Question

Degrees of freedom in a water molecule are (a) 1 (b) 2 (c) 3 (d) 4

Short answer

The total number of degrees of freedom for a water molecule is 9.

Step-by-step solution

Identify molecule type

First identify the type of molecule. Water is a triatomic molecule, meaning it has three atoms.

Calculate translational degrees of freedom

All molecules, regardless of the number of constituent particles (atoms), always have 3 degrees of freedom corresponding to motion (translation) along the 3 spatial dimensions: x, y, and z.

Calculate rotational degrees of freedom

Molecules in three dimensions also have rotation. For nonlinear molecules such as water, there are 3 degrees of freedom corresponding to rotation around the three perpendicular axes.

Calculate vibrational degrees of freedom

The vibrational degrees of freedom for any molecule is given by the formula \(3n-6\) for non-linear molecules, where \(n\) is the number of atoms in the molecule. Substituting \(n=3\) for the water molecule, we get \(3*3-6 = 3\) vibrational degrees of freedom.

Calculate total degrees of freedom

Finally, add up all the degrees of freedom: translational (3), rotational (3), and vibrational (3) to find the total degrees of freedom.

Key concepts

Translational Degrees of Freedom

In physics, the concept of translational degrees of freedom is essential for understanding how molecules move in space. Each molecule can move in three independent directions: along the x, y, and z axes. These movements are translations, meaning the whole molecule shifts without rotating or vibrating. For any molecule, these directions represent the three translational degrees of freedom.

Whether it's a single atom or a complex molecular structure, the translational movement remains the same. It's how molecules traverse through space freely without being hindered by internal structural constraints. In simpler terms, think of it like moving a ball in various directions throughout a room. The ball can go forward, backward, left, right, up, and down. These movements comprise the three degrees of freedom related to translation.

Translational degrees of freedom are a fundamental aspect of thermodynamics and kinetic theory. They help calculate kinetic energy and contribute to understanding molecular behavior in gases, liquids, and solids.

Rotational Degrees of Freedom

Rotational degrees of freedom deal with the ability of a molecule to rotate around its axes. Unlike translational movement, rotation involves spinning action around an axis intersecting the molecule's center.

For non-linear molecules like water, which do not form a straight line, there are three rotational degrees of freedom. These correspond to the molecule's capacity to spin around three perpendicular axes. Think of it like a spinning dreidel or a toy top; the molecule rotates through different angles around its center.

Rotational movement depends on the mass distribution within the molecule. Heavier molecules or those shaped irregularly exhibit different rotational dynamics compared to lighter, symmetrical ones. Understanding rotational degrees is crucial when studying the energy levels and spectra of molecules, particularly in spectroscopy and molecular kinetic energy investigations.

Vibrational Degrees of Freedom

Vibrational degrees of freedom refer to the internal oscillations within a molecule. These oscillations can be thought of as tiny stretches and contractions of bonds between atoms. For non-linear molecules like water, the calculation follows the formula: \(3n - 6\), where \(n\) is the number of atoms. In water's case, \(n = 3\), so it has three vibrational modes.

These vibrations can involve bending, stretching, or twisting motions, leading to unique vibrational patterns. For example, in a water molecule, the atoms may move toward and away from each other, leading to distinct vibrational frequencies.

Vibrational degrees of freedom are pivotal in studies of infrared spectroscopy. They provide insight into the energy levels and help understand molecular structures and interactions. By analyzing vibrational spectra, scientists can deduce information about molecular bonding and structure, applying this knowledge in areas such as chemistry and materials science.