Question

Determine ${\mathbf{\lambda }}_{\mathbf{m}}$, the wavelength at the peak of the Planck distribution, and the corresponding frequency f, at these temperatures: (a) 3.00 K; (b) 300 K; (c)3000 K.

Short answer

1. ${\lambda }_m=9.67\times {10}^{-4};f=3.10\times {10}^{11}Hz$
   
2. ${\lambda }_m=9.67\times {10}^{-6};f=3.10\times {10}^{13}Hz$
   
3. ${\lambda }_m=9.67\times {10}^{-7};f=3.10\times {10}^{14}Hz$

Step-by-step solution

Determine the wavelength and frequency at t = 3.00 K

Use Wien displacement law to find the wavelength

And, the relation between the wavelength and the frequency of a photon is

a. At T = 3.00 K,

The wavelength is

And, frequency is

Therefore, at t = 3.00 K the wavelength at the peak of the Planck’s distribution is and the corresponding frequency is.

Determine the wavelength and frequency at t = 300 K   

b. Substitute the values in the equation found

And, frequency is

Therefore, at t = 300 K the wavelength at the peak of the Planck distribution is and the corresponding frequency is

Determine the wavelength and frequency at t = 3000 K

c. Substitute the values in the equation found

And, frequency is

Therefore, at t = 3000 K the wavelength at the peak of the Planck distribution is , and the corresponding frequency is.