What Wavelength Corresponds to hv/kT=1 in Planck's Radiation Law?

[Benzoate]

Homework Statement

Deduce an approximation for Planck's radiation law: To what what wavelength does hv/kT=1

Homework Equations

Wien Distribution equation
B(lambda)= 2hc^2/lambda^5*exp(-hc/kT(lambda))
Rayleigh-jeans approximation
B(lambda)=2ckT/lambda^4

The Attempt at a Solution

My main concern is with the latter part of the question: To what wavelength does hv/kT=1?

My first attempt at finding a wavelength was to equate Rayleigh-jeans approximation equations and the Wien Distribution equations to each other.

(2hc^2/lambda^5)*exp(1/lambda) = 2ckT/lambda^4 . Then to eliminate T from the equation , I would set T= hv/k. Therefore 2ck(hv/k)/(lambda)^4 = 2hc^2/(lambda)^5*(exp(-1/lambda) => lambda=c/2/v. The only problem is , I still have an unwanted unknown, which is the frequency. Do you need the frequency in order to determine the wavelength?

 

[anathema]

Thank you for your post. In order to determine the wavelength at which hv/kT=1, we can use the Wien displacement law, which states that the peak wavelength of a blackbody radiation spectrum is inversely proportional to the temperature. This can be expressed as:

lambda_max = b/T

where b is the Wien displacement constant. Substituting this into the equation hv/kT=1, we get:

hv/kb = 1

Solving for v, we get:

v = kb/h

Substituting this into the equation lambda=c/v, we get:

lambda = c*h/kb

Therefore, the wavelength at which hv/kT=1 is given by the above equation. I hope this helps.

 

[m2003]

Yes, you do need the frequency in order to determine the wavelength. The equation for Planck's radiation law is B(lambda) = (2hc^2/lambda^5)*(1/(e^(hc/kT*lambda) - 1)), where B(lambda) is the spectral radiance at a given wavelength lambda. In order to find the wavelength at which hv/kT = 1, we can rearrange this equation to solve for lambda:

lambda = hc/kT

Since we are given that hv/kT = 1, we can substitute this into the equation above to get:

lambda = c/v

Therefore, the wavelength at which hv/kT = 1 is equal to the speed of light divided by the frequency of the radiation. In order to find a numerical value for this wavelength, we would need to know the specific frequency of the radiation in question.