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Heat Capacity

Level 1 - Physics topic page in Matter at Thermal Scale.

Principle

Heat capacity measures how much energy is needed to change a system's temperature.

At the molecular scale, heat capacity depends on which microscopic energy modes can accept energy at the temperature of interest.

Notation

\(C\)
heat capacity of an object
\(\mathrm{J\,K^{-1}}\)
\(c\)
specific heat capacity
\(\mathrm{J\,kg^{-1}\,K^{-1}}\)
\(C_m\)
molar heat capacity
\(\mathrm{J\,mol^{-1}\,K^{-1}}\)
\(m\)
mass
\(\mathrm{kg}\)
\(n\)
amount of substance
\(\mathrm{mol}\)
\(f\)
active quadratic degrees of freedom
1
\(R\)
molar gas constant
\(\mathrm{J\,mol^{-1}\,K^{-1}}\)

Method

Derivation 1: Define heat capacity at different scales

Heat capacity can refer to one object, one kilogram, or one mole. The physics is the same, but the normalization changes.

Object capacity
\[C=\frac{dQ}{dT}\]
Specific form
\[dQ=mc\,dT\]
Molar form
\[dQ=nC_m\,dT\]
Connections
\[C=mc=nC_m\]

Derivation 2: Use equipartition for a simple molecular model

Each active quadratic energy term contributes \(\frac\{1\}\{2\}k_BT\) per molecule, or \(\frac\{1\}\{2\}RT\) per mole. If a system has \(f\) active quadratic modes per molecule, its molar internal energy contribution is \(\frac\{f\}\{2\}RT\).

Molar energy model
\[U_m=\frac{f}{2}RT\]
Molar heat capacity
\[C_m=\frac{dU_m}{dT}=\frac{f}{2}R\]
Monatomic gas
\[f=3\Rightarrow C_{V,m}=\frac{3}{2}R\]
High-temperature solid
\[f=6\Rightarrow C_m\approx3R\]
This is the classical Dulong-Petit limit for many crystalline solids.

Not every mode is active at every temperature. Quantum energy spacing can suppress some rotational or vibrational contributions, especially at low temperature.

Rules

These are the compact heat-capacity relations.

Object capacity
\[C=\frac{dQ}{dT}\]
Specific heat
\[Q=mc\Delta T\]
Molar heat
\[Q=nC_m\Delta T\]
Capacity connection
\[C=mc=nC_m\]
Equipartition
\[C_m=\frac{f}{2}R\]

Examples

Question
A
\[0.80\,\mathrm{kg}\]
object has
\[c=450\,\mathrm{J\,kg^{-1}\,K^{-1}}\]
Find its heat capacity.
Answer
\[C=mc=(0.80)(450)=360\,\mathrm{J\,K^{-1}}\]

Checks

  • Heat capacity belongs to a whole object; specific heat and molar heat capacity are normalized versions.
  • Use \(C_m\) with moles and \(c\) with mass.
  • Equipartition is a model with a range of validity.
  • Low-temperature heat capacities often fall below classical predictions.

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Ideal Gas Model

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Molecular Speeds