Scaling of Emission Wavelengths in Balmer Series for ##Li^{2+}##?

[Kara386]

Homework Statement

The emission wavelengths of hydrogen-like atoms are related to nuclear charge. How do they scale as a function of Z? What are the longest and shortest wavelengths in the Balmer series for $Li^{2+}$?

Homework Equations

$E_n = -\frac{R}{n^2}$ (1)

$a_0 = \frac{\hbar^2}{Zme^2}$ (2)

From (2), $\hbar^2 = a_0Zme^2$ (3)

$\alpha = \frac{Ze^2}{\hbar c}$ (4)

$R = \frac{1}{2} mc^2 \alpha^2$ (5)

The Attempt at a Solution

I started by trying to find an expression for the energy levels of hydrogen-like atoms. Substituting (5) into (1) gives
$E_n = -\frac{mc^2\alpha^2}{2n^2}$
Substitute in (4):
$E_n = -\frac{mZ^2e^4}{2\hbar^2 n^2}$
Sub in (3):
$E_n = -\frac{mZ^2e^4}{2Za_0me^2 n^2}$

Simplifying, $E_n = -\frac{Ze^2}{2a_0n^2}$. This can be related to wavlength via $E_n = \frac{hc}{\lambda}$, so emission and absorption wavelengths of hydrogenic atoms are related to nuclear charge Z by the function

$\lambda = -\frac{2hca_0n^2}{Ze^2}$

But I have no idea how to answer the bit about the Balmer series because n=2 and Z=3, so based on my equation isn't there only going to be one answer? How do I get minimum/maximum wavelengths?

 

[Incand]

But I have no idea how to answer the bit about the Balmer series because n=2 and Z=3, so based on my equation isn't there only going to be one answer? How do I get minimum/maximum wavelengths?

In the Balmer series the final state is an $n=2$ state. The initial state may be anything from $n=3$ to $n=\infty$. You need to find the energy difference between these states.

 

[Kara386]

In the Balmer series the final state is an $n=2$ state. The initial state may be anything from $n=3$ to $n=\infty$. You need to find the energy difference between these states.

So I use the equation for $E_n$ I found and do $E_3-E_2$ for the maximum wavelength? This gives a negative answer though, can I just multiply that answer by $-1$? Or do $E_2-E_3$?

 

[Incand]

$E_3-E_2$ is positive from your formula since you got a minus sign in it. That is $-1/9 > -1/4$.
I don't know if all your other constants are correct but you can always compare your answer to the Rydberg formula https://en.wikipedia.org/wiki/Rydberg_formula