Energy Level Diagram Check

So, in summary, the ground-state energy of a hypothetical atom is -10.0eV, and when illuminated with light, only the wavelengths of 207nm and 146nm are absorbed. The energies of these photons were calculated to be 5.99eV and 8.4935eV, respectively. The energy level diagram for these atoms shows the first excited state at -4.01eV and the second excited state at -1.50685eV. The wavelength of the emission line corresponding to the transition from the second excited state to the first excited state was found to be 495.307nm.
  • #1
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1. Problem:
The ground-state energy of a hypothetical atom is at -10.0eV. When these atoms, in the ground state, are illuminated with light, only the wavelengths of 207 nanometers and 146 nanometers are absorbed by the light atoms (1 nanometer = 10^-9meters).

(a)Calculate the energies of the photons of light of the two absorption-spectrum wavelengths.

(b)Complete the energy level diagram shown below for these atoms by showing all the excited energy states.
vpxdut.jpg


(c)Show by arrows on the energy-level diagram all of the possible transitions that would produce emission-spectrum lines.

(d)What would be the wavelength of the emission line corresponding to the transition from the second excited state to the first excited state?


Homework Equations


E=13.6 [(1/ni^2)-(1/n2^2)]eV

En=-(13.6eV)[(Z^2)/(n^2)]

(1/λ)=(2pi^2mk^2e^4/h^3c)[(1/nf^2)-(1/ni^2)]

2pi^2mk^2e^4/h^3c = 1.097 x 10^7 m

E = hc/λ

hc=1.24 x 10^3eV

The Attempt at a Solution


(a)
207nm = 2.07 x 10^-7m
146nm = 1.46 x 10^-7m
E = hc/λ
E1=[(1.24 x 10^3eV)/(2.07 x 10^-7)]
E1=5.99 x 10^9

E2=[(1.24 x 10^3eV)/1.46 x 10^-7)]
E2=8.493 x 10^9

(b)
I assumed it was bohr model...(?)
Ground state is at -10.0eV
First excited state is 1/4 of ground state
=(1/4)(-10.0eV)
first excited state=-2.5eV
Second excited state is 1/9 of ground state
=(1/9)(-10.0eV)
second excited state is -1.11eV

sorry about the terrible picture...

ohty1e.jpg


(c) is included in picture (?) squiggly arrows..

(d)
ni=2
nf=1

(1/λ)=(2pi^2mk^2e^4/h^3c)[(1/nf^2)-(1/ni^2)]
(1/λ)=(1.097 x 10^7 m)[(1/(1)^2)-(1/(2)^2)]
(1/λ)=(1.097 x 10^7 m)[(1/1)-(1/4)]
(1/λ)=(1.097 x 10^7 m)(1-0.25)
(1/λ)=(1.097 x 10^7 m)(0.75)
(1/λ)=8.2275 x 10^6
1=(8.2275 x 10^6)λ
λ=(1/8.2275 x 10^6)
λ=1.2154 x 10^-7m
 
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  • #2
m_physics said:

Homework Equations


E=13.6 [(1/ni^2)-(1/n2^2)]eV

En=-(13.6eV)[(Z^2)/(n^2)]

(1/λ)=(2pi^2mk^2e^4/h^3c)[(1/nf^2)-(1/ni^2)]

2pi^2mk^2e^4/h^3c = 1.097 x 10^7 m

E = hc/λ

hc=1.24 x 10^3eV
Right number, wrong units. It's actually 1.24 x 103 eV·nm

The Attempt at a Solution


(a)
207nm = 2.07 x 10^-7m
146nm = 1.46 x 10^-7m
E = hc/λ
E1=[(1.24 x 10^3eV)/(2.07 x 10^-7)]
E1=5.99 x 10^9

E2=[(1.24 x 10^3eV)/1.46 x 10^-7)]
E2=8.493 x 10^9
Not quite. Try calculating E1 and E2 again, using the correct units for hc (eV·nm) and λ (nm).

(b)
I assumed it was bohr model...(?)
Not a bad idea, but then the absorbed wavelengths would turn out to be different than the ones they said. So it turns out this one does not follow the Bohr model.

Instead, you can use E1 and E2 to figure out where the energy levels are.
 
  • #3
So to try this again.

For part a)
E1 from 207nm lambda comes out to be 5.99 and E2 from 146nm to be 8.4935

For part b)
You draw those two levels as excited states on the diagram
5555w4.jpg


Then for part d)
I keep thinking of using the same (1/lambda) equation to find the wavelength with nf being 2 and ni being 1 but since this isn't a bohr model, that equation doesn't apply..(?)

Meaning... I would have to use I would have to use E1 and E2 ...
Find the energy w/i E1 and E2 by taking E2 - E1
-8.49-(-5.99)
giving me DeltaE = -2.503eV

Then setting that total e and rearranging E=(hc/lambda) to find lambda giving me
Lambda = (hc/E)
Lambda = [(1.24 x 10^3eV*nm)/(-2.503eV)]
Which gives the final answer lambda = -495.406... 495.406nm wavelength of emission??

Correct? Maybe not?
 
  • #4
m_physics said:
So to try this again.

For part a)
E1 from 207nm lambda comes out to be 5.99 and E2 from 146nm to be 8.4935

You're getting there!

Those are the correct energies of the photons.

If a ground state atom absorbs a 207 nm photon, it's energy will be 5.99 eV higher than it was in the ground state. So, what would be the energy level after absorbing a 207 nm photon?
 
  • #5
a) Calculated E1 = 5.99eV and E2 = 8.49eV as energies given off
b) From ground state, -10.0eV + 5.99eV gives me -4.01eV
-10.0eV + 8.49315 gives me = -1.50685eV
Meaning on my diagram, the first excited state would be -4.01eV and second being -1.50685
d) using lambda = (hc/E)
I first find the energy difference from n=2 to n=1
-1.506 - (-4.01)
change in E = 2.5035eV

Then substituting lambda = [(1.24 x 10^3eV*nm)/2.5035eV)]
lambda = 495.307m
 
  • #6
Looks good :smile:
 

1. What is an energy level diagram?

An energy level diagram is a visual representation of the energy levels or states of a system, typically in the form of a graph or chart. It is commonly used in chemistry and physics to show the distribution of electrons in an atom or the energy states of a molecule.

2. How is an energy level diagram used?

An energy level diagram is used to illustrate the energy states or levels of a system, as well as the transitions between these levels. It helps scientists understand the behavior and properties of a system by showing the energy differences between levels and the probability of transitions occurring.

3. What information can be obtained from an energy level diagram?

An energy level diagram provides information about the energy levels or states of a system, the energy differences between these levels, and the transitions that can occur between them. It can also show the energy absorbed or emitted during a transition and the probability of each transition occurring.

4. How do you create an energy level diagram?

To create an energy level diagram, you first need to determine the energy levels or states of the system. This can be done through calculations or experimental data. Then, you can plot these energy levels on a graph or chart, with the lowest level at the bottom and the highest level at the top. The energy differences between levels can be represented by the spacing between them, and the transitions can be shown with arrows.

5. What is the purpose of a "Energy Level Diagram Check"?

The purpose of an "Energy Level Diagram Check" is to verify the accuracy and consistency of an energy level diagram. This can involve checking for errors in the calculations or data used to create the diagram, ensuring that the transitions and energy differences are correctly represented, and identifying any discrepancies or inconsistencies that may need to be addressed.

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