Heat Capacity: Complete Thermodynamics Guide
Heat capacity (or thermal capacity) is an extensive property of matter that describes the amount of heat required to raise the temperature of a given mass of a substance by one unit (Kelvin or Celsius) without undergoing a phase change. It quantifies a material’s ability to store thermal energy.
where \(C\) is the heat capacity, \(\Delta Q\) is the heat supplied, and \(\Delta T\) is the resulting temperature change.
Specific Heat Capacity (s or c)
Specific heat capacity is the heat capacity per unit mass. It is defined as the amount of heat required to raise the temperature of one gram (or one kilogram) of a substance by one degree Celsius (or one Kelvin).
Common units: J·g⁻¹·K⁻¹ or J·kg⁻¹·K⁻¹. Water has a high specific heat (4.184 J·g⁻¹·K⁻¹), making it excellent for cooling and thermal regulation.
Molar Heat Capacity (Cm)
Molar heat capacity is the heat required to raise the temperature of one mole of a substance by one Kelvin. It is related to specific heat by multiplying by the molar mass.
where \(n\) is the number of moles, \(C_m\) is the molar heat capacity (J·mol⁻¹·K⁻¹). For ideal gases, molar heat capacities are often given at constant pressure (\(C_p\)) or constant volume (\(C_v\)).
Molar Heat Capacity at Constant Pressure (Cp)
Cp is the heat required to raise the temperature of one mole of a substance by one Kelvin while keeping pressure constant. Using the first law of thermodynamics:
Molar Heat Capacity at Constant Volume (Cv)
Cv is the heat required to raise the temperature of one mole of a substance by one Kelvin while keeping volume constant. At constant volume, no work is done (\(P\,dV = 0\)), so all heat goes into internal energy.
Detailed Derivation of Cp – Cv = R for an Ideal Gas
This is one of the most important relations in thermodynamics. We derive it step by step.
Ratio of Heat Capacities (γ)
The ratio of \(C_p\) to \(C_v\) is an important dimensionless quantity denoted by \(\gamma\) (gamma).
For monatomic ideal gases, \(\gamma = 5/3 \approx 1.67\); for diatomic gases at moderate temperatures, \(\gamma = 7/5 = 1.4\); for polyatomic gases, \(\gamma\) is closer to 1.33 or lower. This ratio appears in adiabatic processes: \(PV^\gamma = \text{constant}\).
Table of Molar Heat Capacities of Common Gases (at 25°C, 1 atm)
| Gas | \(C_p\) (J·mol⁻¹·K⁻¹) | \(C_v\) (J·mol⁻¹·K⁻¹) | \(\gamma = C_p/C_v\) |
|---|---|---|---|
| Helium (He) – monatomic | 20.79 | 12.47 | 1.67 |
| Argon (Ar) – monatomic | 20.79 | 12.47 | 1.67 |
| Nitrogen (N₂) – diatomic | 29.12 | 20.79 | 1.40 |
| Oxygen (O₂) – diatomic | 29.38 | 20.95 | 1.40 |
| Carbon dioxide (CO₂) – triatomic | 36.94 | 28.46 | 1.30 |
| Methane (CH₄) – polyatomic | 35.69 | 27.38 | 1.30 |
Why Cp is Greater than Cv
When heat is added at constant pressure, the system expands and does work on the surroundings. Therefore, extra energy is required to account for the work done. At constant volume, no work is performed, so all heat goes into increasing internal energy. Hence, \(C_p > C_v\). The difference is exactly \(R\) for ideal gases, as derived above.
Pressure-Volume diagram: Isobaric vs Isochoric heating processes.
[Graph showing constant volume (vertical line) and constant pressure (horizontal line) processes; area under isobaric indicates work done.]
Interactive Calculators
Compute γ and verify Cp – Cv = R
Specific Heat Capacity Calculator
Applications of Heat Capacity
- Calorimetry: Measuring heat changes in chemical reactions (coffee-cup calorimeter, bomb calorimeter).
- Material design: High heat capacity materials (water) used as coolants; low heat capacity metals heat up quickly.
- Atmospheric science: Oceans moderate climate due to high specific heat of water.
- Engineering: Design of heat exchangers, engines, and thermal storage systems.
Summary of Key Equations
Watch this detailed lecture to master the concepts of heat capacity, Cp, Cv, and the relation Cp – Cv = R.
Download Complete Notes Below
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