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Magazine Chapter 10: Q7P (page 260)
(III) The Earth is not a uniform sphere, but has regions of varying density. Consider a simple model of the Earth divided into three regions鈥攊nner core, outer core, and mantle. Each region is taken to have a unique constant density (the average density of that region in the real Earth):
(a) Use this model to predict the average density of the entire Earth. (b) If the radius of the Earth is 6380 km and its mass is 5.98 脳 1024Kg, determine the actual average density of the Earth and compare it (as a percent difference) with the one you determined in (a).
Short Answer
(a) The average density of the earth from the densities of three regions is 5500.5 kg/m3.
(b) The actual density of the earth is 5497.3 kg/m3 and the percentage difference in density is -0.058%.
Step by step solution
Step-1: Density of Earth
The density of the entire earth is the total mass of the earth divided by the total volume of the earth.
Given the data
The density of the inner core is 蟻1 = 13000 kg/m3.
The radius of the inner core is r1 =1220 km.
The density of the outer core is 蟻o =11100kg/m3.
The radius of the outer core is ro = 3480km.
The density of the mantle is 蟻M = 4400 kg/m3.
The radius of the mantle is rM = 6380 Km.
The mass of the earth is mE = 5.98 脳 1024kg.
Step-3- Calculation of the average density of the earth
The average density of the Earth is calculated as;
\begin{aligned}\rho=\frac{{{\rho_{\rm{I}}}{V_{\rm{I}}}+{\rho_{\rm{O}}}{V_{\rm{O}}}+{\rho_{\rm{M}}}{V_{\rm{M}}}}}{{{V_{\rm{I}}}+{V_{\rm{O}}}+{V_{\rm{M}}}}}\\=\frac{{{\rho_{\rm{I}}}\left({\frac{4}{3}\pi r_{\rm{I}}^3}\right)+{\rho_{\rm{O}}}\left({\frac{4}{3}\pi\left({r_{\rm{O}}^3-r_{\rm{I}}^3}\right)}\right)+{\rho_{\rm{M}}}\left({\frac{4}{3}\pi\left({r_{\rm{M}}^3-r_{\rm{O}}^3}\right)}\right)}}{{\left({\frac{4}{3}pi r_{\rm{I}}^3}\right)+\left({\frac{4}{3}\pi\left({r_{\rm{O}}^3-r_{\rm{I}}^3}\right)}\right)+\left({\frac{4}{3}\pi\left({r_{\rm{M}}^3-r_{\rm{O}}^3}\right)}\right)}}\end{aligned}
Here VI is the volume of the inner core, VO is the volume of the outer core, and VM is the volume of the mantle.
On rearranging and substituting the known values of the above equation:
The average density of the Earth is 5500.5 kg/m3.
Calculation of the actual density of the Earth
The density of the Earth from the given mass of the Earth is calculated as;
\begin{aligned}{\rho_{{\rm{actual}}}}=\frac{{{m_{\rm{E}}}}}{{{V_{\rm{E}}}}}\\{\rho_{{\rm{actual}}}}=\frac{{5.98\times{{10}^{24}}{\rm{kg}}}}{{\frac{4}{3}\pi{{\left({6380\;{\rm{km}}\times\frac{{1000{\rm{m}}}}{{1{\rm{km}}}}}\right)}^3}}}\\{\rho_{{\rm{actual}}}}=5497.3{\rm{kg/}}{{\rm{m}}^3}\end{aligned}
Here, VE is the volume of the Earth.
The actual density of the Earth is 5497.3 kg/m3.
The percentage of the difference between the actual density and the calculated density is,
\begin{aligned}\left({\frac{{{\rho_{{\rm{actual}}}}-\rho}}{\rho}}\right)\times100=\frac{{\left({5497.3{\rm{kg/}}{{\rm{m}}^3}-5500.5{\rm{kg/}}{{\rm{m}}^3}}\right)}}{{5500.5{\rm{ kg/}}{{\rm{m}}^3}}}\times100\\=-0.058\%\end{aligned}
The percentage difference in density is -0.058%.
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