Soon after Earth was formed, heat released by the decay of radioactive elements raised the average internal temperature from$\mathbf{300}\mathbf{}\mathbf{\text{\hspace{0.17em}\hspace{0.17em}to}}\mathbf{}\mathbf{3000}\mathbf{}\mathbf{\text{K}}$, at about which value it remains today. Assuming an average coefficient of volume expansion of$3.0\times {10}^{-5}{\text{K}}^{\text{-1}}$, by how much has the radius of Earth increased since the planet was formed?

I) Initial Temperature $T_i=300\text{K}$,

II) Final Temperature $T_f=3000\text{K}$,

III) Coefficient of volume expansion $\beta =3.0\times {10}^{-5}\text{/K}$,

IV) Radius of the Earth, $R_E=6.4\times {10}^3\text{km}$.

The propensity of matter to alter its form, area, volume, and density in reaction to a change in temperature is known as thermal expansion. The coefficient of volumetric expansion and the temperature rise in the earth is given. We have the relation between the coefficient of linear expansion and the coefficient of volumetric expansion. Using this relationship and formula for the coefficient of linear expansion, we can find the increase in the radius of the earth.

Formula:

Linear expansion of a body due to thermal radiation $\Delta L=L\alpha \Delta T$,…(i)

where$\alpha$ is coefficient of thermal expansion $L$, is the length, and $\Delta T$is change in temperature.

The relation of the linear coefficient of expansion to the coefficient of volume expansion is

$\alpha =\frac{1}{3}\left(\beta \right)$ …(ii)

where$\beta$ is the coefficient of volume expansion.

Using equation (ii) and the given value of coefficient of volume expansion, the value of coefficient of linear expansion can be found to be

$\begin{array}{c}\alpha =\frac{3\times {10}^{-5}\text{/K}}{3}\\ =1\times {10}^{-5}/K\end{array}$

The increase in radius of the Earth using equation (i) can be found to be

$\begin{array}{c}\Delta L=(6.4\times {10}^3\text{\hspace{0.17em}km})\times (1\times {10}^{-5}\text{/K})\times (3000\text{\hspace{0.17em}K}-300\text{\hspace{0.17em}K})\\ =1.72\times {10}^2\text{km}\end{array}$

Hence, the increase in the radius of the Earth is $1.72\times {10}^2\text{km}$.