# Understanding Energy of Particles with 'x' Degrees of Freedom

• tanaygupta2000
In summary, the equipartition theorem states that for a particle with x degrees of freedom, the average energy is given by xkT/2. The expression E = ax6 is based on certain assumptions and is proportional to either a coordinate or momentum squared. To calculate the partition function, we can use Z = ∫e-αx6 dx, where α = a/kT. Finally, the average energy can be obtained using E = kT2 d[ln Z]/dT.
tanaygupta2000
Homework Statement
Consider a system with a large number of non interacting particles at temperature T. Each particle has energy given by E = ax^6 associated with the degree of freedom x. Evaluate the mean energy per particle associated with this degree of freedom.
Relevant Equations
E = 3/2 kT
I've learned that for a particle having 3 degrees of freedom, its average energy is 3/2 kT.
So for a particle having 'x' degrees of freedom, its energy should be xkT/2.
So what is the use of given E = ax6 here?

Look up what exactly the equipartition theorem says. The devil is in the details.

vela said:
Look up what exactly the equipartition theorem says. The devil is in the details.
According to Law of Equipartition of Energy, each degree of freedom in a system of particles contributes (1/2)kT to the thermal average energy of the system.
So for x degrees of freedom of each particle in a system of particles, the thermal average energy of the system should be xkT/2.
And it is given that each particle has energy given by E = ax6.

That's based on certain assumptions. What are those assumptions?

tanaygupta2000 and etotheipi
vela said:
That's based on certain assumptions. What are those assumptions?
Okay now I understood.
The above expression holds only when each term is proportional to either a coordinate or momentum squared.
In this case, we similarly have E = ax6 ∝ (x3)2

So we have to use for Partition function,
Z = ∫e-ax6/kTdx = ∫e(-a/kT)× (x6)dx

Let (a/kT) = α
=> Z = ∫e-αx6 dx

Since

So we can obtain Z and finally E using E = kT2 d[ln Z]/dT
Is this correct ?

Last edited:

## 1. What is the concept of 'degrees of freedom' in relation to particles?

The concept of 'degrees of freedom' refers to the number of independent ways in which a particle can move or store energy. In other words, it is the number of variables that are required to fully describe the state of a particle. For example, a particle in a three-dimensional space has three degrees of freedom because it can move along the x, y, and z axes.

## 2. How does the number of degrees of freedom affect the energy of a particle?

The number of degrees of freedom directly affects the energy of a particle. According to the equipartition theorem, each degree of freedom contributes an equal amount of energy to the total energy of the particle. Therefore, a particle with more degrees of freedom will have a higher energy compared to a particle with fewer degrees of freedom.

## 3. What is the relationship between temperature and the energy of particles with 'x' degrees of freedom?

The energy of particles with 'x' degrees of freedom is directly proportional to the temperature. This is because as the temperature increases, the particles gain more kinetic energy and their degrees of freedom also increase, leading to an increase in total energy.

## 4. How does the concept of 'degrees of freedom' apply to different states of matter?

In solids, particles have limited degrees of freedom as they are tightly packed and can only vibrate in fixed positions. In liquids, particles have more degrees of freedom as they can move and slide past each other. In gases, particles have the highest number of degrees of freedom as they can move freely in all directions.

## 5. Can the energy of particles with 'x' degrees of freedom be measured?

Yes, the energy of particles with 'x' degrees of freedom can be measured using various techniques such as calorimetry, spectroscopy, and thermodynamics. These methods allow us to indirectly measure the energy of particles by observing their effects on the surrounding environment.

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