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 calculate the energy levels of a helium atom, we can use the equation for the energy of an electron in an atom based on the Bohr model:
E = -13.6 eV / n²
where E is the energy, -13.6 eV is the ionization energy of hydrogen, and n is the principal quantum number representing the energy level.
a) To calculate the energy levels for a helium atom:
For n = 1:
E1 = -13.6 eV / (1)² = -13.6 eV
For n = 2:
E2 = -13.6 eV / (2)² = -3.4 eV
For n = 3:
E3 = -13.6 eV / (3)² = -1.51 eV
For n = 4:
E4 = -13.6 eV / (4)² = -0.85 eV
For n = 5:
E5 = -13.6 eV / (5)² ≈ -0.55 eV
Thus, the first energy levels for a helium atom are approximately -54.4 eV, -6.04 eV, -3.4 eV, -2.176 eV, and -0.55 eV.
b) To calculate the frequency of a proton emitted from n = 2 to n = 1:
We can use the equation for the frequency of a photon emitted during a transition between energy levels:
ΔE = E2 - E1
where ΔE is the change in energy between the two levels, E2 is the initial energy level, and E1 is the final energy level.
ΔE = -3.4 eV - (-6.04 eV) = 2.64 eV
To convert the energy difference to joules, we can use the conversion factor 1 eV = 1.6 x 10⁻¹⁹ J.
ΔE = 2.64 eV * (1.6 x 10⁻¹⁹ J/eV) = 4.224 x 10⁻¹⁹ J
We can use the equation for the frequency of a photon:
E = hf
where E is the energy, h is Planck's constant (6.626 x 10⁻³⁴ J·s), and f is the frequency.
Solving for f:
f = E / h = (4.224 x 10⁻¹⁹ J) / (6.626 x 10⁻³⁴ J·s) ≈ 9.86 x 10¹⁵ Hz
Therefore, the frequency of the proton emitted from n = 2 to n = 1 is approximately 9.86 x 10¹⁵ Hz.
c) To calculate the wavelength of the emitted photon:
We can use the equation for the speed of light:
c = λf
where c is the speed of light (approximately 3 x 10⁸ m/s), λ is the wavelength, and f is the frequency.
Solving for λ:
λ = c / f = (3 x 10⁸ m/s) / (9.86 x 10¹⁵ Hz) ≈ 3 x 10⁻⁸ m
Therefore, the wavelength of the emitted photon is approximately 3 x 10⁻⁸ m.