The electron in a hydrogen atom in level n= 8 undergoes a transition to a lower leve by emitting a photon of wavelength 956 mom. What is the final level of the electron?
A) 4
B) 8
C) 9
D) 1
E) 3
1/lambda=R*Z^2 (1/n1^2-1/n2^2)
or (1/n1^2-1/n2^2)=1/lambda*R
Lambda*R=956nm*1.097E7=10.48
and the reciprocal of that is .0954
1/n1^2-1/64=.0954
1/n1^2=.111
n1= sqrt (9)=3
To determine the final level of the electron, we can use the formula for the wavelength of emitted photons:
1 / λ = R * (1 / n₁² - 1 / n₂²)
where λ is the wavelength of the emitted photon, R is the Rydberg constant (approximately 1.097 x 10^7 m⁻¹), n₁ is the initial level, and n₂ is the final level.
Given that the initial level (n₁) is 8 and the wavelength (λ) is 956 nm (or 9.56 x 10⁻⁷ m), we can rearrange the formula to solve for n₂:
1 / λ = R * (1 / n₁² - 1 / n₂²)
1 / (9.56 x 10^-7) = 1.097 x 10^7 * (1 / 8² - 1 / n₂²)
Solving this equation will give us the value of n₂, which represents the final level of the electron.
1.097 x 10^7 * (1 / 8² - 1 / n₂²) = 1 / (9.56 x 10^-7)
Using this equation, we can calculate the value of n₂ to determine the final level of the electron.
To determine the final level of the electron in a hydrogen atom when it undergoes a transition from level n = 8 to a lower level by emitting a photon of wavelength 956 nm, we can use the formula for the relationship between energy levels and the wavelength of light emitted or absorbed:
ΔE = E_final - E_initial = (h * c) / λ
Where:
ΔE is the energy difference between the levels
E_final is the energy of the final level
E_initial is the energy of the initial level
h is Planck's constant (6.626 × 10^-34 J s)
c is the speed of light (3.00 × 10^8 m/s)
λ is the wavelength of the photon
First, we need to calculate the energy difference, ΔE, between the levels. We can rearrange the formula to solve for ΔE:
ΔE = (h * c) / λ
Plugging in the values:
ΔE = (6.626 × 10^-34 J s * 3.00 × 10^8 m/s) / (956 × 10^-9 m)
ΔE = 1.981 × 10^-19 J
Next, we know that the energy difference between energy levels in a hydrogen atom is given by the equation:
ΔE = - (13.6 eV) * (1 / n_final^2 - 1 / n_initial^2)
Rearranging this equation to solve for n_final, the final level of the electron:
(13.6 eV) * (1 / n_final^2 - 1 / 8^2) = 1.981 × 10^-19 J
Now, we can solve for n_final:
(1 / n_final^2 - 1 / 8^2) = (1.981 × 10^-19 J) / (13.6 eV)
Converting eV to Joules (1 eV = 1.6 × 10^-19 J):
(1 / n_final^2 - 1 / 8^2) = (1.981 × 10^-19 J) / (13.6 * 1.6 × 10^-19 J)
(1 / n_final^2 - 1 / 64) = 0.0857
To simplify the equation, we can find a common denominator for the fractions:
(64 - n_final^2) / (n_final^2 * 64) = 0.0857
Cross-multiplying and rearranging the equation:
64 - n_final^2 = 0.0857 * n_final^2 * 64
64 - n_final^2 = 5.48 * n_final^2
6.48 * n_final^2 = 64
n_final^2 = 64 / 6.48
n_final^2 ≈ 9.877
Finally, taking the square root of both sides:
n_final ≈ √9.877
n_final ≈ 3.14
The final level of the electron is approximately 3.
Therefore, the correct answer is E) 3.