Degrees of Freedom
The number of independent ways in which a molecule can possess energy, including translational, rotational, and vibrational motions.
What is Degrees of Freedom?
The number of independent parameters required to completely describe the state of a molecule, corresponding to the ways it can store energy (translational, rotational, vibrational).
Key formula / rule: Degrees of Freedom (Monatomic)
Key points
- Define degrees of freedom.
- Identify the types of motion contributing to degrees of freedom.
- Determine the degrees of freedom for different types of molecules (monatomic, diatomic, polyatomic) at various temperatures.
- Understand the implications of the equipartition theorem.
Common exam trap
Assuming diatomic molecules always have 7 degrees of freedom.
Definitions
- Term
Degrees of Freedom
- Meaning
The number of independent parameters required to completely describe the state of a molecule, corresponding to the ways it can store energy (translational, rotational, vibrational).
- Term
Translational Motion
- Meaning
Movement of a molecule along the x, y, and z axes in space.
- Term
Rotational Motion
- Meaning
The spinning of a molecule about its center of mass.
- Term
Vibrational Motion
- Meaning
The oscillation of atoms within a molecule about their equilibrium positions.
- Term
Equipartition Theorem
- Meaning
A theorem stating that in thermal equilibrium, energy is equally divided among all accessible degrees of freedom, with each ° of freedom contributing an average energy of $\frac{1}{2}kT$ per molecule.
Learning objectives
Define degrees of freedom.
Identify the types of motion contributing to degrees of freedom.
Determine the degrees of freedom for different types of molecules (monatomic, diatomic, polyatomic) at various temperatures.
Understand the implications of the equipartition theorem.
Formulae
- Name
Degrees of Freedom (Monatomic)
- Note
Only translational motion.
- Expression
f = 3
- Name
Degrees of Freedom (Diatomic - Room Temp)
- Note
Vibrational modes are not active.
- Expression
f = 3 (translational) + 2 (rotational) = 5
- Name
Degrees of Freedom (Diatomic - High Temp)
- Note
Vibrational modes become active.
- Expression
f = 3 (translational) + 2 (rotational) + 2 (vibrational) = 7
- Name
Degrees of Freedom (Polyatomic - Linear)
- Note
n = number of atoms. Generally, vibrational modes are complex and often simplified in exam problems.
- Expression
f = 3 (translational) + 2 (rotational) + 3n-5 (vibrational)
- Name
Degrees of Freedom (Polyatomic - Non-linear)
- Note
n = number of atoms. Generally, vibrational modes are complex and often simplified in exam problems.
- Expression
f = 3 (translational) + 3 (rotational) + 3n-6 (vibrational)
- Name
Equipartition Theorem (Energy per molecule per ° of freedom)
- Note
k = Boltzmann constant, T = absolute temperature
- Expression
E = $\frac{1}{2}kT$
- Name
Total Internal Energy (Monatomic)
- Note
N = number of molecules, n = number of moles, R = universal gas constant
- Expression
U = $\frac{3}{2}NkT$ or $\frac{3}{2}nRT$
- Name
Total Internal Energy (Diatomic - Room Temp)
- Note
Assumes only translational and rotational modes are active.
- Expression
U = $\frac{5}{2}NkT$ or $\frac{5}{2}nRT$
Prerequisites
Kinetic Theory of Gases
Molecular Structure (Monatomic, Diatomic, Polyatomic)
Basic understanding of energy and motion.
Common mistakes
Assuming diatomic molecules always have 7 degrees of freedom.
Forgetting to include translational degrees of freedom.
Not considering the effect of temperature on vibrational modes.
Confusing degrees of freedom with the number of atoms in a molecule.
Keywords
Degrees of Freedom
Translational Motion
Rotational Motion
Vibrational Motion
Monatomic Gas
Diatomic Gas
Polyatomic Gas
Equipartition Theorem
Kinetic Theory of Gases
Practice preview
What is the degree of freedom for a monatomic gas molecule?…
easy
A diatomic molecule has how many degrees of freedom at moderate temperatures?…
medium
The molar specific heat at constant volume (Cv) for a diatomic gas at moderate temperatures is approximately:…
medium