19-8 Degrees of Freedom and Molar Specific Heats#
Prompts
What is a degree of freedom? State the equipartition theorem. How much energy is associated with each degree of freedom?
Why does a monatomic gas have \(f = 3\)? Why does a diatomic gas have \(f = 5\) (and not 6)?
Write \(E_{\text{int}}\) and \(C_V\) in terms of \(f\). What is \(C_V\) for O\(_2\)? For He?
At very low temperature, why might a diatomic gas behave like a monatomic gas (\(C_V \approx \frac{3}{2}R\))?
Lecture Notes#
Overview#
A degree of freedom is an independent way a molecule can store energy (translation, rotation, vibration).
Equipartition theorem: each degree of freedom has, on average, \(\frac{1}{2}kT\) per molecule (or \(\frac{1}{2}RT\) per mole).
\(E_{\text{int}} = \frac{f}{2}nRT\) and \(C_V = \frac{f}{2}R\), where \(f\) is the number of degrees of freedom.
Equipartition theorem#
Each degree of freedom contributes \(\frac{1}{2}kT\) per molecule to the average energy. For \(n\) moles of \(N = nN_A\) molecules:
Degrees of freedom by molecule type#
Type |
Translational |
Rotational |
Total \(f\) |
\(C_V\) |
|---|---|---|---|---|
Monatomic |
3 |
0 |
3 |
\(\frac{3}{2}R\) |
Diatomic |
3 |
2 |
5 |
\(\frac{5}{2}R\) |
Polyatomic |
3 |
3 |
6 |
\(3R\) |
Translational: All molecules have 3 (motion along \(x\), \(y\), \(z\)).
Monatomic: No rotation (single atom).
Diatomic: 2 rotational degrees (rotation about axes perpendicular to the bond; rotation about the bond axis has negligible moment of inertia).
Polyatomic: 3 rotational degrees (rotation about all three axes).
Poll: Fraction of energy that is translational
For a monatomic ideal gas, what fraction of its total internal energy is translational kinetic energy?
(A) 33.3%
(B) 50.0%
(C) 66.7%
(D) 100%
(E) Depends on temperature
Vibration
Diatomic and polyatomic molecules can also vibrate (atoms oscillate along bonds). At typical room temperatures, vibrational modes are typically frozen (quantum energy gaps too large). At high \(T\), vibration adds more degrees of freedom.
Temperature dependence (quantum effects)#
At very low \(T\), rotational and vibrational energies are quantized. Molecules may not have enough energy to rotate or vibrate—those degrees of freedom are effectively “off.”
H\(_2\) below ~80 K: Only translation → \(C_V \approx \frac{3}{2}R\) (like monatomic).
Room temperature: Translation + rotation → \(C_V \approx \frac{5}{2}R\) (diatomic).
High \(T\): Vibration activates → \(C_V\) increases further.
Poll: Largest \(\Delta T\) for fixed heat—temperature dependence of \(C_V\)
You have 10 J of energy to give to a sample of air via heat. To get the largest temperature change, should you add this heat when the gas is at 50 K or at 300 K?
(A) 50 K
(B) 300 K
(C) No difference
Poll: Degrees of freedom
A diatomic gas at room temperature has \(C_V = \frac{5}{2}R\). How many vibrational degrees of freedom are typically active?
(A) 0
(B) 1
(C) 2
Summary#
Equipartition: \(\frac{1}{2}kT\) per degree of freedom per molecule.
\(E_{\text{int}} = \frac{f}{2}nRT\); \(C_V = \frac{f}{2}R\).
Monatomic \(f = 3\); diatomic \(f = 5\); polyatomic \(f = 6\).
At low \(T\), rotation and vibration can be frozen (quantum effects).