Chapter 9, Problem 16Q

To determine

The fraction of the total volume of the Earth occupied by the crust, the mantle, and the core respectively.

Answer to Problem 16Q

Solution:

$\text{core: }0.17$, $\text{mantle: }0.82$ and $\text{crust: }0.01$.

Explanation of Solution

Introduction:

The Earth is made up of three portions that are named as following:

1. The crust which is the outermost portion of the Earth having depth of about $30\text{ km}$ from the surface.

2. The core which is the innermost portion of the Earth.

3. The mantle which is in between the crust and the core and constitutes maximum volume of the Earth.

The volume of the Earth is proportional to the cube of its radius. This can be expressed as:

$V\propto r^3$

Here $V$ and $r$ are the volume and radius of the Earth respectively.

Explanation:

The Earth consists of three layers beneath its upper surface. These are the core (innermost), the mantle (middle) and the crust (outermost).

The crust does not go beyond $30\text{ km}$ from the Earth’s surface and exist as a very thin layer in structure, comparatively. The radius of the core $\left(r_{\text{core}}\right)$ is known to be $3500\text{ km}$ and that of the Earth $\left(r_{\text{earth}}\right)$ is $6378\text{ km}$.

The volume of the Earth $\left(V_{\text{earth}}\right)$ is proportional to the cube of its radius $\left(r_{\text{earth}}\right)$ and same goes for the core portion of the Earth. The relation between both the volumes is expressed as:

$\frac{V_{\text{core}}}{V_{\text{earth}}}=\frac{{\left(r_{\text{core}}\right)}^3}{{\left(r_{\text{earth}}\right)}^3}$

Substitute $3500\text{ km}$ for $r_{\text{core}}$ and $6378\text{ km}$ for $r_{\text{earth}}$:

$\begin{array}{c}\frac{V_{\text{core}}}{V_{\text{earth}}}=\frac{{\left(3500\text{ km}\right)}^3}{{\left(6378\text{ km}\right)}^3}\\ =0.17\end{array}$

The volume of the mantle $\left(V_{\text{mantle}}\right)$ is the volume of the core subtracted from the volume of the crust. Therefore, radius of the mantle will be equal to the difference between the cubes of radius of the crust and the radius of the core. Therefore, the proportionality of the volume and the radius for the mantle is written as:

$\frac{V_{\text{mantle}}}{V_{\text{earth}}}=\frac{{\left(r_{\text{earth}}-\text{depth of crust}\right)}^3-{\left(r_{\text{core}}\right)}^3}{{\left(r_{\text{earth}}\right)}^3}$

Substitute $6378\text{ km}$ for $r_{\text{earth}}$, $30\text{ km}$ for depth of crust and $3500\text{ km}$ for $r_{\text{core}}$:

$\begin{array}{c}\frac{V_{\text{mantle}}}{V_{\text{earth}}}=\frac{{\left(6378\text{ km}-30\text{ km}\right)}^3-{\left(3500\text{ km}\right)}^3}{{\left(6378\text{ km}\right)}^3}\\ =0.82\end{array}$

And, the average volume of the crust $\left(V_{crust}\right)$ is the difference between the total volume of the Earth and the combibned volume of the core and the mantle. So, the volume of the crust can be expressed as the difference in the cube of the radius of the Earth and the cube of the radius of mantle and core. Therefore, the proportionality relation are written as:

$\frac{V_{\text{crust}}}{V_{\text{earth}}}=\frac{{\left(r_{\text{earth}}\right)}^3-{\left(r_{\text{earth}}-\text{depth of crust}\right)}^3}{{\left(r_{\text{earth}}\right)}^3}$

Substitute $6378\text{ km}$ for $r_{\text{earth}}$ and $30\text{ km}$ for depth of crust:

$\begin{array}{c}\frac{V_{\text{crust}}}{V_{\text{earth}}}=\frac{{\left(6378\text{ km}\right)}^3-{\left(6378\text{ km}-30\text{ km}\right)}^3}{{\left(6378\text{ km}\right)}^3}\\ =0.01\end{array}$

Conclusion:

Hence, the fraction of the crust to the total volume is $0.01$, the fraction of the mantle is $0.82$ and the rest fraction is the core of the Earth which is $0.17$.