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## Homework Statement

I'm a little bit stuck with this exercise.

A small body with temperature T and emissivity ε is placed in a large evacuated cavity with interior walls kept at temperature Tw. When Tw-T is small, show that the rate of heat transfer by radiation is

$$

\frac{dQ}{dt}=4T_{W}^3A\epsilon \sigma (T_{W}-T)

$$

If the body remains at constant pressure, show that the time for the temperature of the body to change from T1 to T2 is given by

$$

t=\frac{C_{P}}{4T_{W}^3A\epsilon \sigma}\ln(\frac{T_{W}-T_{1}}{T_{W}-T_{2}}).

$$

Two small blackened spheres of identical size, one of copper and the other of aluminum, are suspended by silk threads within a large hole in a block of melting ice. It is found that it takes 10 min for the temperature of the aluminum to drop from 276K to 274K, and 14.2 min for the copper to drop the same interval of temperature. What is the ratio of specific heats of aluminum and copper? (The densities of Al and Cu are 2.70 x 10^3 kg/m^3 and 8.96 x 10^3 kg/m^3 at 25°C, respectively.)

## Homework Equations

I think that all relevant equations at given in the problem.

## The Attempt at a Solution

I have already been able to solve the second part. So, I already proofed the equation to obtain the time.

For the first part, I start with: dQ/dt=Aεσ(T

_{W}

^{4}-T

^{4}). I thought for a moment to split (T

_{W}

^{4}-T

^{4}) so that it says (T

_{W}+T)(T

_{W}

^{2}+T

^{2})(T

_{W}-T) but then I do not know how to get the desired result.

And in the third part I'm a little bit stuck with the calculations. I assume that it works out with the equation of the second part, transformed so that it says $C_P=$... But so that it is the specific heat I have to get it "per unit mass". And then I have to divide them to get the ratio of specific heats. So, the Boltzmann constant and $T_W^3$ would cancel out. But I don't know where the densities come into the game and how to go on. I asume that the densities have to do something with the area.

I would appreciate some help.