How much will the average gas density in the ball change if, for the same gas mass, the ball has tripled its radius?

Given:

m1 = m2 = m – the mass of the gas in the ball has not changed;

r2 = 3 * r1 – the radius of the ball has tripled.

It is required to determine ro2 / ro1 – how much the average gas density in the ball has changed.

Find the initial volume of the ball;

V = 4 * n * r1 ^ 3/3, where n is the Pythagorean number (geometric constant).

Then, in the first case, the average gas density will be equal to:

ro1 = m1 / V1 = m / (4 * n * r1 ^ 3/3) = 3 * m / (4 * n * r1 ^ 3).

The changed volume of the ball will be equal to:

V2 = 4 * n * r2 ^ 3/3 = 4 * n * (3 * r1) ^ 3/3 = 4 * 3 * n * r1 ^ 3 = 12 * n * r1 ^ 3.

Then, in the second case, the average gas density will be:

ro2 = m ^ 2 / V ^ 2 = m / (12 * n * r13).

I.e:

ro2 / ro1 = (m / (12 * n * r1 ^ 3)) / (3 * m / (4 * n * r1 ^ 3)) = 4 / (12 * 3) = 1 / (3 * 3) = 1/9, that is, it will decrease by 9 times.

Answer: if the radius is increased by 3 times, the average density of the sphere will decrease by 9 times.