Suppose that 2.091x10^-18 J is absorbed by the electron of a hydrogen atom in the n=1 energy state. Describe the final energy state of the atom.
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To determine the final energy state of the hydrogen atom, we need to calculate the change in energy between the initial and final states.
The energy change can be calculated using the formula:
ΔE = E_final - E_initial
Given that 2.091x10^-18 J of energy is absorbed by the electron, we can assume that the initial energy state is the ground state (n=1) of the hydrogen atom.
For a hydrogen atom, the energy of an electron in the nth energy level is given by the formula:
E_n = -13.6 eV / n^2
Converting the energy absorbed from joules to electron volts (eV), we can use the conversion factor:
1 eV = 1.6x10^-19 J
Hence, the energy absorbed in eV is:
2.091x10^-18 J * (1 eV / 1.6x10^-19 J) = 1.307 eV
Now, we can determine the final energy state (n_final) by rearranging the equation and solving for n:
E_final = -13.6 eV / n_final^2
n_final = sqrt(-13.6 eV / E_final)
Substituting the absorbed energy (1.307 eV) into the equation, we can calculate the final energy state:
n_final = sqrt(-13.6 eV / 1.307 eV)
= sqrt(-13.6 / 1.307)
≈ sqrt(-10.408)
≈ undefined
It appears that the energy absorbed (1.307 eV) does not correspond to a valid energy state for the hydrogen atom. Therefore, it is not possible to determine the final energy state based on the given information.
To determine the final energy state of the hydrogen atom after absorbing 2.091x10^-18 J of energy, we need to understand the energy levels of hydrogen atoms. The energy levels of hydrogen are described by the equation:
E = -13.6 eV / n^2
Where E is the energy level, n is the principal quantum number, and -13.6 eV is the ionization energy of hydrogen. The energy unit in this equation is electron volts (eV), so we need to convert 2.091x10^-18 J to eV.
To do the conversion, we can use the formula:
1 eV = 1.602x10^-19 J
So, we can calculate the energy in electron volts as:
2.091x10^-18 J * (1 eV / 1.602x10^-19 J) = 13 eV
Now we know that the energy absorbed by the electron is 13 eV.
To find the final energy state of the hydrogen atom, we need to find the value of n for which the energy level E matches the absorbed energy of 13 eV.
Using the equation E = -13.6 eV / n^2 and substituting E = 13 eV, we can solve for n:
13 eV = -13.6 eV / n^2
Rearranging the equation, we get:
n^2 = (-13.6 eV) / (13 eV)
Simplifying further, we have:
n^2 = -13.6 / 13
n^2 = -1.046
Since the square of a value cannot be negative, it means that there is no integer value n that satisfies this equation. Therefore, there is no allowed energy state for the hydrogen atom after absorbing 2.091x10^-18 J of energy in the n=1 energy state.