Chapter 15, Problem 97E

Interpretation Introduction

Interpretation:

The amount of heat required to raise the temperature of an alloy from $26°\text{C}$ to $339°\text{C}$ is to be calculated.

Concept introduction:

The amount of energy required to change the state of a substance is known as enthalpy. It is the different in the energy of final and initial state of a substance. The negative and positive sign of enthalpy indicates the energy released and energy absorbed, respectively, during the phase change.

Answer to Problem 97E

The amount of heat required to raise the temperature of an alloy from $26°\text{C}$ to $339°\text{C}$ is $1.02\times {10}^5\text{kJ}$.

Explanation of Solution

The amount of heat required to raise the temperature of an alloy from $26°\text{C}$ to $264°\text{C}$ is calculated by the formula shown below.

${\text{q}}_{\text{1}}=\text{mc}\left({\text{T}}_2-{\text{T}}_1\right)$…(1)

Where,

• $\text{m}$ is the mass of the sample.

• $\text{c}$ is the specific heat of an alloy.

• ${\text{T}}_{\text{2}}$ is the final temperature $\left(264°\text{C}\right)$.

• ${\text{T}}_{\text{1}}$ is the initial temperature $\left(26°\text{C}\right)$.

The specific heat of an alloy (solid) is $0.27\text{J}/\text{g}\cdot °\text{C}$.

Substitute the mass, final, initial temperature and specific heat of an alloy in equation (1).

$\begin{array}{rll}{\text{q}}_{\text{1}}& =& 941\times {10}^3\text{g}\times 0.27\text{J}/\text{g}\cdot °\text{C}\left(264°\text{C}-\left(26°\text{C}\right)\right)\\ & =& 6.0\times {10}^7\text{J}\end{array}$

The amount of energy required for phase transformation is calculated by the formula shown below.

${\text{q}}_{\text{2}}=\text{m}\Delta {\text{H}}_{\text{fus}}$…(2)

Where,

• is the heat of fusion.

The heat of fusion of an alloy is $29\text{J}/\text{g}$.

Substitute the mass and heat of fusion in equation (2).

$\begin{array}{rll}{\text{q}}_{\text{2}}& =& 941\times {10}^3\text{g}\times 29\text{J}/\text{g}\\ & =& 2.7\times {10}^7\text{J}\end{array}$

The amount of heat required to raise the temperature of an alloy from $264°\text{C}$ to $339°\text{C}$ is calculated by the formula shown below.

${\text{q}}_{\text{3}}=\text{mc}\left({\text{T}}_2-{\text{T}}_1\right)$…(3)

Where,

• $\text{m}$ is the mass of the sample.

• $\text{c}$ is the specific heat of n-pentane.

• ${\text{T}}_{\text{2}}$ is the final temperature $\left(339°\text{C}\right)$.

• ${\text{T}}_{\text{1}}$ is the initial temperature $\left(264°\text{C}\right)$.

The specific heat of an alloy (liquid) is $0.21\text{J}/\text{g}\cdot °\text{C}$.

Substitute the mass, final, initial temperature and specific heat of an alloy in equation (3).

$\begin{array}{rll}{\text{q}}_{\text{3}}& =& 941\times {10}^3\text{g}\times 0.21\text{J}/\text{g}\cdot °\text{C}\left(339°\text{C}-\left(264°\text{C}\right)\right)\\ & =& 1.5\times {10}^7\text{J}\end{array}$

The total amount of energy sample gained when temperature is changed from $26°\text{C}$ to $339°\text{C}$ is shown below.

$\begin{array}{rll}\text{q}& =& {\text{q}}_1+{\text{q}}_2+{\text{q}}_3\\ & =& 6.0\times {10}^7\text{J}+2.7\times {10}^7\text{J}+1.5\times {10}^7\text{J}\\ & =& 1.02\times {10}^8\text{J}\end{array}$

Convert $1.02\times {10}^8\text{J}$ to $\text{kJ}$.

$\begin{array}{rll}1.02\times {10}^8\text{J}& =& \frac{1.02\times {10}^8}{1000}\text{kJ}\\ & =& 1.02\times {10}^5\text{kJ}\end{array}$

Conclusion

The amount of heat required to raise the temperature of an alloy from $26°\text{C}$ to $339°\text{C}$ is $1.02\times {10}^5\text{kJ}$.