The relationship between temperature and heat in a solid or liquid

In solids and liquids, which are not undergoing a phase change, the relation between heat in and Temperature change is

đ 𝑄 = 𝑚 𝑐 𝑑 𝑇

where $𝑐$ is the specific heat of the object. This is sometimes given as

đ 𝑄 = 𝜈 𝐶 𝑑 𝑇

where $𝐶$ is the molar specific heat of the object.

The values of $𝐶$ (and $𝑐$ ) are different depending on whether the heating process occurs at constant volume or constant pressure.

𝐶 𝑝 > 𝐶 𝑉

Explanation

By the First law of thermodynamics

đ 𝑄 = 𝑑 𝐸 + 𝑝 𝑑 𝑉

so if $𝑉$ is constant, $đ 𝑄 = 𝑑 𝐸$ . However if we keep $𝑝$ constant, $𝑑 𝑉 > 0$ so some heat gets spent as work.

Note that at Phase change boundaries, it is impossible to increase temperature. See Heat of transformation.

Ideal gas

For an Ideal gas

𝐶 𝑝 = 𝐶 𝑉 + 𝑅

Proof

For constant $𝜈$ , we have $𝐸 = 𝛼 𝜈 𝑅 𝑇$ . It follows $𝑑 𝐸 = 𝛼 𝜈 𝑅 𝑑 𝑇$ . Hence for a quasistatic process ( $đ 𝑄 = 𝑑 𝐸 + 𝑝 𝑑 𝑉$ ), it follows $đ 𝑄 = 𝛼 𝑅 𝜈 𝑑 𝑇 + 𝑝 𝑑 𝑉$ .

First we consider constant volume, whence

đ 𝑄 = 𝛼 𝑅 𝜈 𝑑 𝑇 = 𝑑 𝐸 ⟹ 𝐶 𝑉 = 1 𝜈 đ 𝑄 𝑑 𝑇 = 𝛼 𝑅

hence $𝐸 = 𝜈 𝐶 𝑉 𝑇$ .

Now consider constant pressure, whence

đ 𝑄 = 𝜈 𝐶 𝑉 𝑑 𝑇 + 𝑑 (𝑝 𝑉) = 𝜈 𝐶 𝑉 𝑑 𝑇 + 𝑑 (𝜈 𝑅 𝑇) = 𝜈 𝐶 𝑉 𝑑 𝑇 + 𝜈 𝑅 𝑑 𝑇 = 𝜈 (𝐶 𝑉 + 𝑅) 𝑑 𝑇

whence $𝐶 𝑝 = 𝐶 𝑉 + 𝑅$ .

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