Suppose an amount of heat \(\Delta Q\) supplied to a substance changes its temperature from \(T\) to \(T+\Delta T\). We define the heat capacity of a substance to be
\(
S=\frac{\Delta Q}{\Delta T} \dots(12.4)
\)
We expect \(\Delta Q\) and, therefore, heat capacity \(S\) to be proportional to the mass of the substance. Further, it could also depend on the temperature, i.e., a different amount of heat may be needed for a unit rise in temperature at different temperatures. To define a constant characteristic of the substance and independent of its amount, we divide \(S\) by the mass of the substance \(m\) in \(\mathrm{kg}\) :
\(
s=\frac{S}{m}=\left(\frac{1}{m}\right) \frac{\Delta Q}{\Delta T} \dots(12.5)
\)
\(s\) is known as the specific heat capacity of the substance. It depends on the nature of the substance and its temperature. The unit of specific heat capacity is \(\mathrm{J} \mathrm{kg}^{-1} \mathrm{~K}^{-1}\).

If the amount of substance is specified in terms of moles \(\mu\) (instead of mass \(m\) in kg ), we can define heat capacity per mole of the substance by
\(
C=\frac{S}{\mu}=\frac{1}{\mu} \frac{\Delta Q}{\Delta T} \dots(12.6)
\)
\(C\) is known as the molar-specific heat capacity of the substance. Like \(s, C\) is independent of the amount of substance. \(C\) depends on the nature of the substance, its temperature, and the conditions under which heat is supplied. The unit of \(C\) is \(\mathrm{J} \mathrm{mol}^{-1} \mathrm{~K}^{-1}\). 

Table 12.1 lists measured specific and molar heat capacities of solids at atmospheric pressure and ordinary room temperature.

Consider a solid of \(N\) atoms, each vibrating about its mean position. An oscillator in one dimension has an average energy of \(2 \times 1 / 2 k_B T=k_B T\). In three dimensions, the average energy is \(3 k_B T\). For a mole of a solid, the total energy is
\(
U=3 k_B T \times N_A=3 R T\left(\because k_B T \times N_A=R\right)
\)
Now, at constant pressure, \(\Delta Q=\Delta U+P \Delta V \cong\) \(\Delta U\), since for a solid \(\Delta V\) is negligible. Therefore,
\(
C=\frac{\Delta Q}{\Delta T}=\frac{\Delta U}{\Delta T}=3 R \dots(12.7)
\)

Specific heat capacity of water

The old unit of heat was calorie. One calorie was earlier defined to be the amount of heat required to raise the temperature of \(1 g\) of water by \(1^{\circ} \mathrm{C}\). With precise measurements, it was found that the specific heat of water varies slightly with temperature. Figure \(12.5\) shows this variation in the temperature range 0 to \(100^{\circ} \mathrm{C}\).

One calorle is defined to be the amount of heat required to raise the temperature of \(1 \mathrm{~g}\) of water from \(14.5^{\circ} \mathrm{C}\) to \(15.5^{\circ} \mathrm{C}\). Since heat is just a form of energy, it is preferable to use the unit joule, J. In SI units, the specific heat capacity of water is \(4186 \mathrm{~J} \mathrm{~kg}^{-1} \mathrm{~K}^{-1}\) 1.e. \(4.186 \mathrm{~J} \mathrm{~g}^{-1} \mathrm{~K}^{-1}\). The so-called mechanical equivalent of heat defined as the amount of work needed to produce 1 cal of heat is in fact just a conversion factor between two different units of energy: calorie to joule. Since in SI units, we use the unit joule for heat, work, or any other form of energy.

As already remarked, the specific heat capacity depends on the process or the conditions under which heat capacity transfer takes place. For gases, for example, we can define two specific heats: specific heat capacity at constant volume and specific heat capacity at constant pressure. For an ideal gas, we have a simple relation.
\(
C_p-C_v=R \dots(12.8)
\)
where \(C_p\) and \(C_v\) are molar-specific heat capacities of an ideal gas at constant pressure and volume respectively and \(R\) is the universal gas constant. To prove the relation, we begin with Eq. (12.3) for 1 mole of the gas :
\(
\Delta Q=\Delta U+P \Delta V
\)
If \(\Delta Q\) is absorbed at constant volume, \(\Delta V=0\)
\(
C_{\mathrm{v}}=\left(\frac{\Delta Q}{\Delta T}\right)_{\mathrm{v}}=\left(\frac{\Delta U}{\Delta T}\right)_{\mathrm{v}}=\left(\frac{\Delta U}{\Delta T}\right) \dots(12.9)
\)
where the subscript \(v\) is dropped in the last step, since \(U\) of an ideal gas depends only on temperature. (The subscript denotes the quantity kept fixed.) If, on the other hand, \(\Delta Q\) is absorbed at constant pressure,
\(
C_{\mathrm{p}}=\left(\frac{\Delta Q}{\Delta T}\right)_{\mathrm{p}}=\left(\frac{\Delta U}{\Delta T}\right)_{\mathrm{p}}+P\left(\frac{\Delta V}{\Delta T}\right)_{\mathrm{p}} \dots(12.10)
\)
The subscript \(p\) can be dropped from the first term since \(U\) of an ideal gas depends only on \(T\). Now, for a mole of an ideal gas
\(
P V=R T
\)
which gives
\(
P\left(\frac{\Delta V}{\Delta T}\right)_p=R \dots(12.11)
\)