Physic Labs

Classical statistical mechanics

Internal energy and enthalpy

Internal energy U is a system’s microscopic energy; enthalpy H=U+pV is a thermodynamic potential useful for constant-pressure processes.

Internal energy U is a system’s microscopic energy; enthalpy H=U+pV is a thermodynamic potential useful for constant-pressure processes.

H=U+pV,dU=δQ−δWH=U+pV, \qquad dU=\delta Q-\delta W

Definition: Quantities and meaning

U includes microscopic kinetic and potential energies, not bulk kinetic or potential energy. For a simple compressible system, dU=T dS−p dV (with only reversible pV work); dH=T dS+V dp. Thus at constant pressure, enthalpy change equals heat exchanged when only pV work occurs.

Đổi nhiệt và công pV

Quantitative relation

CV=(∂U/∂T)V,Cp=(∂H/∂T)pC_V=(\partial U/\partial T)_V, \qquad C_p=(\partial H/\partial T)_p

Example: Worked example

One mole of monatomic ideal gas is heated at constant pressure by ΔT=10 K. Find ΔH, given Cₚ=5R/2.

Solution

ΔH=nCₚΔT=(1)(5R/2)(10 K)≈208 J.

Example: Example: gas through a heater

One mole of monatomic ideal gas is heated at constant pressure from 300 K to 360 K. With Cₚ=5R/2 and only pV work, the supplied heat is Qₚ=ΔH=nCₚΔT. Using R=8.314 J·mol⁻¹·K⁻¹ gives Qₚ≈1×(5/2)×8.314×60≈1.25 kJ. This energy both raises internal energy and supplies expansion work.

At fixed temperature and volume, the constant-volume heat capacity directly tracks changes in internal energy, whereas the constant-pressure heat capacity tracks enthalpy. For an ideal gas, heating at constant pressure also expands the gas, so extra heat supplies expansion work and Cₚ−Cᵥ=R per mole. This relation also reminds us that enthalpy is not “heat stored in a body”; it is a state function chosen because it is convenient when pressure is controlled. In liquids and solids, thermal expansion is small, so the two heat capacities are often close, though their difference still depends on compressibility and expansion. Steam tables and flow-device calculations use enthalpy to simplify energy balances. This is why enthalpy is central in turbines and heat exchangers.

Quick check

For a simple system at constant pressure with only pV work, what equals the enthalpy change?

Which equation defines enthalpy?

References

  1. Charles Kittel and Herbert Kroemer (1980). Thermal Physics
  2. L. D. Landau and E. M. Lifshitz (1980). Statistical Physics