Problem 51 - 43 Years JEE ADVANCED (1978-2020) + JEE MAIN Chapterwise & Topicwise Solved Papers Chemistry

Question

\(C_{v}\) value of \(\mathrm{He}\) is always \(3 R / 2\) but \(C_{v}\) value of \(\mathrm{H}_{2}\) is \(3 R / 2\) at low temperature and \(5 R / 2\) at moderate temperature and more than \(5 R / 2\) at higher temperature. Explain in two to three lines.

Step-by-Step Solution

Answer

\(C_v\) is constant for He due to translational modes; for \(H_2\), it increases with temperature as rotational and vibrational modes activate.

Step 1: Understanding Heat Capacity

Here, we need to understand the arithmetic of heat capacity under constant volume, denoted as \(C_{v}\). It indicates the amount of heat required to raise the temperature of a mole of a gas by one degree at constant volume. \(R\) is the ideal gas constant.

Step 2: Degres of Freedom Analysis

For \( ext{He}\), it is a monatomic gas and hence has \(C_{v} = \frac{3}{2} R\) due to its three translational degrees of freedom. This value doesn't change because monatomic gases like helium do not have vibrational or rotational modes that can be excited at higher energies.

Step 3: Degrees of Freedom in \( ext{H}_2\)

\( ext{H}_2\) is diatomic, and at low temperatures, it rotates minimally, contributing \(3 R/2\) as only translational modes exert an effect. As the temperature rises, rotational modes become active, increasing \(C_{v}\) to \(5 R/2\). At even higher temperatures, vibrational modes also gain energy, leading \(C_{v}\) to exceed \(5R/2\).

Key Concepts

Degrees of Freedom

Degrees of freedom refer to the independent ways in which a molecule can store energy. For gases, these degrees of freedom usually include:

• Translational modes – movement in three-dimensional space.
• Rotational modes – rotation around its center of mass.
• Vibrational modes – back-and-forth motion along the axis between two atoms.

Understanding degrees of freedom is crucial in determining the heat capacity of gases, such as in the case of monatomic and diatomic gases. The more degrees of freedom available, the more ways energy can be distributed.

Monatomic Gases

Monatomic gases consist of single atoms, like helium (\( ext{He}\)). These gases are characterized by having only translational degrees of freedom since they lack the complex structure needed for rotational and vibrational modes.
As a result, all energy is stored in the movement of particles in space, leading to a heat capacity under constant volume (\(C_v\)) of \(\frac{3}{2} R\). This means, that regardless of the temperature, monatomic gases will primarily have only three translational modes, and their heat capacity remains constant.

Diatomic Gases

Diatomic gases, like hydrogen (\(\text{H}_2\)), possess more structure, having two atoms connected by a bond. At lower temperatures, their heat capacity resembles that of monatomic gases, \(3 R/2\), due to minimal rotation.
However, as temperature increases, rotational and eventually vibrational modes begin to contribute to the degrees of freedom. This shift in active modes causes an increase in the heat capacity:

• Moderate temperatures activate rotational modes, reaching \(5 R/2\)
• High temperatures activate vibrational modes, leading to values exceeding \(5 R/2\)

Translational Modes

Translational modes refer to the movement of gas molecules in three-dimensional space. These modes are responsible for the basic motion of individual gas particles and are present in both monatomic and diatomic gases.
For monatomic gases, translational motion represents all their degrees of freedom, expressed through the formula:\[C_v = \frac{3}{2}R\]Each degree of translational freedom contributes \(R/2\) to the heat capacity, which explains why monatomic gases, such as helium, have this definitive value. Translational modes are a fundamental form in which energy is stored in a gas.

Rotational Modes

In addition to translational modes, diatomic gases have rotational modes. These modes allow diatomic molecules to store energy through rotation about their center of mass.
At moderate temperatures, these modes become accessible, increasing the heat capacity. When these rotational modes are active, two additional degrees of freedom become involved, raising the value of \(C_v\) up to \(5 R/2\).
Rotational degrees of freedom can significantly affect the behavior of a gas as they become accessible at certain temperature ranges, important in understanding thermal dynamics.

Vibrational Modes

Vibrational modes occur when atoms within a molecule move relative to each other. In diatomic gases, vibrations involve back-and-forth oscillations of the atoms in the bond.
These modes only become significant at higher temperatures when the energy threshold for vibration is overcome. This increase further raises the heat capacity as more energy is stored in vibrational motion.
Understanding vibrational modes is crucial when analyzing systems that operate under varying thermal conditions, particularly in the case of gases that can exhibit complex molecular motion at elevated temperatures.