Blackbody radiation - derive expression for T

[accountkiller]

Homework Statement

Derive an expression for the temperature of an ideal blackbody so that its radiated light at the peak intensity wavelength has exactly enough energy to excite the ground state to fourth excited state.

Homework Equations

I = σ*T⁴
λ_m*T = 2.9 * 10^-3 m * K
E = hf = hc/λ

The Attempt at a Solution

Well, the peak intensity wavelength is at λ_m = 2.9*10^-3 / T.
I'm not sure what to do with the energy levels... the energy difference between levels is hf... do I use that Rydberg formula with n's?

I've got bits and pieces of information but I can't pull it all together. Could someone guide me in the right direction? Thanks!

 

[D H]

The ground state of what?

 

[accountkiller]

See, it doesn't say the ground state of what, that's also what confused me, but I assumed it was hydrogen since that's the simplest and that's the only thing we've done so far.

But I worked through it and got:

T = -2.9*10^-3 * R * (1/1² - 1/4²)

Can anyone confirm that is correct?

Also, how do I delete a post? I can't seem to find the 'delete' button.

 

[Redbelly98]

T = -2.9*10^-3 * R * (1/1² - 1/4²)

Can anyone confirm that is correct?

Looks good, assuming you use appropriate units and that they meant hydrogen.

Also, how do I delete a post? I can't seem to find the 'delete' button.

Members can delete their own posts for some limited time after posting. (I should know what that time limit is, but I don't ) Otherwise, you can hit the Report button and request that a moderator delete the post -- but you should provide a reasonable justification for doing so, or we will simply leave it intact.

 

[rwisz]

Hello there,

Your attempt at a solution is on the right track. Let's break it down step by step.

First, we know that the peak intensity wavelength is given by λm = 2.9*10^-3 / T, where T is the temperature of the blackbody. This is the wavelength at which the blackbody emits the most energy.

Next, we need to consider the energy levels of the blackbody. The energy difference between the ground state and the fourth excited state is given by hf, where h is Planck's constant and f is the frequency. We can also express this in terms of wavelength using the formula E = hc/λ, where c is the speed of light. So, the energy difference between the ground state and the fourth excited state can also be written as hc/λ.

Now, we want the energy of the emitted light at the peak intensity wavelength to be just enough to excite the ground state to the fourth excited state. This means that the energy of the emitted light (given by hc/λm) must be equal to the energy difference between the ground state and the fourth excited state (given by hc/λ). So, we can set these two equations equal to each other and solve for T:

hc/λm = hc/λ
2.9*10^-3 / T = hc/λ
T = λ/2.9*10^-3

Finally, we can substitute the value of λm = 2.9*10^-3 / T into this equation to get our final expression for T:

T = λm/2.9*10^-3

And there you have it! This is the expression for the temperature of an ideal blackbody so that its radiated light at the peak intensity wavelength has exactly enough energy to excite the ground state to the fourth excited state. I hope this helps guide you in the right direction. Keep up the good work!