1.6 Concept of heat capacity

The heat capacity of a body is defined as the quantity of heat required to raise the temperature of the body by one degree.

It follows from this definition that the heat capacity of substance is an extensive property of a body. Indeed, the heat capacity of a given body is the greater, the greater the amount of substance contained in this body. For instance, the heat capacity of 10 kg of water is five times greater than the heat capacity of 2 kg of water.

The heat capacity of a unit mass of substance is given the name of specific heat. In accordance with what was said in Sec. 1.2, it is clear that specific heat is an intensive property of a substance, i.e. its magnitude does not depend on the amount of substance in the system.

Average heat capacity and true heat capacity

Heat capacity will be denoted by c[1]. From the definition of heat capacity, it follows that

(1.64)

where t₁ is the initial temperature, t₂ the final temperature, and q₁₂ the heat added to unit mass of substance while heating it from temperature t₁ to temperature t₂.

Heat capacity is not a constant quantity. It changes with temperature, the dependence being quite substantial in a number of cases. Therefore, the heat capacity determined from formula (1.64) for the temperature interval t₂ - t₁ is called the average heat capacity[2], as distinguished from the so-called true heat capacity which is defined as the derivative of the quantity of heat added to the body with respect to the temperature of this body:

(1.65)

whence

(1.66)

If we plot the amount of heat added to the body versus temperature (Fig. 1.5), it is clear that due to the variation of heat capacity the dependence will be represented by the curve q = f (t) and not by a straight line. In accordance with Eq. (1.66), on the curve q = f (t) the average heat capacity can be considered to be equal to the slope of the secant passing through points 1 and 2, i.e. to tan β, and the true heat capacities in the states 1 and 2; to tan α₁ and tan α₂, respectively.

Fig. 1.5

Knowing the dependence of the true capacity on temperature, we can easily determine the average heat capacity in a given temperature interval:

(1.67)

Mass, molar and volume heat capacities

Mass heat capacity, c, is defined as the amount of heat required to raise the temperature of unit mass of substance (usually 1 kg or 1 g) by 1 °C.

Molar heat capacity, μc, is the name given to the heat capacity reduced to one mole (or kilomole) of substance.

The volume heat capacity, C, is defined as the heat capacity reduced to unit volume of substance (usually to 1 m³).

It is obvious that

(1.68)

In practice, use is made of the mass heat capacity, which will be referred to below as heat capacity.

Dependence of heat capacity on process

Depending on the nature of the process of adding heat, the amount of heat to be added to a body so as to raise its temperature by 1 degree will differ. When speaking of heat capacity, therefore, it is necessary to specify which particular process of adding heat to a body is involved.

To put it differently, the quantity q, present in Eq. (1.65), depends not only on the temperature interval, but also on the kind of the process of adding heat. In Eq. (1.65), therefore, the amount of added heat q must be given a subscript characterizing the kind of process involved,

(1.69)

where the subscript x indicates the property which is preserved constant in the given process.

In practice, the heat capacities of the isobaric (x = p = const) and isochoric (x = v = const) processes are usually employed. These heat capacities are referred to as the isobaric and isochoric heat capacities and they are denoted by c_p (heat capacity at constant pressure) and c_v (heat capacity at constant volume).

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[1] Not to be confused with the notation of the mass fraction of a mixture.

[2] The average heat capacity is usually denoted by c_m or (the second symbol indicates the temperature interval for the particular average heat capacity).