For a helium atom containing two protons, two neutrons and two electrons:
A)Calculate the first energy levels(n=1,2,3,4,5)(answers (-54.4eV, -6.04eV, -3.4eV, -2.176eV)
b)Calculate the frequency in Hz of a proton emitted from n=2 to n=1 (answer 9.86x 10^15Hz)
c)calculate the wavelength in meters of this emitted photon (answer 3 x 10^-8 m)
To answer these questions, we need to understand the energy levels and transitions within an atom.
a) Calculate the first energy levels (n=1,2,3,4,5):
The energy levels of an atom can be calculated using the equation:
E = -13.6 eV / n^2
Where E is the energy level in electron volts (eV), and n is the principal quantum number.
For a helium atom (with two protons, two neutrons, and two electrons), we'll calculate the energy levels for n=1, 2, 3, 4, and 5:
For n=1:
E1 = -13.6 eV / (1^2) = -13.6 eV = -1.096 x 10^-18 Joules
For n=2:
E2 = -13.6 eV / (2^2) = -13.6 eV / 4 = -3.4 eV = -2.176 x 10^-18 Joules
For n=3:
E3 = -13.6 eV / (3^2) = -13.6 eV / 9 ≈ -1.511 eV ≈ -9.64 x 10^-19 Joules
For n=4:
E4 = -13.6 eV / (4^2) = -13.6 eV / 16 ≈ -0.85 eV ≈ -5.44 x 10^-19 Joules
For n=5:
E5 = -13.6 eV / (5^2) = -13.6 eV / 25 ≈ -0.544 eV ≈ -3.469 x 10^-19 Joules
So, the first energy levels for a helium atom are approximately -54.4 eV, -6.04 eV, -3.4 eV, -2.176 eV, and so on.
b) Calculate the frequency in Hz of a photon emitted from n=2 to n=1:
The frequency of a photon emitted during a transition between energy levels can be calculated using the equation:
ΔE = E2 - E1
Where ΔE is the difference in energy levels between the initial (E2) and final (E1) states.
For the n=2 to n=1 transition in a helium atom, we'll use the energy levels calculated above:
ΔE = E2 - E1 = (-2.176 x 10^-18 Joules) - (-1.096 x 10^-18 Joules) = -1.08 x 10^-18 Joules
The frequency (ν) of the emitted proton can be calculated using the equation:
ν = ΔE / h
Where h is Planck's constant (approximately 6.626 x 10^-34 J·s).
ν = (-1.08 x 10^-18 Joules) / (6.626 x 10^-34 J·s) ≈ 1.63 x 10^15 Hz
Therefore, the frequency of the proton emitted from n=2 to n=1 is approximately 1.63 x 10^15 Hz or 1.63 THz.
c) Calculate the wavelength in meters of this emitted photon:
The wavelength (λ) of a photon can be calculated using the equation:
λ = c / ν
Where c is the speed of light (approximately 3 x 10^8 m/s) and ν is the frequency of the photon.
λ = (3 x 10^8 m/s) / (1.63 x 10^15 Hz) ≈ 1.84 x 10^-7 m
Therefore, the wavelength of the emitted photon is approximately 1.84 x 10^-7 meters or 1.84 x 10^-8 cm.