow a direct proportionality (uniform ratio), this clues us that this is an isochoric process (constant volume) with an internal characteristic depends directly on external influence like a linear combination or known links to work like isothermal steps and thus guides Molar heat capacity interpretation.

Step 4: Determine the Molar Heat Capacity

For an isothermal process in an ideal gas, the internal energy is constant, and thus no heat is absorbed or emitted with change. However, the reference to linear stipulates an adiabatic evaluation implying that system changes can interpolate usual shifts because changes are proportionate or cause zero intersection. In such a thermal characterization, Coefficients of volumetric constraints via molar capacity were misleading due to the absence of heat return between transformations. Hence characteristic results being that molar equals polymorphic appearance matrix understanding then \( C = 0 \) hence it is spontaneous nulling scope. Therefore the correct value of molar heat capacity is \( 0 \) which matches option (c).

Key Concepts

Ideal Gas Law

When you explore how gases behave, the Ideal Gas Law is your go-to formula. It's a real gem of physics and chemistry, woven together with simplicity and power. The Ideal Gas Law is expressed as \( PV = nRT \). Here’s what those symbols mean:

• \( P \) stands for pressure
• \( V \) is volume
• \( n \) represents the number of moles of gas
• \( R \) is the ideal gas constant
• \( T \) refers to temperature in Kelvin

This equation states that the product of pressure and volume for a gas is proportional to the temperature. What makes it really versatile is that it applies to ideal gases under non-extreme conditions. This law helps us understand crucial relationships such as how gas pressure can change with temperature if the volume remains constant (among other things). Always remember it simplifies our understanding of gas behaviors in a way that captures the essence without the clutter of real-world complexities.

Adiabatic Process

In the vast world of thermodynamics, an adiabatic process is a fascinating phenomenon. It's one where no heat is transferred into or out of the system. Imagine a perfectly insulated container where the gas inside could expand or compress, but no heat enters or leaves it.

• In an adiabatic process, energy change occurs solely due to work done on or by the gas.
• This typically results in changes in temperature and pressure, sometimes causing the temperature to drop or rise rapidly depending on the compression or expansion.

A classic equation for adiabatic processes in an ideal gas is \( PV^\gamma = ext{constant} \), where \( \gamma \) is the heat capacity ratio \( C_p/C_v \). This frames how volume and pressure interrelate in this thermally isolated environment. Adiabatic processes help us understand natural phenomena like atmospheric pressure changes and are crucial in the design of engines and turbines where quick, isolated changes are beneficial.

Isothermal Process

In contrast to the adiabatic process, an isothermal process keeps things cool by maintaining a constant temperature. When a gas undergoes an isothermal process, it happens in an environment where thermal equilibrium is maintained, which means neither the system's temperature nor the surrounding changes.

• The key characteristic of an isothermal process is that any heat absorbed by the system is perfectly balanced by the work done by or on the system.
• Thanks to the constant temperature, the internal energy of the system remains unchanged.

For an ideal gas, during an isothermal process, pressure and volume change inversely while maintaining \( PV = ext{constant} \). In isothermal processes, systems can exchange heat slowly enough with surroundings to keep the temperature level. This process is exemplified in mechanisms like refrigerators that require a thorough understanding of energy exchanges without altering thermal states.

Mono-atomic Gas

Mono-atomic gases like helium, neon, and argon, stand out in the periodic table as gases that consist of single atoms. These gases are considered ideal when learning about gases because of their simple molecular structure.

• Their thermodynamic properties are easier to predict, making them ideal candidates for exploring basic laws and processes.
• Such gases have simpler expressions for their specific heat capacities. For instance, a mono-atomic gas has a constant-volume heat capacity \( C_v = \frac{3R}{2} \) and constant-pressure heat capacity \( C_p = \frac{5R}{2} \).

Due to their simplicity, these gases perfectly illustrate how energy distributions work in gases. They also help in visualizing theoretical processes like those described by the Ideal Gas Law and in understanding theoretical adiabatic or isothermal processes with minimal interaction complexities.