Law of Equipartition of Energy

One atom is unrestricted in its movement in space along the X, Y, and Z axes. Each of these actions, however, uses energy. This is derived from the energy that each atom possesses. Energy is allocated to each atomic motion according to the Law of Equipartition of Energy (translational, rotational, and vibrational). Let’s examine the notion of the Law of Equipartition of Energy in more detail.

Law of Equipartition of Energy

The full molecule splits along the direction of their degrees of freedom when an atom or particle is maintained under constant thermal conditions, in accordance with the law of energy partition. This indicates that a particle can travel freely in its separate directions of free space or when subjected to external pressure. Students can go freely toward their respective houses after school, for instance. Their dwellings serve as a measure of freedom in this situation.

In complex molecular systems, the total internal energy is governed by the law of energy partition. It provides an explanation to queries regarding, for example, why a complicated gas’s specific heat rises as the number of atoms per molecule rises. The internal energy and high specific heat content of diatomic molecules are higher than those of monatomic gas particles. This is due to the fact that while monatomic particles only have three degrees of freedom in translation, diatomic particles have five degrees of freedom, including two rotational degrees.

In relation to the x, y, and z axes, an object’s kinetic energy is given by

Along x-axis, ½mv_x²

Along y-axis, ½mv_y²

Along z-axis, ½mv_z²

Kinetic Energy = ½mv_x² + ½mv_y² + ½mv_z²

According to the kinetic theory of gases, an item, body, or molecule’s average kinetic energy is directly inversely related to its temperature. You could indicate it as:

½ mv_rms² = 3/2 k_B T

where,

• v_rms = Root mean square velocity of molecules,
• k_B = Boltzmann constant,
• T = Temperature of gas.

For a gas at a temperature T, the average kinetic energy per molecule denoted as <K.E.> is

<K.E.> = <½mv_x²> + <½mv_y²> + <½mv_z²> = ½k_B T

The overall translational energy contribution of the molecule is consequently 3/2k_B T since the mean energy associated with each component of translational kinetic energy, which is quadratic in the velocity components in the x, y, and z directions, is ½k_B T.

A monatomic molecule only experiences translational motion, hence each motion requires ½ KT of energy. Divide the molecule’s total energy by the number of degrees of freedom to get this value:

K.E. = 3/2 k T ÷ 3 = ½ k T

Motions in translation, vibration, and rotation are all present in diatomic molecules. A diatomic molecule’s energy is represented by the following:

For Translational motion, 

K.E. = ½ m_x² + ½ m_y² + ½ m_z²

For Vibrational motion,

K.E. = ½ m (dy / dt)² + ½ k y²

where,

• k = oscillator’s force constant,
• y = vibrational coordinate.

For Rotational motion,

K.E. = ½ (I₁ω₁) + ½ (l₂ω₂)

where,

• I₁ and I₂ = Moments of inertia,
• ω₁ and ω₂ = angular speeds of rotation.

You should be aware that vibrational motion consists of both kinetic and potential energies.

The entire energy of the system is allocated evenly among the many energy modes present in the system under thermal equilibrium circumstances, in accordance with the law of energy partition. The motion’s total energy is contributed by the translational, rotational, and vibrational motions, each of which contributes a ½ k T of energy. The vibrational motion, which possesses both kinetic and potential energy, provides a whole 1 k T of energy.

Degree of Freedom

In the discussion above, the entire molecule is unrestricted in its ability to move around in three dimensions. If it can only travel on a two-dimensional plane surface, then only two coordinates, such as x and y, will be necessary to describe its location, and two components, v_x and v_y, will be adequate to describe its motion in the plane. If a molecule is moving in a straight line, its location and motion can be described using only the x coordinate and one velocity component, v_x.

In the aforementioned cases, we claim that the molecule is free to carry out translational motion in 3, 2, and 1. To put it another way, the molecule in these examples has 3, 2, and 1 degree of freedom, respectively.

The total number of coordinates or independent variables needed to fully represent the position and configuration of a system is known as its degree of freedom.

Diatomic Molecules

The He atoms are found in monoatomic gases like helium. Three degrees of freedom exist in translation for a He atom. Think about an O₂ or N₂ molecule, for instance, where the two atoms are positioned along the x-axis. The molecule has three degrees of freedom during translation. It may also revolve around the z-axis and y-axis. We consider a molecule’s rotation about the z-axis. Similar to that, rotation about the y-axis is conceivable. (Note that a rotation about the x-axis is not a rotation because the positions of the two atoms in the molecule are not changed.)

Generally, a diatomic molecule has two perpendicular rotational orientations about its centre of mass. As a result, it is claimed that molecules like O₂ have two additional degrees of freedom – two rotational degrees of freedom. The rotating kinetic energy is influenced by each of these two degrees of freedom. The rotational kinetic energies for rotation about the two axes will be ½I_zω_z² and ½I_yω_y², respectively, if I_z and I_y are moments of inertia about the z and y axes and w_z and w_y are the corresponding angular speeds. As a result, the total energy owing to the degrees of freedom for translation and rotation in a diatomic molecule is,

E = ½mv_x² + ½mv_y² + ½mv_z² + ½I_zω_z² + ½I_yω_y²

The quadratic terms in the aforementioned expression are related to the several degrees of freedom that a diatomic molecule can have. Each of them adds ½k_B T to the molecule’s overall energy. It was implied in the explanation above that the rotating molecule is a rigid rotator. Real molecules, on the other hand, have covalent links between their atoms, which allows them to execute extra motion, namely atomic vibrations about their mean locations, similar to a one-dimensional harmonic oscillator. As a result, these molecules have an extra degree of freedom that corresponds to their various vibrational modes. Only along the internuclear axis may the atoms oscillate in diatomic molecules like O₂, N₂ and CO. The vibrational energy associated with this motion is added to the molecule’s overall energy.

E = E(translational) + E(rotational) + E(vibrational)

Both the kinetic energy term and the potential energy term contribute to the word E(vibrational), which is composed of two components.

E(vibrational) = ½mu² + ½kr²

Where u denotes the rate of vibration of the molecule’s atoms, r denotes the distance between the oscillating atoms, and k is the force constant. Each of the quadratic velocity and position terms in equation 1 will contribute ½k_B T. The total internal energy is thus increased by 2 × ½k_B T for each mode or degree of freedom for vibrational motion.

Thus, for a non-rigid diatomic gas in thermal equilibrium at a temperature T, the mean kinetic energy associated with molecular translation along three directions is 3 × ½kB T, and the mean kinetic energy associated with molecular rotation about two perpendicular axes is 2 × ½k_B T, and the total vibrational energy is 2 × ½k_B T, which corresponds to kinetic and potential energy terms. The average energy for each molecule associated with each quadratic term is ½k_B T when the law of energy partition is applied to gas in thermal equilibrium at a temperature T. In contrast to extremely low temperatures, where quantum effects are significant, the law of energy equilibria only applies to high temperatures.

FAQs on Equipartition of Energy Principle

Question 1: What is the Equipartition of Energy Principle?

Answer:

The total internal energy of intricate molecular systems is defined by the law of equipartition of energy. It clarifies the idea that the specific heat of complicated gases rises as the number of atoms per molecule increases. In comparison to monatomic gas molecules, diatomic gas molecules have a higher internal energy and molar specific heat content. This is due to the diatomic gas molecule’s five degrees of freedom compared to the monatomic gas molecule’s three.

Question 2: What relationship exists between gas kinetic energy and pressure?

Answer:

The internal energy per unit volume, or internal energy density (u=U/V), or pressure of the gas, can be calculated using the formula as u=U/V. The equation states that pressure is equal to 2/3 of mean kinetic energy per unit volume.

Question 3: What does the term “three degrees of freedom” mean?

Answer:

Six degrees of freedom are included in total, three of which are associated with rotational movement and the remaining three with translational movement. Pitch, yaw, and roll are three terms used to describe the rotational degrees of freedom along the x, y, and z axes. On the other hand, the translational degrees of freedom along the x, y, and z axes can be shifted upward, downward, left to right, or in any other direction.

Question 4: What kind of force is required to hit a baseball? Describe in terms of the impact of kinetic energy.

Answer:

Kinetic energy is also referred to as the motional energy. Kinetic energy can be found in any moving thing. A lot of kinetic energy is present in baseball. When the pitcher delivers the ball, kinetic energy is transferred to the ball. The bat will gain kinetic energy when Batman swings because of the motion. The ball’s direction and speed will change when it is struck by the bat, according to the principles of kinetic energy.

Question 5: What does rotational motion mean?

Answer:

It is referred to as having a rotational motion when a body rotates around a fixed axis. When a rigid body rotates, it means that all of its constituent parts move in a circular manner around a single axis. The motion of the wheel, the earth’s rotation, a spinning top, and other objects are some instances of rotational motion.

Question 6: What does the Maxwell-Boltzmann theory entail?

Answer:

The Maxwell-Boltzmann equation, which serves as the cornerstone of the kinetic theory of gases, represents the distribution of different speeds for a gas at a given temperature. According to this equation, the likelihood of discovering the molecule changes exponentially with the energy when K_B is divided by the energy at a given state. The distribution function, average speed, and root-mean-square speed can all be deduced using this equation to determine the speed that is most likely to occur.

Question 7: How might the degree of freedom be generated?

Answer:

The quantity of gas particles and the quantity of restrictions can determine the degree of freedom.

Question 8: What does translational motion refer to?

Answer:

The term “translational motion of the body” refers to a body’s movement when it shifts from one place in space to another. Every part of the body that is moving is moving uniformly and in the same direction. Examples of translational motion include a train moving along a track, a person strolling along a street, birds flying in the sky, and so forth. If a body is doing translational motion, the orientation in relation to the fixed point remains unchanged.