Chapter 15, Problem 95E

Interpretation Introduction

Interpretation:

The amount of energy released when the molten zinc metal cools, solidifies and further cools to $21°\text{C}$ 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 difference in the energy of the 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 95E

The amount of energy released cooling, solidifying and further cooling zinc metal to $21°\text{C}$ is $-2.3\times {10}^2\text{kJ}$.

Explanation of Solution

Zinc solidifies at $420°\text{C}$. Hence, the temperature of zinc is first changed from $552°\text{C}$ to $420°\text{C}$.

The amount of energy released when temperature changes from $552°\text{C}$ to $420°\text{C}$ 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 zinc.

• ${\text{T}}_{\text{2}}$ is the final temperature $\left(420°\text{C}\right)$.

• ${\text{T}}_{\text{1}}$ is the initial temperature $\left(552°\text{C}\right)$.

The specific heat of zinc is $0.51\text{J}/\text{g}\cdot °\text{C}$.

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

$\begin{array}{rll}{\text{q}}_{\text{1}}& =& \text{689}\text{g}\times 0.51\text{J}/\text{g}\cdot °\text{C}\left(420°\text{C}-\left(552°\text{C}\right)\right)\\ & =& -46383.48\text{J}\end{array}$

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 zinc is $-112\text{J}/\text{g}$.

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

$\begin{array}{rll}{\text{q}}_{\text{2}}& =& \text{689}\text{g}\times -112\text{J}/\text{g}\\ & =& -77168\text{J}\end{array}$

The amount of energy required to raise the temperature of zinc from $420°\text{C}$ to $21°\text{C}$ 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 zinc.

• ${\text{T}}_{\text{2}}$ is the final temperature $\left(21°\text{C}\right)$.

• ${\text{T}}_{\text{1}}$ is the initial temperature $\left(420°\text{C}\right)$.

The specific heat of zinc is $0.39\text{J}/\text{g}\cdot °\text{C}$.

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

$\begin{array}{rll}{\text{q}}_{\text{3}}& =& \text{689}\text{g}\times 0.39\text{J}/\text{g}\cdot °\text{C}\left(21°\text{C}-\left(420°\text{C}\right)\right)\\ & =& -107215.29\text{J}\end{array}$

The total amount of energy required to cool, solidify and further cool metal at $21°\text{C}$ is calculated as shown below.

$\begin{array}{rll}\text{q}& =& {\text{q}}_1+{\text{q}}_2+{\text{q}}_3\\ & =& -46383.48\text{J}+\left(-77168\text{J}\right)+\left(-107215.29\text{J}\right)\\ & =& -230766.77\text{J}\end{array}$

Convert $-230766.77\text{J}$ to $\text{kJ}$.

$\begin{array}{rll}-230766.77\text{J}& =& \frac{-230766.77}{1000}\text{kJ}\\ & =& -2.3\times {10}^2\text{kJ}\end{array}$

Conclusion

The amount of energy released cooling, solidifying and further cooling zinc metal to $21°\text{C}$ is $-2.3\times {10}^2\text{kJ}$.