Calculate the physical properties of a Bohr atom

[Benzoate]

Homework Statement

What is the radius of the n=1 orbit in C^5+ ? What is the energy of the electron in that orbit? What is the wavelength of the radiation emitted by C^5+ in the lyman alpha transition

Homework Equations

1/lambda=R*(1/(n^2)-1/(m^2) ); a(0)=(h/(2*pi))/(m*k*e^2);a(0)=.0529e-9m =5.29e-7 mE=-k*Z^2*e^2/2/((r)); r= a(0)*n^2/Z

The Attempt at a Solution

to find the radius , I know r=a(0)*n^2/Z ; Z is the atomic number. The atomic number of C^+5 is 5. r=(5.29e-7 m)*(1)^2/(5)= 1.06e-7 m

to Find energy E=-k*Z^2*e^2/(2*r)= (9e9)((5)^2)((1.609e-19 J)^2)/(2*(1.06e-7m))=2.74e-20=.170 eV

Now I'm asked to find the wavelength in the lyman alpha transition. Since I'm in the lyman transition , should I say one the transition state n=1? Maybe I don't need to find the other transition state. I could used the fact that 1/lambda=(1/hc)*(E(f)-E(i)) . I know from earlier in the problem that E(i) = .170 eV. For a lyman series E(f)= 13.6 eV . E(f)-E(i)/(hc)=1/lambda => lambda= hc/(E(f)-E(i))

 

[learningphysics]

The Lyman alpha transition is from m = 2... intial orbit, to n = 1... final orbit...

so substitute these values into the formula and you'll get the wavelength.

 

[Benzoate]

how do you know one of the transitions is m=2?

 

[learningphysics]

how do you know one of the transitions is m=2?

I looked it up here:

http://en.wikipedia.org/wiki/Lyman-alpha_line

 

[Benzoate]

so the rest of my calculations are okay? . USing the fact that E(f)-E(i)/(hc)=1/lambda is an option for finding lambda to right since E(f)=13.6 eV for the lyman series

 

[learningphysics]

so the rest of my calculations are okay? . USing the fact that E(f)-E(i)/(hc)=1/lambda is an option for finding lambda to right since E(f)=13.6 eV for the lyman series

I'm not sure... what does C^5+ mean?

 

[Benzoate]

I'm not sure... what does C^5+ mean?

C^5+ is Carbon and it reads that carbon has an atomic number Z +5. I finding some confusion here: according to the periodic table, the atomic number of carbon is 6.

 

[learningphysics]

so the rest of my calculations are okay? . USing the fact that E(f)-E(i)/(hc)=1/lambda is an option for finding lambda to right since E(f)=13.6 eV for the lyman series

Does this problem involve the Bohr model of the atom? In that case En = -13.6ev*Z^2/n^2

the energy at n = 1, comes out to: -13.6*5^2/1^2 = -340eV

I think your calculation was wrong: it should be -(1/2)*k(5e)e/r (this is the potential energy + kinetic energy)

and r =(5.29e-11 m)*(1)^2/(5)= 1.06e-11 m

using these the energy also comes out to: -340eV

I think the best way to get energy here is using En = -13.6ev*Z^2/n^2

So yes, get the energy difference from n=2 to n =1... then set that equal to hc/lambda... then solve for lambda.

 

[Benzoate]

Does this problem involve the Bohr model of the atom? In that case En = -13.6ev*Z^2/n^2

the energy at n = 1, comes out to: -13.6*5^2/1^2 = -340eV

I think your calculation was wrong: it should be -(1/2)*k(5e)e/r (this is the potential energy + kinetic energy)

and r =(5.29e-11 m)*(1)^2/(5)= 1.06e-11 m

using these the energy also comes out to: -340eV

I think the best way to get energy here is using En = -13.6ev*Z^2/n^2

So yes, get the energy difference from n=2 to n =1... then set that equal to hc/lambda... then solve for lambda.

Yes but doesn't E(0)=-13.6 eV only when we are talking about the hydrogen atom

 

[learningphysics]

Yes but doesn't E(0)=-13.6 eV only when we are talking about the hydrogen atom

Ah... z = 6 not 5. sorry about that!

I think the bohr model is for 1-electron atoms... I understand now that carbon 5+ refers to a carbon ion... a carbon atom that has lost 5 of its electrons, and just left with 1...

like you found out carbon has atomic number 6. it lost 5 electrons, so it has just one left... So the bohr model still applies:

En = -13.6ev*Z^2/n^2

here if z = 1, then we're dealing with hydrogen... if we use Z = 6 then we have a carbon nucleus with 1 electron...

E_1 = -13.6*6^2/1^2 = -489.6eV

and using:

-(1/2)*k(6e)e/r will also give -489.6eV

so you can find E_2 etc...

 

[Benzoate]

Ah... z = 6 not 5. sorry about that!

I think the bohr model is for 1-electron atoms... I understand now that carbon 5+ refers to a carbon ion... a carbon atom that has lost 5 of its electrons, and just left with 1...

like you found out carbon has atomic number 6. it lost 5 electrons, so it has just one left... So the bohr model still applies:

En = -13.6ev*Z^2/n^2

here if z = 1, then we're dealing with hydrogen... if we use Z = 6 then we have a carbon nucleus with 1 electron...

E_1 = -13.6*6^2/1^2 = -489.6eV

and using:

-(1/2)*k(6e)e/r will also give -489.6eV

so you can find E_2 etc...

I still have two unknowns: E_2 and lambda. How would I find E_2? Why isn't E_1 -13.6 eV since the wavelength supposed to be in the lyman series

I think I have it : E_1=E_0/n^2= 13.6 eV/1^2=13.6 eV. E_2=E_0/n^2=13.6/2^2=3.4 eV. Only problems I see with this line of reason is how do you know the other transition state is n=2?

 

[learningphysics]

I still have two unknowns: E_2 and lambda. How would I find E_2? Why isn't E_1 -13.6 eV since the wavelength supposed to be in the lyman series

I think I have it : E_1=E_0/n^2= 13.6 eV/1^2=13.6 eV. E_2=E_0/n^2=13.6/2^2=3.4 eV.

this is only for hydrogen... for other atoms:

En = -13.6ev*Z^2/n^2

so for z = 6, E1 = -489.6 eV... using the same formula E2 = -122.4eV

Only problems I see with this line of reason is how do you know the other transition state is n=2?

I think that is just what "Lyman alpha transition" means.