Chapter 5, Problem 5.24P

Interpretation Introduction

Interpretation:

The mean heat capacities ${\langle C_P\rangle }_H$ and ${\langle C_P\rangle }_S$ are inherently positive should be proved, whether $T<T_0$ or $T>T_0$and comment on why they are well defined for $T=T_0$

Concept Introduction:

Enthalpy change with non constant heat capacity is defined by:

  $\Delta H={\langle C_P\rangle }_H\left(T-T_0\right)$

The entropy change with non constant heat capacity is defined by:

  $\Delta S={\langle C_P\rangle }_S\ln \frac{T}{T_0}$

Hence the mean heat capacities are defined from equation (5.12) as

  ${\langle C_P\rangle }_S=\frac{{\displaystyle \underset{T_0}{\overset{T}{\int }}{C}_P\frac{dT}{T}}}{\ln \frac{T}{T_0}}=\frac{{\displaystyle \underset{T}{\overset{T_0}{\int }}{C}_P\frac{dT}{T}}}{\ln \frac{T_0}{T}}$   .....(1)

And

  ${\langle C_P\rangle }_H=\frac{{\displaystyle \underset{T_0}{\overset{T}{\int }}{C}_{P}dT}}{\left(T-T_0\right)}=\frac{{\displaystyle \underset{T}{\overset{T_0}{\int }}{C}_{P}dT}}{\left(T_0-T\right)}$   .....(2)