Question

The degrees of freedom for triatomic gas \(1 \mathrm{~s}\) (At room temperature) (A) 8 (B) 6 (C) 4 (D) 2

Short answer

The correct answer is (B) 6 degrees of freedom. At room temperature, a triatomic gas molecule has 3 translational degrees of freedom and for a nonlinear triatomic molecule, it has 3 rotational degrees of freedom. Vibrational degrees of freedom are not considered, as they require a high amount of energy.

Step-by-step solution

Understand Degrees of Freedom

Degrees of freedom represent the number of independent ways a system can store energy, which can be categorized into three main modes - translational, rotational, and vibrational. For a gas molecule, the total degrees of freedom are the sum of all the possible modes.

Translational Degrees of Freedom

In a 3D space, every molecule can move in three independent directions - along the x-axis, y-axis, and z-axis. So, a gas molecule has 3 translational degrees of freedom.

Rotational Degrees of Freedom

For a triatomic gas, there are two types of molecules: linear and nonlinear. A linear triatomic molecule has a single axis passing through all three atoms, which restricts its rotation to only two axes. So, it has 2 rotational degrees of freedom. On the other hand, a nonlinear triatomic molecule (such as H2O) has 3 rotational degrees of freedom.

Vibrational Degrees of Freedom

At room temperature, vibrational degrees of freedom are usually not activated for most gases, as they require a high amount of energy. Therefore, at room temperature, we will not consider vibrational degrees of freedom.

Calculate Total Degrees of freedom

Now we can sum up the degrees of freedom for each mode to find the total degrees of freedom. For a linear triatomic molecule: 3 (translational) + 2 (rotational) + 0 (vibrational) = 5 For a nonlinear triatomic molecule: 3 (translational) + 3 (rotational) + 0 (vibrational) = 6

Answer

Comparing our results with the given multiple-choice options, we can see that the most appropriate answer is 6 degrees of freedom, which corresponds to option (B). So the correct answer is (B) 6.

Key concepts

Translational Motion

Translational motion refers to the movement of molecules in three-dimensional space. Each molecule can travel in the x, y, and z directions, giving it the ability to move freely in space. This type of motion is straightforward to visualize: imagine a ball that can move forward, backward, left, right, up, or down. These movements result in 3 translational degrees of freedom. All gas molecules exhibit translational motion, which allows them to spread throughout a container and collide with the walls, contributing to pressure. This unrestricted movement is a fundamental aspect of kinetic theory, which describes how molecules move and interact in gases.

Rotational Motion

Rotational motion involves the spinning or rotation of molecules around an axis. The degrees of freedom for rotational motion depend on the shape and structure of the molecule. For example, linear triatomic molecules, like carbon dioxide (CO₂), can rotate about two axes perpendicular to their bond axis, giving them 2 rotational degrees of freedom.
In contrast, nonlinear molecules, such as water (H₂O), are more complex. They can rotate about three perpendicular axes, thus possessing 3 rotational degrees of freedom. This extra degree corresponds to the fact that their structure allows for additional rotational motion, making them more versatile in terms of energy distribution.

Vibrational Motion

Vibrational motion involves the oscillation of atoms within a molecule. Unlike translational and rotational motions, vibrational motion requires significant energy changes. At room temperature, many simple gases have minimal vibrational activity due to insufficient thermal energy support.
Vibrational degrees of freedom often become prominent at higher temperatures or through energy input from external sources. Although triatomic molecules have potential vibrational modes, these do not usually contribute to their degrees of freedom at ambient conditions. Understanding vibrational motion, however, becomes crucial when studying molecular interactions at elevated temperatures or in spectroscopy.

Linear and Nonlinear Molecules

Molecules can be broadly categorized into linear and nonlinear configurations based on their atomic arrangement. Linear molecules have all their atoms aligned in a straight line. This alignment affects both rotational and vibrational properties, as in the case of oxygen (O₂) or hydrogen cyanide (HCN), resulting in a specific set of degrees of freedom. Linear molecules generally have 5 degrees of freedom at room temperature (3 translational and 2 rotational).
Nonlinear molecules, such as water (H₂O), differ in that their atoms are angled or bent, allowing them 3 rotational degrees of freedom and potentially more complex vibrational motions. Despite having the same number of atoms as some linear molecules, their nonlinear configuration enables them to have a greater degree of freedom, notably 6 at room temperature, encompassing translational and rotational movements. These differences are crucial when analyzing molecular motion and energy distribution in various chemical and physical environments.