Chapter 1, Problem 113CP

On October 21, 1982, the Bureau of the Mint changed the composition of pennies (see Exercise 120). Instead of an alloy of 95% Cu and 5% Zn by mass, a core of 99.2% Zn and 0.8% Cu with a thin shell of copper was adopted. The overall composition of the new penny was 97.6% Zn and 2.4% Cu by mass. Does this account for the difference in mass among die pennies in Exercise 120? Assume the volume of the individual metals that make up each penny can be added together to give the overall volume of the penny, and assume each penny is the same size. (Density of Cu = 8.96 g/cm³; density of Zn = 7.14 g/cm³).

Interpretation Introduction

Interpretation:

The density of the old and new pennies has to be calculated and the reason for the change in mass with change in alloy used has to be explained.

Concept introduction:

The quantity density the amount of substance per unit volume of the substance.  Density is a unique property of a substance.

$\text{Density=}\frac{\text{Mass}}{\text{Volume}}$

Answer to Problem 113CP

The density of the old penny is $\text{8}{\text{.85g/cm}}^{\text{3}}$

The density of the new penny is $\text{7}{\text{.14g/cm}}^{\text{3}}$

From the density measurements of the old and new pennies, we have found that there is a change in density along with the change in the composition.  Hence, we can conclude that the change in mass is due the difference in the alloys that is used for the preparation of pennies.

Explanation of Solution

To find the density of the old penny

Given,

Density of one mole of copper $\text{=8}{\text{.96g/cm}}^{\text{3}}$

Density of one mole of zinc $\text{= 7}{\text{.14g/cm}}^{\text{3}}$

Let's assume there are $100g$ of the given penny.  The mixture contains $95g$ of copper and $5g$ of zinc.

$\begin{array}{l}\text{The volume of total amount of the copper in the mixture= }\frac{\text{95g }}{\text{8}{\text{.96g/cm}}^{\text{3}}}\\ \text{=10}{\text{.6cm}}^{\text{3}}\end{array}$

$\begin{array}{l}\text{The volume of total amount of the zinc in the mixture= }\frac{\text{5g }}{\text{7}{\text{.14g/cm}}^{\text{3}}}\\ \text{=0}{\text{.70cm}}^{\text{3}}\end{array}$

The total volume is the sum of the volume of copper and zinc.

$\begin{array}{l}\text{Total volume of the penny =10}{\text{.6cm}}^{\text{3}}\text{+0}{\text{.70cm}}^{\text{3}}\\ \text{=11}{\text{.3cm}}^{\text{3}}\end{array}$

$\begin{array}{l}\text{The average density of the old penny=}\frac{\text{100g}}{\text{11}{\text{.3cm}}^{\text{3}}}\\ \text{=8}{\text{.85g/cm}}^{\text{3}}\end{array}$

To find the density of the new penny

Given,

Density of one mole of copper $\text{=8}{\text{.96g/cm}}^{\text{3}}$

Density of one mole of zinc $\text{= 7}{\text{.14g/cm}}^{\text{3}}$

Let's assume there are $100g$ of the given penny.  The mixture contains $97.6g$ of zinc and $2.4g$ of copper.

$\begin{array}{l}\text{The volume of total amount of the zinc in the mixture= }\frac{\text{97}\text{.6g }}{\text{87}{\text{.14g/cm}}^{\text{3}}}\\ \text{=13}{\text{.7cm}}^{\text{3}}\end{array}$

$\begin{array}{l}\text{The volume of total amount of the copper in the mixture= }\frac{\text{2}\text{.4g }}{\text{8}{\text{.96g/cm}}^{\text{3}}}\\ \text{=0}{\text{.27cm}}^{\text{3}}\end{array}$

The total volume is the sum of the volume of copper and zinc.

$\begin{array}{l}\text{Total volume of the penny =13}{\text{.7cm}}^{\text{3}}\text{+0}{\text{.27cm}}^{\text{3}}\\ \text{=14}{\text{.0cm}}^{\text{3}}\end{array}$

$\begin{array}{l}\text{The average density of the new penny=}\frac{\text{100g}}{\text{14}{\text{.0cm}}^{\text{3}}}\\ \text{=7}{\text{.14g/cm}}^{\text{3}}\end{array}$

From the density measurements of the old and new pennies, we have found that there is a change in density along with the change in the composition.  Hence, we can conclude that the change in mass is due the difference in the alloys that is used for the preparation of pennies.

Conclusion

The density of the old and new pennies has been calculated and the reason for the change in mass with change in alloy used has been explained.