Blackbody radiation - derive expression for T

In summary, the conversation discusses deriving an expression for the temperature of an ideal blackbody so that its radiated light at the peak intensity wavelength can excite the ground state to the fourth excited state. The equation for peak intensity is given as I = σ*T4 and the equation for energy is E = hf = hc/λ. After some confusion about the ground state being referenced to, the final solution is T = -2.9*10-3 * R * (1/12 - 1/42) with the assumption that the ground state is referring to hydrogen. The process of deleting a post is also briefly mentioned.
  • #1
accountkiller
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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
λm*T = 2.9 * 10-3 m * K
E = hf = hc/λ

The Attempt at a Solution


Well, the peak intensity wavelength is at λm = 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!
 
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  • #2
The ground state of what?
 
  • #3
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.
 
  • #4
mbradar2 said:
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 :redface:) 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.
 
  • #5


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!
 

FAQ: Blackbody radiation - derive expression for T

1. What is blackbody radiation?

Blackbody radiation is the electromagnetic radiation emitted by a perfect absorber and emitter of energy, known as a blackbody. It is a type of thermal radiation that is emitted by any object that has a non-zero temperature.

2. What is the expression for temperature (T) in blackbody radiation?

The expression for temperature (T) in blackbody radiation is given by the Wien displacement law, which states that the peak wavelength of blackbody radiation is inversely proportional to the temperature, T. This can be expressed as T = b/λ_max, where b is a constant known as Wien's displacement constant and λ_max is the peak wavelength of the radiation.

3. How is the expression for T in blackbody radiation derived?

The expression for T in blackbody radiation is derived using Planck's law, which is an equation that describes the spectral intensity of blackbody radiation. Planck's law states that the spectral intensity is directly proportional to the temperature and inversely proportional to the fourth power of the wavelength. By equating this law with the Wien displacement law, we can derive the expression for T.

4. What is the significance of T in blackbody radiation?

The temperature (T) in blackbody radiation is significant because it determines the peak wavelength of the emitted radiation. This means that the temperature of an object can be determined by the wavelength at which its blackbody radiation is most intense. T also affects the overall intensity of the radiation, with higher temperatures leading to a greater overall intensity.

5. How does the expression for T in blackbody radiation relate to the Stefan-Boltzmann law?

The expression for T in blackbody radiation is related to the Stefan-Boltzmann law, which states that the total energy radiated from a blackbody is directly proportional to the fourth power of the temperature. This means that as T increases, the total energy emitted by the blackbody also increases. The expression for T in blackbody radiation is used in the Stefan-Boltzmann law to calculate the total energy emitted from a blackbody at a given temperature.

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