Chapter 2, Problem 2.42P

Interpretation Introduction

Interpretation:

The steady state temperature at section two $T_2$ is measured as the function of power $P$ is shown in the table below. Calculate the average specific heat of the substance tested over the temperature range from ${\text{0}}^{\text{o}}\text{C}$ to ${\text{10}}^{\text{o}}\text{C}$ along with the temperature range from ${\text{90}}^{\text{o}}\text{C}$ to ${\text{100}}^{\text{o}}\text{C}$ and over the entire range tested. Derive an expression for the specific heat as a function of temperature.

$T_2$ (   ${}^{\text{o}}\text{C}$ )	10	20	30	40	50	60	70	80	90	100
$P$ (   $W$ )	5.5	11	16.6	22.3	28.0	33.7	39.6	45.4	51.3	57.3

Concept Introduction:

Since for average specific heat calculations, the specific heat use in heat calculations under the integration sign is taken as constant. So, for average heat calculations, transfer of heat in section two shall be:

For normal specific heat calculations, correlation for the specific heat is:

  $Q={\displaystyle \underset{T_1}{\overset{T_2}{\int }}\frac{\Delta C_P}{R}dT}$

For average heat calculations, transfer of heat in section two shall be:

  $Q=m{C}_p(T_2-T_1)$

Or

  $P=\stackrel{•}{m}{C}_p(T_2-T_1)$

Now for calorimeter with no shaft work

  $\frac{Q}{t}=\frac{\Delta H}{t}=P$

So,

  $\frac{\Delta H}{t}=\stackrel{•}{m}{C}_p(T_2-T_1)$...........(1)

As heat is given per unit, given time helping fluid to raise its temperature by $\text{1}\text{K}$ as transferring its thermal energy to fluid rate to increase its heat capacity. So heat capacity is always function of temperature, more will be raise in temperature, more heat capacity is required to store its thermal energy.