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.
To estimate the wavelength of the Ká characteristic X-ray using the Bohr model, we need to calculate the energy difference between the K and á electron orbits, and then use that energy difference to calculate the wavelength. Here are the steps to follow:
Step 1: Determine the atomic number (Z) of the element. In this case, Z = 50, which represents tin (Sn).
Step 2: Calculate the energy difference between the K and á electron orbits using the formula:
ΔE = 13.6 * (Z^2 / n₁^2 - Z^2 / n₂^2) eV
Where:
ΔE = Energy difference in electron volts (eV)
Z = Atomic number of the element
n₁ = Principal quantum number of the K electron orbit (n₁ = 1)
n₂ = Principal quantum number of the á electron orbit (n₂ = 2)
Plugging in the values, we get:
ΔE = 13.6 * (50^2 / 1^2 - 50^2 / 2^2) eV
Simplifying the equation:
ΔE = 13.6 * (2500 - 625) eV
ΔE = 13.6 * 1875 eV
ΔE = 25500 eV
Step 3: Convert the energy difference from electron volts to joules using the conversion factor:
1 eV = 1.602 × 10^(-19) J
So, the energy difference ΔE in joules is:
ΔE = 25500 eV * (1.602 × 10^(-19) J / 1 eV)
ΔE = 4.086 × 10^(-15) J
Step 4: Calculate the wavelength (λ) of the X-ray using the equation:
E = h * c / λ
Where:
E = Energy of the X-ray in joules (E = ΔE)
h = Planck's constant, approximately 6.626 × 10^(-34) J·s
c = Speed of light, approximately 3 × 10^8 m/s
λ = Wavelength of the X-ray
Rearranging the equation to solve for λ:
λ = h * c / E
Plugging in the values, we get:
λ = (6.626 × 10^(-34) J·s * 3 × 10^8 m/s) / (4.086 × 10^(-15) J)
λ = (1.9878 × 10^(-25) J·m) / (4.086 × 10^(-15) J)
λ = 4.861 × 10^(-11) m
Therefore, the estimated wavelength (λ) of the Ká characteristic X-ray emission for an atom of tin (Z = 50) is approximately 4.861 × 10^(-11) meters.