Physics · Ch 10 — Thermal Properties of Matter

Specific Heat Capacity

10.6

Specific Heat Capacity

The Meaning of Heat Capacity

When you supply heat to a substance, its temperature usually rises. The amount of heat needed to raise the temperature of a given body by one degree (1 °C or 1 K) is called its heat capacity (often denoted by $C$ ). If a body absorbs an amount of heat $\Delta Q$ and its temperature changes by $\Delta T$ , then its average heat capacity over that interval is

C = \frac{\Delta Q}{\Delta T}

In the limit of an infinitesimally small temperature change, we define the heat capacity at a given temperature as

C = \frac{dQ}{dT}

Heat capacity is an extensive property — it depends on the amount of substance present. A large iron rod has a larger heat capacity than a small iron nail, even though both are made of the same material.

Specific Heat Capacity

To compare materials independently of the amount of substance, we use the specific heat capacity (often simply called specific heat). It is the heat capacity per unit mass. If a body of mass $m$ undergoes a temperature change $\Delta T$ upon absorbing heat $\Delta Q$ , its average specific heat capacity $s$ (or $c$ ) is

s = \frac{\Delta Q}{m \Delta T}

In the infinitesimal limit,

s = \frac{1}{m} \frac{dQ}{dT}

The SI unit of specific heat capacity is $\text{J kg}^{-1} \text{K}^{-1}$ (joule per kilogram per kelvin). The dimensional formula is $[L^2 T^{-2} K^{-1}]$ .

Note

The symbol $c$ is also widely used for specific heat capacity. In this chapter, the textbook uses $s$ to avoid confusion with the speed of light. In many problems and other contexts, you will see $c$ — both are acceptable.

Table 10.3 Specific heat capacity of some substances

Substance	Specific heat capacity ( $\text{J kg}^{-1}\text{K}^{-1}$ )
Aluminium	900.0
Carbon	506.5
Copper	386.4
Lead	127.7
Silver	236.1
Tungsten	134.4
Water	4186.0
Ice	2060
Glass	840
Iron	450
Kerosene	2118
Edible oil	1965
Mercury	140

Molar Specific Heat Capacity

For gases, it is often more convenient to work with one mole of the substance rather than one kilogram. The molar specific heat capacity $C$ (capital C) is the heat capacity per mole:

C = \frac{1}{\mu} \frac{dQ}{dT}

where $\mu$ is the number of moles. Its SI unit is $\text{J mol}^{-1} \text{K}^{-1}$ .

Watch out

Do not confuse the symbol $C$ for molar specific heat capacity with the earlier $C$ for heat capacity. The context (units and whether mass or moles appear) tells you which is meant. In many textbooks, molar specific heat is denoted $C_m$ or $C_{molar}$ to avoid ambiguity.

The Crucial Distinction: $C_p$ and $C_v$ for Gases

For solids and liquids, the change in volume upon heating is usually negligible, so it hardly matters whether the heating is done at constant pressure or constant volume. But for gases, the volume changes significantly with temperature, and the amount of heat required to raise the temperature by 1 K depends strongly on whether the gas is allowed to expand (constant pressure) or is confined to a fixed volume (constant volume).

This leads to two distinct molar specific heats for a gas:

Molar specific heat at constant volume, $C_V$ : the heat required to raise the temperature of 1 mole of the gas by 1 K while keeping its volume constant.
Molar specific heat at constant pressure, $C_p$ : the heat required to raise the temperature of 1 mole of the gas by 1 K while keeping its pressure constant.

Since at constant pressure the gas does work on its surroundings as it expands, more heat must be supplied to achieve the same temperature rise. Therefore, $C_p$ is always greater than $C_V$ for a gas.

$C_p$ and $C_V$ : Mayer's Relation

For an ideal gas, the two molar specific heats are related by a simple and important result, known as Mayer's relation:

C_P - C_V = R

This shows that the difference between the two molar specific heats is exactly the universal gas constant, $R = 8.31 \text{ J mol}^{-1} \text{K}^{-1}$ . Physically, at constant volume no work is done, so all the heat supplied goes into raising the internal energy; at constant pressure, the gas additionally does work against its surroundings as it expands, so more heat is needed for the same temperature rise — and that extra heat works out to exactly $R\Delta T$ . The full step-by-step derivation of this result, built on the first law of thermodynamics $\Delta Q = \Delta U + \Delta W$ , is developed in Chapter 11 (Thermodynamics); this chapter only needs the result.

Important

Mayer's formula holds only for an ideal gas. For real gases, the difference is close to $R$ but not exactly equal, especially at high pressures or near the liquefaction point.

Table 10.4 Molar specific heat capacities of some gases

Gas	$C_p$ ( $\text{J mol}^{-1}\text{K}^{-1}$ )	$C_v$ ( $\text{J mol}^{-1}\text{K}^{-1}$ )
He	20.8	12.5
H $_2$	28.8	20.4
N $_2$	29.1	20.8
O $_2$	29.4	21.1
CO $_2$	37.0	28.5

Specific Heat Capacity of Water

Water has an unusually high specific heat capacity: $s_{\text{water}} = 4186 \text{ J kg}^{-1} \text{K}^{-1}$ (or approximately $4.2 \text{ J g}^{-1} \text{°C}^{-1}$ ). This means water can absorb or release a large amount of heat with only a modest change in its temperature. This property has profound consequences:

It moderates the climate of coastal regions (land heats and cools faster than the sea).
It makes water an excellent coolant in car radiators and industrial processes.
It explains why the human body, which is mostly water, maintains a relatively stable internal temperature.

The Calorimeter and the Principle of Calorimetry

A calorimeter is a device used to measure the amount of heat transferred in a process. The simplest form is a well-insulated copper vessel with a stirrer and a thermometer.

The principle of calorimetry is a direct application of the law of conservation of energy: in an isolated system, the total heat lost by the hotter bodies equals the total heat gained by the colder bodies, provided no heat is exchanged with the surroundings. …

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Specific Heat Capacity