Question

Calculate the wavelength and frequency at which the intensity of the radiation is a maximum for a black body at 298 K.

Answer 1

Answer:

wavelength (λ) = 2.73 E-4 m

frecuency (f) = 1.10 E12 s-1

Explanation:

Stefan-Boltzmann law:

• intensity of the radiation for a black body (W) = σ.(T)∧4

∴ σ = 5.670 E-8  W/m²(K)∧4

∴ T = 298 K

⇒ W = (5.670 E-8 W/m²K∧4)×(298 K)∧4

⇒ W = 447.145 W/m² = 447.145 J/s.m²

assuming an area of  1 m²

⇒ W = 447.145 J/s

Rayleigh-Jeans's law:

∴ Black body radiation at a specific wavelength (Bλ(T)):

⇒ Bλ(T) = ( 2 c.k.T ) / (λ)∧4

∴ c = 3 E8 m/s.....velocity of speed

∴ k = 1.38065 E -23 J/K.........Boltzmann's constant

∴ Bλ(T) = 447.145 J/s.m²

⇒ (λ)∧4 = ((2)(3 E8 m/s)(1.38065 E-23 J/K)(298 K)) / (447.145 J/s)

⇒ λ = (5.521 E-15)∧(1/4)

⇒ λ = 2.73 E-4 m

• f = c/λ

⇒ f = (3 E8 m/s) / (2.73 E-4 m)

⇒ f = 1.10 E12 s-1

Answer 2

Answer:

λ(max)  = = 9.7 *10^-6m

Frequency(max) = 3.09 * 10^13 /s = 3.09 * 10^13 Hz

Explanation:

Step 1: Data given

Temperature =  298 K

In order to find maximum wavelength and maximum frequency, apply Wien's Law of maximum wavelength * Temperature = 2.9*10^-3 meters * Kelvin

Step 2: Calculate wavelength

λ(max) * T = 2.9 *10^-3 m*K

λ(max)  = (2.9 *10^-3 m*K)/T

λ(max)  = (2.9 *10^-3 m*K)/298

λ(max)  = = 9.7 *10^-6m

Step 3: Calculate frequency

Frequency(max) = c/λ(max)

Frequency(max) = 3*10^8 m/s  / 9.7 *10^-6m

Frequency(max) = 3.09 * 10^13 /s = 3.09 * 10^13 Hz