What Is the Law of Equipartition of Energy in Physics?
The Law of Equipartition of Energy states that the total energy in thermal equilibrium for any dynamic system gets divided equally among the degrees of freedom.
The kinetic energy along the x-axis, the y-axis, and the z-axis for a single molecule is given by-
Along x-axis \[\frac {1} {2}\] mvx2
Along y-axis \[\frac {1} {2}\] mvy2
Along z-axis \[\frac {1} {2}\] mvz2
In the case of thermal equilibrium the average kinetic energy of the gas is given by-
Along x-axis ( \[\frac {1} {2}\] mvx2)
Along y-axis ( \[\frac {1} {2}\] mvy2)
Along z-axis ( \[\frac {1} {2}\] mvz2)
Now, the average kinetic energy of a molecule according to the kinetic theory of gases is represented as-
\[\frac {1} {2}\]mvrms2=32KbT
where the root-mean-square velocity is represented by vrms
&
the Boltzmann constant is represented by Kb
&
T is temperature
Degree of Freedom
There are three degrees of freedom in the case of the monoatomic gas. Thus, the average kinetic energy per degree of freedom is represented as-
KEx= \[\frac {1} {2}\] KbT
A molecule possesses three translational degrees of freedom, which is free to move in space and hence needs three coordinates in order to specify its location. The molecule possesses two degrees of freedom if it is constrained to move in a plane and if it is in a straight line, there is one translational degree of freedom. On the contrary in the case of a molecule that is triatomic, the translational degree of freedom is 6 and in this case, the kinetic energy of the per molecule is given by-
6 x N x \[\frac {1} {2}\] x KbT= 3 x \[\frac {R} {N}\] x NKbT=3 RT
The translational degree of freedom in the case of the molecules of mono-atomic gases such as helium and argon is one. Then, the kinetic energy per molecule is given by-
3 x N x \[\frac {1} {2}\] x KbT=3 x \[\frac {R} {N}\] x NKbT= \[\frac {3} {2}\] RT
The translational degree of freedom in the case of diatomic gases such as oxygen and nitrogen is 3.
FAQs on Law of Equipartition of Energy: Concepts, Examples & FAQs
1. What is the Law of Equipartition of Energy as per the Class 11 syllabus?
The Law of Equipartition of Energy states that for any dynamic system in thermal equilibrium, the total energy is distributed equally among all its available degrees of freedom. The energy associated with each degree of freedom, whether translational or rotational, is exactly the same, averaging to ½ kBT per molecule, where kB is the Boltzmann constant and T is the absolute temperature.
2. In which chapter of the CBSE Class 11 Physics syllabus is this topic covered?
The Law of Equipartition of Energy is a key concept within the CBSE Class 11 Physics syllabus, specifically in Chapter 13: Kinetic Theory. It builds upon the concepts of the kinetic theory of gases and degrees of freedom to explain the specific heat capacities of different types of gases.
3. How is the Law of Equipartition of Energy applied to a monoatomic gas?
A monoatomic gas, such as Helium (He) or Argon (Ar), has only 3 translational degrees of freedom (motion along the x, y, and z axes). It has no rotational degrees of freedom. According to the equipartition theorem, the average energy per molecule is the sum of energies from these three modes:
Total Energy (U) = 3 × (½ kBT) = ³/₂ kBT.
4. What is the difference between translational and rotational degrees of freedom?
The primary difference lies in the type of motion they represent:
- Translational degrees of freedom describe the motion of a molecule's centre of mass through space. A molecule can move along three independent axes (x, y, z), so it always has 3 translational degrees of freedom.
- Rotational degrees of freedom describe the rotation of a molecule around an axis passing through its centre of mass. A linear molecule (like O₂) can rotate about two perpendicular axes, while a non-linear molecule (like H₂O) can rotate about three.
5. How does the Law of Equipartition of Energy help calculate the ratio of specific heats (γ)?
This law provides a direct method to find the ratio of specific heats (γ = Cₚ/Cᵥ). The steps are:
- First, determine the total internal energy (U) for one mole of a gas using U = (f/2)RT, where 'f' is the total degrees of freedom.
- Next, find the molar specific heat at constant volume using Cᵥ = dU/dT = (f/2)R.
- Then, use Mayer's relation (Cₚ = Cᵥ + R) to find the molar specific heat at constant pressure.
- Finally, calculate the ratio γ = Cₚ/Cᵥ, which simplifies to 1 + 2/f.
6. Why is the Law of Equipartition of Energy important for understanding gas behaviour?
Its importance lies in connecting the microscopic properties of molecules (like their structure and degrees of freedom) to the macroscopic properties of the gas (like its internal energy and specific heat capacity). It successfully explains why diatomic gases have a higher specific heat capacity than monoatomic gases—because they have more degrees of freedom to store thermal energy.
7. What are the major limitations of the Law of Equipartition of Energy?
The law is based on classical mechanics and fails when quantum effects become significant. Its main limitations are:
- It does not hold true at very low temperatures, where some degrees of freedom get 'frozen' and do not contribute to internal energy.
- It cannot explain the temperature dependence of specific heats. For example, it predicts a constant specific heat for diatomic gases, but experimentally it is found to increase at higher temperatures when vibrational modes get activated.
- It is inapplicable to systems where energy is quantised, failing to explain phenomena like the specific heat of solids at low temperatures.
8. Why are vibrational degrees of freedom ignored for diatomic gases at room temperature?
Vibrational modes are ignored at room temperature because they require a significantly higher amount of energy to become active compared to translational or rotational modes. The energy gaps between vibrational levels are large. At moderate temperatures (like room temperature), there isn't enough thermal energy (kBT) to excite these modes. They only start contributing to the specific heat at very high temperatures (typically >1000 K).
