Question

The density of mercury (the element) is 13.69 g/cm3. The density of Mercury (the planet) is 5.43 g/cm3. If Mercury (the planet) was made of mercury (the element) and had the same mass, what would its diameter be?

Answer 1

Answer:

The diameter of Mercury would be 0.735 times smaller than the actual diameter i.e., it would become 3590 km from 4879 km

Explanation:

For the sake of simplicity, let us consider Mercury spherical and uniform. The provided data shows that if the Mercury planet is purely composed of mercury then it will require less space as compared to the actual planet, due to the higher density of the mercury. The ratio of the two densities shows that the volume of Mercury would reduce 2.52 times. From the change in volume we can determine the change in diameter by taking the ratio of initial and final states, shown in the following derivation:

$V=(\frac{4}{3})(\pi )(r^{3})$,

Here, V is the volume and r is the radius of the Mercury.

From the ratio of initial and final states, we obtain the following relation

$\frac{V_{2} }{V_{1} } = \frac{r_{2} ^{3}}{r_{1} ^{3}}$

as,

$V_{2}=\frac{V_{1}}{2.52}$

so,

$\frac{1}{2.52} = \frac{r_{2} ^{3}}{r_{1} ^{3}}$

$r_{2}^{3}=\frac{r_{1}^{3}}{2.52}$

$r_{2}= 0.735.r_{1}$

The factor will remain same for the diameter i.e.,

$D_{2}= 0.735 D_{1}$

For absolute diameter of Mercury:

Actual diameter of Mercury = 4879.4 km

Hence, the diameter of Mercury, if it was made of mercury would become 3590 km.

P.S.: The element mercury first letter is kept small while the other is capped