Chapter 7, Problem 118IAE

In the Are You Wondering 7-1 box, the temperature variation of enthalpy is discussed, and the equation q_p= heat capacity × temperature change $=C_0\times \Delta T$ 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) wth temperature, that is
$C_P=\frac{dH}{dT}$ (at constant pressue)
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_pm. the heat capacity of one mole of substance, which is 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,
$dH=C_{P}dT$ (at constant pressue)
To evaluate the change in enthalpy for a particular temperature change, from $T_1$ to $T_2$ we write
${\displaystyle {\int }_{H\left(T_1\right)}^{H\left(T_2\right)}dH=H\left(T_2\right)-H\left(T_1\right)={\displaystyle {\int }_{T_1}^{T_2}{C}_{P}dT}}$
If we assume that $C_0$ , is independent of temperature, then we recover equation (7.5)
$q_p=\Delta H=Cp\Delta T$
On the other hand, we often find that the heat capacity is a function of temperature; a convenient empirical expression is
$C_{P,m}=a+bT+\frac{c}{T^2}$
What is the change in molar enthalpy of $N_2$ when it is heated from 25-0°C to 100.0 °C? The molar heat capacity of nitrogen is given by
$C_{P,m}=\left(28.58+3.77\times {10}^{-3}T-\frac{0.5\times {10}^5}{T^2}\right)Jmo{l}^{-1}{K}^{-1}$