Question

A diatomic molecule has how many degrees of freedom (For rigid rotator) (A) 4 (B) 3 (C) 6 (D) 5

Short answer

A diatomic molecule, as a rigid rotator, has 2 degrees of freedom for rotation around the X and Y axes, which are perpendicular to the internuclear axis. However, if we consider 3 translational degrees of freedom, the total degrees of freedom for a diatomic molecule would be 5 (2 rotational + 3 translational). Therefore, the closest correct answer is (D) 5.

Step-by-step solution

Recall the definition of degrees of freedom

Degrees of freedom are the independent ways a system (in this case, a diatomic molecule) can store energy or change its motion. For the rigid rotator model, we only consider the rotational motion.

Consider the rotational axes for the diatomic molecule

A diatomic molecule can rotate about the three axes in space: the X, Y, and Z axes. However, the rotation around the axis passing through the bond (internuclear axis) doesn't contribute to any energy change; this is because it is a rigid molecule, and the bond length does not change.

Determine the degrees of freedom

Since we only need to take into account two of the three rotational axes (the ones not passing through the bond), the diatomic molecule, as a rigid rotator, has two degrees of freedom: - Rotation around the X-axis (perpendicular to the internuclear axis) - Rotation around the Y-axis (perpendicular to the internuclear axis)

Choose the correct answer

From the given options, we can select the answer that corresponds to 2 degrees of freedom. Unfortunately, there is no option that matches 2 as the correct answer. However, if we consider the question in the more general sense (not specifying only the rigid rotator), the diatomic molecule has 3 translational degrees of freedom and 2 rotational degrees of freedom. In this sense, the answer would be: \(3 + 2 = 5\) So, selecting the option (D) 5 would be the closest correct answer in this case.

Key concepts

Diatomic Molecule

A diatomic molecule consists of two atoms bonded together. Examples include hydrogen (H₂) and oxygen (O₂). These molecules are fundamental in studying physical chemistry due to their simplicity and prevalence.
A diatomic molecule exhibits movement in translational, rotational, and vibrational modes. However, for simpler models, such as the rigid rotator model, we often consider only the translational and rotational motions.
Understanding these molecules helps us grasp more extensive molecular systems and predict their behavior under different conditions. Diatomic molecules are also used to test key theories in statistical mechanics and quantum mechanics.

Rigid Rotator Model

The rigid rotator model is a simplified physical model used to describe the rotational motion of a diatomic molecule. In this model, the two atoms are considered point masses connected by a fixed, invariable bond length. This abstraction allows us to ignore vibrational motions, simplifying calculations significantly.
- In terms of mathematical description, the model accounts for rotational energy in quantum mechanical terms, which is especially insightful for predicting energy levels and spectral lines.
- The model assumes no forces act to change the bond length, allowing it to focus solely on how the molecule rotates in three-dimensional space.
Leveraging this model helps us better understand rotational spectra and molecular symmetry, essential tools in molecular spectroscopy.

Rotational Motion

Rotational motion in physics refers to the movement of a particle or system around a central axis. For diatomic molecules, this aspect involves considering the molecule's ability to spin around axes other than the bond axis.
In the context of the rigid rotator model:

• A diatomic molecule can rotate about two axes perpendicular to the molecular bond (the X and Y axes, whereas rotation about the Z-axis, the internuclear axis, does not contribute to energy states).
• This rotational motion gives rise to the rotational degrees of freedom, contributing to the molecule's overall energy state.

Exploring rotational motion is crucial for understanding molecular behavior, especially in chemical reactions and physical transformations.

Translational Degrees of Freedom

Translational degrees of freedom refer to the number of independent movements a molecule can make in space without rotation. For a diatomic molecule, it can move along the three spatial dimensions: X, Y, and Z. This feature is common to all molecules, regardless of complexity.
- Each translational movement allows the diatomic molecule to change position in any given direction, contributing to its energy in the form of kinetic energy.
- This concept plays a vital role in understanding how molecules interact with external forces and each other.
By combining translational and rotational degrees of freedom, we get a more comprehensive understanding of a molecule's kinetic behavior, particularly in gases, where these movements are unrestricted.