Lara N.

asked • 07/09/16

A one electron species, Xm, where m is the charge of the one electron species and X is the element symbol, loses its one electron from its ground state when it

absorbs 1.96E-17 J of energy. Using the prior information, the charge of the one electron species is ______.

1 Expert Answer

By:

Steven W. answered • 07/09/16

Tutor
4.9 (4,376)

Physics Ph.D., professional, easygoing, 11,000+ hours tutoring physics
About this tutor ›
About this tutor ›
The two key pieces of information here are that:
 
(a)  The atom has only one electron, and is thus "hydrogen-like," or hydrogenic.  This means Bohr theory applies very well to it
 
(b)  If by "loses its electron out of the ground state," it is meant that the electron is liberated from the atom altogether, then that means 1.96E-17 J is the ionization energy from the ground (n=1, in Bohr theory) state.
 
I don't know how much Bohr theory you have been given, and how much you have been expected to derive.  But there is a direct relationship in Bohr theory between ground state ionization energy and the charge Z of the nucleus of the atom, which I believe is what the problem asks you to calculate.  It looks like:
 
E = (Z2(13.6 eV))/n2 
 
There is also a minus sign in the definition, but the main thing is that this is an expression for the energy required to remove the single electron in a given state n from a hydrogenic atom.  Convert the ionization energy to eV and put in n=1 for the ground state, and you can solve for Z.
 
If you would like to check an answer, I could do that.  You should effectively get an integer for Z, and I did.
Upvote • 0 Downvote
More
Report

Lara N.

How do you convert it to eV? 
Report

07/09/16

Lara N.

And how would finding the integer of Z allow me to know what the charge is? The equation we were given is 
 
delta E = -2.178E-18 J * Z^2 * ((1/nfinal^2)-(1/ninitial^2)
Report

07/09/16

Steven W.

tutor
There are Joules to electron-volt conversions in a lot of places, but to do it from the ground up, use the fact that an electron volt is the energy needed to accelerate one electron through a potential difference of 1 volt. 
 
That E = qV = (1.6x10-19 C)(1 V) = 1.6x10-19 J.  Thus, 1 eV = 1.6x10-19 J.  That should allow you to do the conversion.
Report

07/09/16

Steven W.

tutor
Because Z is the atomic number, the number of protons in the nucleus.  If you multiply the atomic number by the charge of the proton, that will be the positive charge of the nucleus.  If you then subtract the negative charge of the one electron, that will be the net total charge of the one-electron species.
Report

07/09/16

Lara N.

Is this correct: 
 
13.6 ev * (1.6E-19 J/1ev) = 2.298E-14 J
 
How do I proceed? Can you help me with rearranging the equation? 
Report

07/09/16

Steven W.

tutor
If you have your equation in Joules, as you showed above, then you do not have to convert to eV (I usually use the expression in eV).  If I read the question right, nfinal = "infinity" (because the electron is fully removed from the atom, so it has gone to the "infinite" energy level, and ninitial = 1, since it starts in the ground state.  Then solve for Z and continue.
Report

07/09/16

Lara N.

However, if I solve for Z^2 then how will I solve without a final E? 
 
It would be -2.178E-18 * Z^2 * ((1-(1.69E-19)^2 - (1)) = E
 
 
Report

07/09/16

Steven W.

tutor
I think there was a math error in your conversion.  But if you do it correctly, you still only derive the equation you already showed me.  It will look like:
 
ΔE = -2.188E-18*Z2 *((1/nfinal)-(1/ninitiial))  where nfinal = "infinity" (very large) as I mentioned above, and ninitial = 1, because it starts in the ground state.
 
Then solve for Z and use it to find the total net charge on Xm
Report

07/09/16

Steven W.

tutor
You have ΔE.  It is given in the problem.
Report

07/09/16

Steven W.

tutor
In this equation, ΔE is the ionization energy, as I mentioned above, the energy needed to take an electron from the ground state to being totally liberated from the atom.
Report

07/09/16

Steven W.

tutor
I need to head out for a while, but I can swing back later and check an answer, or we can arrange to meet online in a session to talk in more detail.
Report

07/09/16

Lara N.

Yes, either one works. I'm just also confused as to whether what n final would be then. 
Report

07/09/16

Steven W.

tutor
Turns out I had a bit longer than I thought.  The key about nfinal in this case is not its exact value, but rather that it is very large, and what that means for 1/nfinal.  Think of it this way:  1/10 is pretty small, 1/100 is smaller, 1/1000 is much smaller, and 1/1,000,000 is very much smaller.  The key is, as n gets really big, 1/n tends toward zero.  So, as a shorthand in a case like this where nfinal is really large, it just means 1/nfinal is effectively zero.
Report

07/09/16

Lara N.

So I got 1.96E-17 = -2.178E-18 J * M * 1
 
M = 8.999 or 9; is that correct? 
Report

07/09/16

Steven W.

tutor
Yes, that is what I get for the atomic number. 
Report

07/09/16

Lara N.

So if I figure out that the element is Fluorine, how do I know what the charge is? Would it be -1 because it is group 7A? 
Report

07/09/16

Steven W.

tutor
Oops, I mistyped. Your M is actually the atomic number squared. So first get the atomic number. 
Report

07/09/16

Lara N.

So 9^2 = 81 so it would be TI? Would that mean the charge is -5? 
Report

07/09/16

Steven W.

tutor
It depends on how your class states charge. In physics, we would say that you have an amount of positive charge equal to the charge of one proton times the number of protons (the atomic number). Then you have a negative charge equal to the charge ob a single electron. The net charge is then the total positive charge minus the total negative charge. 
Report

07/09/16

Steven W.

tutor
No, the other way. 9 = atomic number squared, so atomic number is 3
Report

07/09/16

Steven W.

tutor
If your class lists charge as just +1 for each proton and -1 for each electron, then the charge would just be (atomic number - 1). 
Report

07/09/16

Lara N.

So since the atom is Li and is in Group IA, the right answer is a +1 charge??
Report

07/09/16

Lara N.

So since the atom is Li and is in Group IA, the right answer is a +1 charge??
Report

07/09/16

Lara N.

So since the atom is Li and is in Group IA, the right answer is a +1 charge??
Report

07/09/16

Steven W.

tutor
You have to adjust for the fact that this is Li with only one electron. So three protons, one electron. 
Report

07/09/16

Lara N.

So if there are three protons (3+) and one electron (1-) does that mean the correct answer is 2+
Report

07/09/16

Steven W.

tutor
That is what I would say, yes. 
Report

07/09/16

Still looking for help? Get the right answer, fast.

Ask a question for free

Get a free answer to a quick problem.
Most questions answered within 4 hours.

OR

Find an Online Tutor Now

Choose an expert and meet online. No packages or subscriptions, pay only for the time you need.