the following values are the only energy levels of a hypothetical one electron atom:
E6= -2x10^-19J,
E5= -7x10^-19L
E4= -11x10^-19J,
E3= -15x10^-19J
E2= -17x10^-19J,
E1= -20x10^-19J
a: if the electon were in the n = 3 level, what would be the highest frequency and minimum wavelength of radiation that could be emitted?
b: what is the ionization energy in kJ/mol of the atom in its ground state?
Highest frequency and lowest wavelength. Small wavelength means highest energy. Emitted means given off. So the electron must fall from the n = 3 to the n = 1 level.
delta E = E3 - E1 = hc/wavelength
Solve for wavelength.
b. For ionization energy that is the energy required to remove the outside electron to infinity where E = 0. Assuming the single electron is in the E = 1 level (the ground state) then delta E = 20 x 10^-19 J. That is for the 1 electron in a single atom. Multiply by 6.02E23 to find for a mole and convert to kJ.
a) To find the highest frequency and minimum wavelength of radiation that could be emitted when the electron is in the n = 3 level, we need to calculate the energy difference between the n = 3 level and the lowest level, and then use the relationship between energy, frequency, and wavelength.
Step 1: Calculate the energy difference between the n = 3 level and the lowest level (n = 1).
ΔE = E3 - E1 = -15x10^-19J - (-20x10^-19J) = 5x10^-19J
Step 2: Use the equation E = hν, where E is the energy, h is Planck's constant (6.626x10^-34 J·s), and ν is the frequency.
ν = ΔE / h = (5x10^-19J) / (6.626x10^-34 J·s) ≈ 7.56x10^14 Hz
Step 3: Convert the frequency to the wavelength using the equation c = λν, where c is the speed of light (3x10^8 m/s) and λ is the wavelength.
λ = c / ν ≈ (3x10^8 m/s) / (7.56x10^14 Hz) ≈ 3.97x10^-7 m
So, the highest frequency of radiation that could be emitted is approximately 7.56x10^14 Hz, and the minimum wavelength of radiation is approximately 3.97x10^-7 m.
b) The ionization energy is the energy required to remove an electron from an atom in its ground state. In this case, we want to find the ionization energy of the atom in its ground state.
Step 1: Identify the energy level of the ground state, which is the lowest level (n = 1) in the given values.
E1 = -20x10^-19J (in the problem statement)
Step 2: The ionization energy is the difference between the energy of the ground state and the energy of a completely ionized atom (i.e., when the electron is no longer bound to the atom).
Ionization energy = E atom - E ionized atom
In this case, since the electron is completely removed when the atom is ionized, the energy of the completely ionized atom is zero.
Ionization energy = E1 - E ionized atom = E1 - 0 = -20x10^-19J
Step 3: Convert the energy to kilojoules (kJ) and per mole (mol).
Ionization energy = (-20x10^-19J) * (1 kJ / 1000 J) * (1 mol / 1 J) ≈ -2x10^-23 kJ/mol
Therefore, the ionization energy of the atom in its ground state is approximately -2x10^-23 kJ/mol.