A doubly ionised lithium atom is hydrogen-like with atomic number 3.
(a) Find the wavelength of the

Q: Why do we use the hydrogen formula for Lithium?

The question specifies a *doubly ionised* lithium atom ($\text{Li}^{2+}$). This means it has lost two of its three electrons, leaving only one. Any atom or ion with a single electron behaves like hydrogen, just with a stronger nuclear charge ($Z$).

Q: Why is the energy negative?

The negative sign indicates that the electron is bound to the nucleus. You have to supply positive energy to free it (bring its energy to zero).

Q: Should I use 12400 or 12375 for the $hc$ value?

Both are approximations for $hc$. $12400$ is easier for quick mental math, but $12375$ is often used in JEE for slightly more precise calculations. Always check the options or the given constants in the exam.

Q: How do we know the electron will fall back down?

Excited states are inherently unstable. The electron will naturally seek the lowest possible energy state (the ground state), emitting the excess energy as photons.

Q: What if the electron was excited to $n=4$? How many lines would there be?

You would use the same formula: $\frac{n(n-1)}{2}$. For $n=4$, it would be $\frac{4(3)}{2} = 6$ spectral lines.

Question

A doubly ionised lithium atom is hydrogen-like with atomic number 3.
(a) Find the wavelength of the radiation required to excite the electron in $	ext{Li}^{2+}$ from the first to the third Bohr orbit. (Ionisation energy of the hydrogen atom equals $13.6	ext{ eV}$.)
(b) How many spectral lines are observed in the emission spectrum of the above excited system?

Sigma Percentile · Accepted Answer

### The Quantum Leap: Exciting a Lithium Ion

Imagine you are an electron, bound to a nucleus. The stronger the nucleus pulls on you, the deeper the energy well you sit in. In this problem, we are dealing with a doubly ionised lithium atom ($	ext{Li}^{2+}$). Because it has lost two of its three electrons, it is a single-electron system, meaning it behaves exactly like a hydrogen atom, but with a much stronger nuclear charge ($Z=3$).

### The Energy Jump

To move an electron from a lower orbit to a higher one, we must supply it with a very specific amount of energy. The energy of an electron in the $n^{	ext{th}}$ orbit of a hydrogen-like atom is given by the Bohr model formula:

$$E_n = -13.6 \frac{Z^2}{n^2} 	ext{ eV}$$

We want to excite the electron from the ground state ($n=1$) to the third orbit ($n=3$). Let's calculate the energy of these two states for our lithium ion ($Z=3$):

$$E_1 = -13.6 	imes \frac{3^2}{1^2} = -122.4 	ext{ eV}$$
$$E_3 = -13.6 	imes \frac{3^2}{3^2} = -13.6 	ext{ eV}$$

The energy required for this quantum leap is simply the difference between the final and initial states:

$$\Delta E = E_3 - E_1 = -13.6 - (-122.4) = 108.8 	ext{ eV}$$

### The Wavelength

Now that we know the energy required, we need to find the wavelength of the photon that can deliver exactly this amount of energy. We can use the handy shortcut formula relating energy in electron-volts to wavelength in Angstroms:

$$\lambda (	ext{in \AA}) = \frac{12375}{\Delta E (	ext{in eV})}$$

Plugging in our calculated energy:

$$\lambda = \frac{12375}{108.8} = 113.74 	ext{ \AA}$$

This is the wavelength of the radiation required to excite the electron.

### The Spectral Lines

Once the electron is in the excited state ($n=3$), it won't stay there forever. It will eventually fall back down to the ground state, emitting photons as it drops. It can take multiple paths to get back down. The total number of possible emission lines from the $n^{	ext{th}}$ state is given by the combination formula:

$$	ext{Number of lines} = \frac{n(n-1)}{2}$$

For our electron in the $n=3$ state:

$$	ext{Number of lines} = \frac{3(3-1)}{2} = 3$$

These three lines correspond to the transitions $3 	o 1$, $3 	o 2$, and $2 	o 1$. Each transition releases a photon of a specific wavelength, creating a distinct line in the emission spectrum.

Visualized Solution

Visualizing the Energy Levels

Energy of an Electron in $n^{th}$ Orbit

Excitation Energy Setup

Calculating Energy Levels

Calculating $Δ E$

Wavelength Formula

Calculating Wavelength

Emission Spectrum Setup

Number of Spectral Lines Formula

Calculating Spectral Lines

Visualizing the Transitions

The Sigma Insight: Bohr's Atomic Model and Energy Levels

The Quantum Leap

The Energy Jump

The Wavelength

The Spectral Lines

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