Using the Bohr model, estimate the wavelength ë of the Ká characteristic X-ray for a metal an atom of which contains Z = 50 protons.
1/L = 1.1*10^7(49)^2 * (1-1/4)
= 1980*10^7
so L = 5.05*10^-4*10^-7
= 5.05 * 10^-11 m
Elena showed you how to do this sort of problem. Try it.
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old problem:
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http://www.jiskha.com/display.cgi?id=1335130177
It was wrong and I still don't get how to do it!
for Ka series m = 2 though I think
physics help! - Elena, Sunday, April 22, 2012 at 5:58pm
Moseley's law
1/λ = R• (Z –σ)^2•(1/n^2 -1/m^2),
where
R=1.1•10^7 m^-1 is the Rydberg constant,
Z = 50,
σ = 1 for K-series,
for K-β: n = 1, m = 3.
So
1/λ = 1.1•10^7 • (50 –1)^2•(1 - 1/9) =…
λ = 4.26•10^-11 m
try using m = 2, small difference of 1/4 instead of 1/9
That makes so much more sense! Thanks so much!!!
To estimate the wavelength (λ) of the Ká characteristic X-ray using the Bohr model, we need to use the following formula:
λ = hc / (E_final - E_initial)
Where:
λ is the wavelength of the X-ray
h is Planck's constant (6.62607015 × 10^-34 J·s)
c is the speed of light (3.00 × 10^8 m/s)
E_final is the energy of the final state
E_initial is the energy of the initial state
In the Bohr model, the energy levels of an atom are given by:
E_n = -13.6 eV / n^2
Where:
E_n is the energy of the nth energy level
eV is electron volt (1.602 × 10^-19 J)
n is the principal quantum number
For the Ká characteristic X-ray, we have to consider the transition from the M shell (n=3) to the K shell (n=1). So, using the above formulas, we can calculate the wavelength:
E_final = -13.6 eV / 1^2 = -13.6 eV
E_initial = -13.6 eV / 3^2 = -1.51 eV
Converting the energies to joules:
E_final = -13.6 eV * (1.602 × 10^-19 J/eV) = -2.179 × 10^-18 J
E_initial = -1.51 eV * (1.602 × 10^-19 J/eV) = -2.418 × 10^-19 J
Now we can substitute these values into the formula:
λ = (6.62607015 × 10^-34 J·s * 3.00 × 10^8 m/s) / (-2.179 × 10^-18 J - (-2.418 × 10^-19 J))
Simplifying the equation:
λ = (1.987 × 10^-25 J·m) / (1.937 × 10^-18 J)
Calculating the wavelength:
λ ≈ 1.025 × 10^-7 m
Therefore, the estimated wavelength (λ) of the Ká characteristic X-ray for a metal atom with 50 protons is approximately 1.025 × 10^-7 meters.