Chapter 1, Problem 1KSP

Given that the density of gold is $\text{19}{\text{.3 g/cm}}^{\text{3}}$, calculate the volume $\left({\text{in cm}}^{\text{3}}\right)$ of a gold nugget with a mass of 5.98 g.

$\begin{array}{l}\left(\text{a}\right)\text{3}{\text{.23 cm}}^{\text{3}}\hfill \\ \left(\text{b}\right)\hfill \\ \text{5}{\text{.98 cm}}^{\text{3}}\hfill \\ \left(\text{c}\right)\hfill \\ {\text{115 cm}}^{\text{3}}\hfill \\ \left(\text{d}\right)\hfill \\ \text{0}{\text{.310 cm}}^{\text{3}}\hfill \\ \begin{array}{l}\left(\text{e}\right)\\ \text{13}{\text{.3 cm}}^{\text{3}}\end{array}\hfill \end{array}$

Interpretation Introduction

Interpretation:

The volume (in ${\text{cm}}^3$) of the gold nugget with given massis to be determined.

Concept Introduction:

The relationship between density, volume, and mass is as:

$d=\frac{m}{V}$ …… (1)

Here, $d$ is the density, $V$ is the volume,and $m$ is the mass of the given sample.

Density is defined as the ratio of mass of the sample to its volume. It is generally expressed in $\frac{\text{g}}{{\text{cm}}^{\text{3}}}$.

Answer to Problem 1KSP

Correct answer:

The volume of the given gold nugget is $0.310{\text{cm}}^3$. Thus, option (d) is correct.

Explanation of Solution

Reason for correct option:

Use equation (1) to calculate the density of the gold nugget as follows:

$d=\frac{m}{V}$

Now, rearrange equation (1) so that the volume of the gold nugget is calculated as:

$V=\frac{m}{d}$

Substitute, $19.3\frac{\text{g}}{{\text{cm}}^{\text{3}}}$ for density $\left(d\right)$ and $5.98\text{ g}$ for mass $\left(m\right)$ in the above equation as:

$\begin{array}{c}V=\frac{5.98\cancel{\text{g}}}{19.3\frac{\cancel{\text{g}}}{{\text{cm}}^3}}\\ =0.30984{\text{ cm}}^3\\ \approx 0.310{\text{ cm}}^3\end{array}$

Hence, option (d) is correct.

Reasons for incorrect options:

Option (a) is incorrect because according to the above calculation, the value $3.23{\text{ cm}}^3$ cannot be obtained. So, it is a wrong answer.

Option (b) is incorrect because according to the above calculation, the value $\text{5}{\text{.98 cm}}^3$ cannot be obtained. So, it is a wrong answer.

Option (c) is incorrect because according to the above calculation, the value ${\text{115 cm}}^3$ cannot be obtained. So, it is a wrong answer.

Option (e) is incorrect because according to the above calculation, the value $\text{13}{\text{.3 cm}}^3$ cannot be obtained. So, it is a wrong answer.

Hence, options ( a), (b), (c), and (e) are incorrect.