Answer:
6.03x 10^-10 m
Explanation:
Given that
E= hc/ wavelength
And also
E= h²n2/8mL²
Equating the two and if we say the transition was from energy level 1 to 2 then
E2 - E1 = h²2/(8mL²) x ( 2² - 1²) = 3h²2/(8mL²)
So
L² = 3 h lambda / (8mc)
= 3 x6.626 10^-34 kg m^2/s x 400 10^-9 m /( 8 x 9.11 x10^-31 kg x3.00 10^8 m/s)
= 36.4 x 10^-20 m^2
L = 6.03 x 10^-10 m
The length of the box that absorbs the light is;
L = 6.03 × 10^(-10) m
We are given;
Longest wavelength of spectrum; λ = 400 nm = 400 × 10^(-9) m
Now, the formula for energy of quantization is;
E = h²n²/8mL²
Also, Energy of a photon is;
E = hc/λ
Thus;
hc/λ = h²n²/8mL²
h will cancel out to give;
c/λ = hn²/8mL²
Where;
h is Planck's constant = 6.626 × 10^(-34) m².kg/s
c is speed of light = 3 × 10^(8) m/s
λ is wavelength = 400 × 10^(-9) m
L is length of box
m is mass of electron = 9.11 × 10^(-31) kg
n² is difference in energy levels = (2² - 1²) = 3
Making L the subject gives;
L = √(hn²λ/8mc)
Thus;
L = √((6.626 × 10^(-34) × 3 × 400 × 10^(-9))/(8 × 9.11 × 10^(-31) × 3 × 10^(8))
L = √(3.636663007683 × 10^(-19))
L = 6.03 × 10^(-10) m
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