the electron in a ground state h atom absorbs a photon of wavelength 102.57nm to what energy level does the electron move

To determine the energy level to which the electron moves, we can use the formula for the energy of a photon:

E = hc / λ

Where:
E = energy of the photon
h = Planck's constant (6.626 x 10^-34 Js)
c = speed of light (3.00 x 10^8 m/s)
λ = wavelength of the photon

First, let's convert the given wavelength from nanometers to meters:

102.57 nm = 102.57 x 10^-9 m

Now, we can substitute the values into the formula and solve for the energy:

E = (6.626 x 10^-34 Js) * (3.00 x 10^8 m/s) / (102.57 x 10^-9 m)

Calculating the expression gives us:

E ≈ 1.220 x 10^-18 J

Next, we need to find the energy levels of the hydrogen atom. The energy levels are given by the formula:

E = -13.6 eV / n^2

Where:
E = energy level
n = quantum number

To find the value of n, we can rearrange the formula:

n = √(-13.6 eV / E)

Now, let's substitute the value of E we calculated earlier:

n = √(-13.6 eV / 1.220 x 10^-18 J)

Note: 1 eV = 1.602 x 10^-19 J

n = √(-13.6 eV / (1.220 x 10^-18 J / (1.602 x 10^-19 J/eV)))

Calculating the expression gives us:

n ≈ √(-13.6 eV / 0.762 eV)

n ≈ √(-17.8)

Since we're dealing with positive values for n, we ignore the negative sign. Therefore:

n ≈ √17.8

n ≈ 4.216

The electron moves to approximately the 4th energy level (n = 4) in the hydrogen atom when it absorbs a photon with a wavelength of 102.57 nm.

To determine the energy level to which the electron in a hydrogen atom moves when it absorbs a photon of a specific wavelength, we need to make use of the Rydberg formula:

1/λ = R * (1/n1^2 - 1/n2^2)

where:
- λ is the wavelength of the absorbed photon
- R is the Rydberg constant (approximately 1.097 x 10^7 m^-1)
- n1 is the initial energy level of the electron (which is 1, since it is in the ground state)
- n2 is the final energy level we need to find

First, we need to convert the given wavelength from nm to meters:
102.57 nm = 102.57 x 10^-9 m

Now we can substitute the values into the formula and solve for n2:

1/(102.57 x 10^-9) = (1.097 x 10^7) * (1/1^2 - 1/n2^2)

Simplifying the equation, we get:

1.097 x 10^7 * (1 - 1/n2^2) = 1/(102.57 x 10^-9)

Now we can solve for n2 by rearranging the equation:

1 - 1/n2^2 = 1/(102.57 x 10^-9) / (1.097 x 10^7)

1 - 1/n2^2 = 0.00908678

Simplifying further:

1/n2^2 = 1 - 0.00908678

1/n2^2 = 0.99091322

Taking the square root of both sides:

1/n2 = 0.99492237

Solving for n2:

n2 = 1 / 0.99492237

n2 ≈ 1.00511

Therefore, the electron in a ground state hydrogen atom will move to the energy level approximately equal to 1.00511.

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