Problem 88 - A Complete Resource Book in Chemistry for JEE Main

Question

In monoatomic gases, ratio of specific heat at constant pressure to that at constant volume is (a) \(3 / 5\) (b) \(5 / 3\) (c) \(7 / 5\) (d) \(4 / 5\)

Step-by-Step Solution

Answer

The ratio is \(5/3\) which corresponds to option (b).

Step 1: Identify the given problem

The problem requires us to find the ratio of specific heat at constant pressure \((C_p)\) to that at constant volume \((C_v)\) for monoatomic gases. This ratio is also known as the adiabatic index or gamma \((\gamma)\).

Step 2: Recall the formula for monoatomic gases

For monoatomic gases, the specific heat ratio \( \gamma \) is given by the formula \( \gamma = \frac{C_p}{C_v} \). We need to recall that \( \gamma = \frac{5}{3} \) for monoatomic gases.

Step 3: Match the calculated formula to the options

The calculated value of \( \gamma = \frac{5}{3} \) corresponds to option (b).

Key Concepts

Specific Heat

Specific heat is a core concept when studying thermodynamics. It is the quantity of heat required to change the temperature of a substance by one degree Celsius. Think of it as how much energy a substance needs to get warmer. For different substances, this amount can vary greatly.

There are two main contexts when talking about specific heat:

• Specific heat at constant volume \(C_v\): This is when the substance is in a fixed volume, such as in a sealed container. Here, the energy goes primarily into raising the temperature of the gas.
• Specific heat at constant pressure \(C_p\): Here, the pressure is kept steady, and extra energy is needed because the gas can expand and do work on its surroundings.

Understanding the difference between these helps explain how gases behave under different conditions. For monoatomic gases, we use these to find the adiabatic index, crucial in many physical processes.

Adiabatic Index

The adiabatic index, or gamma \(\gamma\), is an important ratio in thermodynamics, particularly for gases. It connects the concepts of specific heat at constant pressure \(C_p\) and specific heat at constant volume \(C_v\) through the formula:

• \( \gamma = \frac{C_p}{C_v} \)

For monoatomic gases, \gamma\ is usually equal to \(\frac{5}{3}\).

This ratio is important because it impacts how gases expand and compress—processes known as adiabatic processes. These are changes that happen without heat exchange with the surroundings. \gamma\ influences things like the speed of sound in gases and the efficiency of engines.

In more practical terms, when dealing with monoatomic gases like helium or argon, knowing this ratio helps predict how the gas responds to changes in pressure and temperature without losing or gaining heat from its surroundings.

Monoatomic Gases

Monoatomic gases are gases composed of single atoms. Common examples include noble gases like helium (He), neon (Ne), and argon (Ar). These gases are simple and lack molecular bonds, which makes them unique in thermodynamics.

Their simplicity leads to distinct thermodynamic properties:

• Fewer degrees of freedom compared to molecules made of two or more atoms.
• Their energy is mostly translational—moving in straight lines.
• This results in specific heat values that are significantly different from diatomic or polyatomic gases.

In the context of specific heats, monoatomic gases have well-defined values. Their specific heat at constant volume \(C_v\) is \(\frac{3}{2} R\), and specific heat at constant pressure \(C_p\) is \(\frac{5}{2} R\). Thus the ratio of these, or the adiabatic index \(\gamma\), is \(\frac{5}{3}\), making them special cases in ideal gas behavior.