Specific heat capacity

Heat capacity S = ΔQ/ΔT for a temperature rise ΔT. It is proportional to the mass of the body and may vary with temperature.

Specific heat capacity s = S/m = (1/m)(ΔQ/ΔT), in J kg⁻¹ K⁻¹, depends on the substance and its temperature but not on its amount.

Molar specific heat capacity C = S/μ = (1/μ)(ΔQ/ΔT), in J mol⁻¹ K⁻¹, for μ moles. It also depends on the conditions under which the heat is supplied.

Table 11.1, solids at room temperature and atmospheric pressure, specific heat (J kg⁻¹ K⁻¹) then molar specific heat (J mol⁻¹ K⁻¹): aluminium 900.0, 24.4; carbon 506.5, 6.1; copper 386.4, 24.5; lead 127.7, 26.5; silver 236.1, 25.5; tungsten 134.4, 24.9.

Equipartition for a solid: each atom oscillates about its mean position; a one-dimensional oscillator has average energy 2 × ½k_BT = k_BT, so in three dimensions 3k_BT. For one mole, U = 3k_BT × N_A = 3RT.

At constant pressure ΔQ = ΔU + PΔV ≈ ΔU for a solid, since its ΔV is tiny. So C = ΔU/ΔT = 3R ≈ 24.9 J mol⁻¹ K⁻¹, which matches the measured values at ordinary temperatures. Carbon (6.1) is the exception, and the agreement fails at low temperatures.

The calorie was the old unit of heat. Since water's specific heat varies slightly with temperature (Fig. 11.5, 0–100 °C), 1 cal is defined as the heat that takes 1 g of water from 14.5 °C up to 15.5 °C. In SI units water's specific heat is 4186 J kg⁻¹ K⁻¹ = 4.186 J g⁻¹ K⁻¹, so 1 cal = 4.186 J.

The 'mechanical equivalent of heat' (work needed to produce 1 cal) is only a conversion factor between calorie and joule; with joules used for every form of energy the term is no longer needed.

For gases the conditions matter, so two specific heats are defined: at constant volume (C_v) and at constant pressure (C_p). For an ideal gas, C_p − C_v = R.

Proof for 1 mole: at constant volume ΔV = 0, so C_v = ΔU/ΔT. At constant pressure C_p = ΔU/ΔT + P(ΔV/ΔT). U of an ideal gas depends only on T, so ΔU/ΔT is the same in both; and PV = RT gives P(ΔV/ΔT) at constant P = R. Hence C_p − C_v = R.