Math 2.0

Classical thermodynamics, heat transfer Definition

Heat capacity

Builds on HeatTemperatureInternal energyEnthalpyIsochoric processIsobaric processPartial derivative

Definition

Let a closed system in equilibrium undergo a quasistatic process in which it receives the heat δQ\delta Q and its temperature changes by dT≠0dT \ne 0. The heat capacity of the system along that process is C=δQ/dTC = \delta Q / dT. For a simple system with internal energy U(T,V)U(T, V) and enthalpy H(T,p)H(T, p) the two standard heat capacities are:

  1. the heat capacity at constant volume, along an isochoric process: CV=(∂U/∂T)VC_V = (\partial U / \partial T)_V;
  2. the heat capacity at constant pressure, along an isobaric process: Cp=(∂H/∂T)pC_p = (\partial H / \partial T)_p.

These are partial derivatives.

Remarks

The two formulas follow from the first law: at constant volume δQ=dU\delta Q = dU, and at constant pressure δQ=dU+p dV=dH\delta Q = dU + p\,dV = dH. Equivalently, with the entropy SS, CV=T(∂S/∂T)VC_V = T (\partial S / \partial T)_V and Cp=T(∂S/∂T)pC_p = T (\partial S / \partial T)_p. Heat capacity is extensive; the specific heat capacity is C/mC/m for the mass mm, and the molar heat capacity is C/nC/n for the amount nn.

Example: for nn moles of a monatomic ideal gas, U=32nRTU = \tfrac{3}{2} nRT with the gas constant RR. Hence CV=32nRC_V = \tfrac{3}{2} nR, and since H=U+pV=52nRTH = U + pV = \tfrac{5}{2} nRT, also Cp=52nRC_p = \tfrac{5}{2} nR.

Non-example: during melting at fixed pressure, the system absorbs heat while dT=0dT = 0. The quotient δQ/dT\delta Q / dT is not defined there, and the heat absorbed is a latent heat, not a heat capacity.

Reading path

To understand this page from the ground up, read these 12 pages in order. The foundations come first.

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Enthalpy, concept, no evidence levelEnthalpyconceptEuclidean space, object, no evidence levelEuclidean spaceobjectFunction, concept, no evidence levelFunctionconceptHeat, concept, no evidence levelHeatconceptInternal energy, concept, no evidence levelInternal energyconceptIsobaric process, concept, no evidence levelIsobaric processconceptIsochoric process, concept, no evidence levelIsochoric processconceptTemperature, concept, no evidence levelTemperatureconceptFunction of several variables, concept, no evidence levelFunction of several v…conceptLimit of a function of several variables, concept, no evidence levelLimit of a function o…conceptPartial derivative, definition, no evidence levelPartial derivativedefinitionHeat capacity, definition, no evidence levelHeat capacitydefinition
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Enthalpy concept
  2. Euclidean space object
  3. Function concept
  4. Heat concept
  5. Internal energy concept
  6. Isobaric process concept
  7. Isochoric process concept
  8. Temperature concept
  9. Function of several variables concept
  10. Limit of a function of several variables concept
  11. Partial derivative definition
  12. Heat capacity definition

Further reading

  1. Web https://en.wikipedia.org/wiki/Heat_capacity
About this page
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Kinddefinition
WritingW0 stub
MSC80A10
DefinesHeat capacity
UsesHeat, Temperature, Internal energy, Enthalpy, Isochoric process, Isobaric process, Partial derivative
Changed3 November 2026

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