In the Are You Wondering \(7-1\) box, the temperature variation of enthalpy is
discussed, and the equation \(q_{P}=\) heat capacity \(\times\) temperature change
\(=C_{P} \times \Delta T\) was introduced to show how enthalpy changes with
temperature for a constant-pressure process. Strictly speaking, the heat
capacity of a substance at constant pressure is the slope of the line
representing the variation of enthalpy (H) with temperature, that is
$$C_{P}=\frac{d H}{d T} \quad(\text { at constant pressure })$$
where \(C_{P}\) is the heat capacity of the substance in question. Heat capacity
is an extensive quantity and heat capacities are usually quoted as molar heat
capacities \(C_{P, \mathrm{m}},\) the heat capacity of one mole of substance; an
intensive property. The heat capacity at constant pressure is used to estimate
the change in enthalpy due to a change in temperature. For infinitesimal
changes in temperature,
$$d H=C_{p} d T \quad(\text { at constant pressure })$$
To evaluate the change in enthalpy for a particular temperature change, from
\(T_{1}\) to \(T_{2}\), we write
$$\int_{H\left(T_{1}\right)}^{H\left(T_{2}\right)} d
H=H\left(T_{2}\right)-H\left(T_{1}\right)=\int_{T_{1}}^{T_{2}} C_{P} d T$$
If we assume that \(C_{P}\) is independent of temperature, then we recover
equation (7.5) $$\Delta H=C_{P} \times \Delta T$$ On the other hand, we often
find that the heat capacity is a function of temperature; a convenient
empirical expression is $$C_{P, \mathrm{m}}=a+b T+\frac{c}{T^{2}}$$ What is
the change in molar enthalpy of \(\mathrm{N}_{2}\) when it is heated from
\(25.0^{\circ} \mathrm{C}\) to \(100.0^{\circ} \mathrm{C} ?\) The molar heat
capacity of nitrogen is given by$$C_{P, \mathrm{m}}=28.58+3.77 \times 10^{-3}
T-\frac{0.5 \times 10^{5}}{T^{2}} \mathrm{JK}^{-1} \mathrm{mol}^{-1}$$
