Problem 40 - A Complete Resource Book in Chemistry for JEE Main

Question

Which of the following gas molecule has the maxi mum specific heat at constant pressure? (a) helium (b) argon (c) nitrogen (d) oxygen

Step-by-Step Solution

Answer

Nitrogen or oxygen has the maximum specific heat at constant pressure.

Step 1: Understand the Concept

To determine which gas molecule has the maximum specific heat at constant pressure, we need to consider the degree of freedom of the gas molecules. For monoatomic gases like helium and argon, the degree of freedom is 3. For diatomic gases like nitrogen and oxygen, it is generally 5 at room temperature due to translational and rotational motions. The specific heat at constant pressure is given by the formula \( C_p = \frac{f + 2}{2} R \), where \( f \) is the degree of freedom and \( R \) is the universal gas constant.

Step 2: Calculate Specific Heat at Constant Pressure

Use the formula \( C_p = \frac{f + 2}{2} R \) to calculate for each gas: - **Helium (monatomic):** \( C_p = \frac{3 + 2}{2} R = \frac{5}{2} R \)- **Argon (monatomic):** \( C_p = \frac{3 + 2}{2} R = \frac{5}{2} R \)- **Nitrogen (diatomic):** \( C_p = \frac{5 + 2}{2} R = \frac{7}{2} R \)- **Oxygen (diatomic):** \( C_p = \frac{5 + 2}{2} R = \frac{7}{2} R \)

Step 3: Identify the Gas with Maximum Specific Heat

Compare the calculated specific heats: Helium and Argon have \( \frac{5}{2} R \), while Nitrogen and Oxygen both have \( \frac{7}{2} R \). This means that diatomic gases (nitrogen and oxygen) have higher specific heats compared to monoatomic gases.

Step 4: Conclude with the Correct Answer

Among the given options (helium, argon, nitrogen, oxygen), both nitrogen and oxygen have the same specific heat at constant pressure, but they are higher than those for helium and argon. Therefore, we can conclude that either nitrogen or oxygen has the maximum specific heat at constant pressure; since the specific heat values are often identical for similar diatomic molecules, they are both correct.

Key Concepts

Degree of Freedom

In the realm of physics, the degree of freedom of a gas molecule plays a crucial role in understanding its properties, especially its specific heat capacity at constant pressure. The degree of freedom refers to the number of independent ways in which a molecule can move or store energy. Each degree of freedom corresponds to a different form of motion or energy storage:

• Translational: Movement along the x, y, and z axes.
• Rotational: Spinning about different axes.
• Vibrational: Molecule parts moving relative to each other (significant in polyatomic gases).

For monoatomic gases, like helium or argon, molecules can only move translationally, thus having 3 degrees of freedom. In diatomic gases such as nitrogen or oxygen, they possess five degrees of freedom at room temperature, as they also rotate, storing additional energy in rotational motion. More degrees of freedom means more ways to store energy, impacting the specific heat capacity.

Monoatomic Gases

Monoatomic gases are those composed of single atoms, like helium and argon. They are the simplest form of gases, possessing only translational degrees of freedom. Consequently, monoatomic gases have fewer degrees of freedom compared to more complex molecules.

Monoatomic gases only have 3 degrees of freedom due to their ability to move in three-dimensional space. As a result, their specific heat at constant pressure, denoted as \( C_p \), can be precisely calculated using the formula: \[ C_p = \frac{f + 2}{2} R \] where \( f = 3 \) for monoatomic gases, resulting in \( C_p = \frac{5}{2} R \).

This limited capacity for energy storage makes their specific heat capacity lower compared to gases with more complex structures and movement capabilities, such as diatomic gases.

Diatomic Gases

Diatomic gases consist of molecules with two atoms, like nitrogen and oxygen. Unlike monoatomic gases, diatomic molecules are capable of more complex movements, including both translational and rotational motions.

The translational and rotational movements together provide diatomic gases with 5 degrees of freedom at room temperature:

• 3 translational degrees of freedom due to movement along x, y, and z axes.
• 2 rotational degrees of freedom around perpendicular axes passing through their center of mass.

This higher degree of freedom is reflected in their specific heat at constant pressure:\[ C_p = \frac{f + 2}{2} R \] For diatomic gases, \( f = 5 \), leading to a \( C_p = \frac{7}{2} R \).

The ability to store more energy in these various modes of movement results in diatomic gases having a higher specific heat compared to their monoatomic counterparts.

Universal Gas Constant

The universal gas constant, denoted by \( R \), is a critical factor in thermodynamics that appears in equations governing the behavior of gases. It's a constant value that relates the energy scale in physics to temperature and amount of substance in a system.

The value of the universal gas constant \( R \) is approximately 8.314 J/(mol·K). This constant helps in calculating everything from the specific heat capacities of gases to predicting their behavior under various pressure and temperature conditions.

In specific heat calculations, such as \( C_p \), this constant is essential for determining the amount of energy required to increase the temperature of one mole of a gas by one degree Kelvin at constant pressure. Its presence in the equation \[ C_p = \frac{f + 2}{2} R \] shows its importance in quantifying the relationship between energy, temperature, and molecular structure of gases.