Chapter 7.4, Problem 93PPE

To determine

To find: The density of Neptune in grams per cubic meter.

Answer to Problem 93PPE

The density of Neptune in grams per cubic meter is $1.6\times {10}^6\frac{\text{g}}{{\text{m}}^{\text{3}}}$ .

Explanation of Solution

Given information: The mass of the Neptune is $1.02\times {10}^{26}\text{kg}$ and the radius of Neptune is $2.48\times {10}^4\text{km}$ .

Calculation:

Let the mass and radius of Neptune be $m$ and $r$ respectively.

The formula of the volume of sphere is:

  $V=\frac{4}{3}\pi r^3$

Substitute $2.48\times {10}^4\text{km}$ for $r$ in the above formula.

  $\begin{array}{c}V=\frac{4}{3}\pi {\left(2.48\times {10}^4\text{km}\right)}^3\left[\therefore {\left(ab\right)}^n=a^n{b}^n\right]\\ =\frac{4}{3}\pi {\left(2.48\right)}^3\times {10}^{12}{\text{km}}^3\\ =\frac{4}{3}\left(3.14\right)\left(15.253\right)\times {10}^{12}{\text{km}}^3\\ =63.9173\times {10}^{12}{\text{km}}^3\end{array}$

The conversion of cubic kilometer into cubic meter is:

  $\begin{array}{c}1{\text{km}}^{\text{3}}=1,000,000,000{\text{m}}^{\text{3}}\\ ={10}^9{\text{m}}^{\text{3}}\end{array}$

To find the conversion of $63.9173\times {10}^{12}{\text{km}}^3$ , multiply both sides of above conversion by $63.9173\times {10}^{12}$ .

  $63.9173\times {10}^{12}\times 1{\text{km}}^3=63.9173\times {10}^{12}\times {10}^9{\text{m}}^3$

Apply the property of addition of exponent of same base on the right hand side of the above expression.

  $\begin{array}{c}63.9173\times {10}^{12}\times 1{\text{km}}^{\text{3}}=63.9173\times {10}^{12}\times {10}^9{\text{m}}^3\left[\therefore a^m\times a^n=a^{m+n}\right]\\ =63.9173\times {10}^{21}{\text{m}}^3\end{array}$

So, the volume of Neptune in cubic meter is $63.9173\times {10}^{21}{\text{m}}^3$ .

The conversion of kilogram into gram is:

  $1\text{kg}=1000\text{g}$

To find the conversion of $1.02\times {10}^{26}\text{kg}$ , multiply both sides of above conversion by $1.02\times {10}^{26}\text{kg}$ .

  $\begin{array}{c}1.02\times {10}^{26}\times 1\text{kg}=1.02\times {10}^{26}\times 1000\text{g}\\ =\text{1}\text{.02}\times {10}^{26}\times {10}^3\text{g}\end{array}$

Apply the property of addition of exponent of same base on the right hand side of the above expression.

  $\begin{array}{c}1.02\times {10}^{26}\times 1\text{kg}=\text{1}\text{.02}\times {10}^{26}\times {10}^3\text{g}\\ =1.02\times {10}^{29}\text{g}\end{array}$

So, the mass of Neptune in grams is $1.02\times {10}^{29}\text{g}$ .

Formula to determine density of Neptune is:

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

Substitute $1.02\times {10}^{29}\text{g}$ for mass and $63.9173\times {10}^{21}{\text{m}}^3$ for volume in the above expression.

  $\begin{array}{c}\text{Density}=\frac{1.02\times {10}^{29}\text{g}}{63.9173\times {10}^{21}{\text{m}}^3}\\ =\frac{1.02}{63.9173}\times \frac{{10}^{29}}{{10}^{21}}\frac{\text{g}}{{\text{m}}^{\text{3}}}\end{array}$

Apply the property of subtraction of exponent of same base on the right hand side of the above expression.

  $\begin{array}{c}\text{Density}=\frac{1.02}{63.9173}\times {10}^{29-21}\frac{\text{g}}{{\text{m}}^{\text{3}}}\\ =0.01596\times {10}^8\frac{\text{g}}{{\text{m}}^{\text{3}}}\\ =1.596\times {10}^6\frac{\text{g}}{{\text{m}}^{\text{3}}}\\ \approx 1.6\times {10}^6\frac{\text{g}}{{\text{m}}^{\text{3}}}\end{array}$

Therefore, the density of Neptune in grams per cubic meter is $1.6\times {10}^6\frac{\text{g}}{{\text{m}}^{\text{3}}}$ .