Degrees of Freedom and Specific Heats
Equipartition of energy and Cp, Cv for mono-, di- and polyatomic gases. (Physics › Kinetic Theory of Gases, NEET UG syllabus.)
What is Degrees of Freedom and Specific Heats?
The total number of independent ways in which a system (like a gas molecule) can possess energy (translational, rotational, vibrational).
Key formula / rule: Internal Energy of n moles of gas
Key points
- Define degrees of freedom and state the law of equipartition of energy.
- Determine the degrees of freedom for monatomic, diatomic, and polyatomic gases.
- Derive expressions for internal energy, Cv, Cp, and γ in terms of 'f' and R.
- Calculate Cv, Cp, and γ for different types of gases.
Common exam trap
Confusing molar specific heats with specific heats per unit mass.
Definitions
- Term
Degrees of Freedom (f)
- Meaning
The total number of independent ways in which a system (like a gas molecule) can possess energy (translational, rotational, vibrational).
- Term
Law of Equipartition of Energy
- Meaning
For a system in thermal equilibrium, the total energy is equally distributed among its various degrees of freedom, with each ° of freedom contributing (1/2)kT of energy per molecule.
- Term
Molar Specific Heat at Constant Volume (Cv)
- Meaning
The amount of heat required to raise the temperature of one mole of a gas by one ° Celsius (or Kelvin) at constant volume.
- Term
Molar Specific Heat at Constant Pressure (Cp)
- Meaning
The amount of heat required to raise the temperature of one mole of a gas by one ° Celsius (or Kelvin) at constant pressure.
- Term
Adiabatic Index (γ)
- Meaning
The ratio of the molar specific heat at constant pressure (Cp) to the molar specific heat at constant volume (Cv), i.e., γ = Cp/Cv.
Learning objectives
Define degrees of freedom and state the law of equipartition of energy.
Determine the degrees of freedom for monatomic, diatomic, and polyatomic gases.
Derive expressions for internal energy, Cv, Cp, and γ in terms of 'f' and R.
Calculate Cv, Cp, and γ for different types of gases.
Apply these concepts to solve problems related to thermal processes in gases.
Formulae
- Name
Internal Energy of n moles of gas
- Note
f is degrees of freedom, R is universal gas constant, T is absolute temperature.
- Expression
U = n * (f/2) * R * T
- Name
Molar Specific Heat at Constant Volume
- Note
For one mole of gas.
- Expression
Cv = (f/2) * R
- Name
Molar Specific Heat at Constant Pressure
- Note
For one mole of gas.
- Expression
Cp = ((f+2)/2) * R
- Name
Mayer's Relation
- Note
Relates specific heats at constant pressure and volume.
- Expression
Cp - Cv = R
- Name
Ratio of Specific Heats (Adiabatic Index)
- Note
Gamma is crucial for adiabatic processes.
- Expression
γ = Cp / Cv = 1 + (2/f)
- Name
Energy per molecule per ° of freedom (Equipartition Theorem)
- Note
k is Boltzmann's constant.
- Expression
Eavg = (1/2) * k * T
Prerequisites
Basic understanding of kinetic theory of gases postulates.
Concept of internal energy.
First Law of Thermodynamics.
Ideal gas equation (PV=nRT).
Molar specific heat definition.
Common mistakes
Confusing molar specific heats with specific heats per unit mass.
Incorrectly counting degrees of freedom, especially for diatomic and polyatomic molecules (e.g., forgetting rotational or including vibrational at low T).
Not applying Mayer's relation correctly.
Using k instead of R in molar specific heat calculations or vice-versa.
Forgetting that vibrational degrees of freedom contribute 2 (kinetic + potential) to 'f'.
Keywords
Degrees of freedom
equipartition of energy
specific heat
molar specific heat
Cv
Cp
γ
monatomic
diatomic
polyatomic
translational
rotational
vibrational
internal energy
Mayer's relation
Practice preview
How many degrees of freedom does a monoatomic gas molecule possess?…
easy
According to the Law of Equipartition of Energy, the energy associated with each degree of freedom for a gas molecule in thermal equilibrium is:…
easy
The molar specific heat at constant volume (Cv) for an ideal diatomic gas at moderate temperatures is:…
medium