# Black body balloon in vacuum

1. Mar 12, 2012

### eric.mercer92

1. The problem statement, all variables and given/known data
There is a perfectly spherical balloon with surface painted black. It is placed in a perfect vacuum. It is gently inflated with an ideal mono-atomic gas at Kelvin temperature Ti, slowly enough so that it reaches thermal equilibrium with the gas, and then it is sealed off. It has radius ri at this time and contains N atoms. The vacuum is large, so radiation from its walls can be ignored.

a) Show that T/Ti = (r/ri)3 if the pressure inside the balloon is independent of its radius.

b) How much energy does the balloon radiate per second when it is at radius r? Express your answer in terms of r and constants.

c) What is the rate of change of the internal energy of the gas? Express you answer in terms of r, r', and constants.

2. Relevant equations
PV = nRT
dQ/dt = σT4 * (4$\pi$r2)

3. The attempt at a solution

I already showed a), so I don't need help with that.
In part b), I wrote pretty much
dQ/dt = σT4 * (4pi*r2)
so that should be the speed of radiation of energy.
I also have no idea how to do part c), so any help on c) is welcome.

EDIT: For c) I am supposed to get:
(9NkBTir2r') / (2ri3)
I am unsure how to get that point.

Last edited: Mar 13, 2012
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