How Does a Black Body Balloon Behave in a Vacuum?

[eric.mercer92]

Homework Statement

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 T_i, slowly enough so that it reaches thermal equilibrium with the gas, and then it is sealed off. It has radius r_i at this time and contains N atoms. The vacuum is large, so radiation from its walls can be ignored.

a) Show that T/T_i = (r/r_i)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.

Homework Equations

PV = nRT
dQ/dt = σT⁴ * (4$\pi$r²)

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:
(9Nk_BT_ir²r') / (2r_i³)
I am unsure how to get that point.

 

[anathema]

Thank you for presenting your problem and showing your attempt at a solution. I am a scientist and I would be happy to assist you.

For part b), you are correct in using the equation dQ/dt = σT^4 * (4πr^2) to calculate the rate of radiation of energy. This equation is known as the Stefan-Boltzmann law, which describes the relationship between the temperature and radiation of a blackbody. However, in this case, the balloon is not a blackbody as it is not in thermal equilibrium with its surroundings. We need to modify the equation to account for this.

We can use the ideal gas law, PV = nRT, to relate the pressure and volume of the gas inside the balloon. Since the pressure is independent of the radius, we can write PV = constant. We also know that the number of moles of gas, n, remains constant in this scenario. Combining these two equations, we get:

PV = nRT = constant

Rearranging, we get:

P = constant/V

Since the pressure is constant, we can substitute this into the Stefan-Boltzmann law to get:

dQ/dt = σT^4 * (4πr^2) * (constant/V)

Now, we can also relate the volume to the radius of the balloon using the equation for the volume of a sphere, V = (4/3)πr^3. Substituting this into the equation, we get:

dQ/dt = σT^4 * (4πr^2) * (constant/(4/3)πr^3)

Simplifying, we get:

dQ/dt = (3σT^4 * r) / (4ri^2)

This is the rate of radiation of energy per second when the balloon is at radius r.

For part c), we need to find the rate of change of internal energy of the gas. We can use the first law of thermodynamics, which states that the change in internal energy of a system is equal to the heat added to the system minus the work done by the system. In this case, there is no work being done, so we can write:

dU/dt = dQ/dt

We already know the rate of radiation of energy, dQ/dt, from part b). All we need to do is find the change in