Q.A hydrogen atom initially in the ground level absorbs a photon, which excites it to the $n = 4$ level. Determine the wavelength and frequency of photon.

Using $hc = 1240\ \text{eV·nm}$: $\lambda = \dfrac{1240}{12.75} \approx 97.3\ \text{nm}$. …

Energy of the transition

For hydrogen, $E_n = -\dfrac{13.6\ \text{eV}}{n^2}$.

Since the atom is excited from $n=1$ to $n=4$, the photon must supply exactly this energy gap:

Converting to frequency

Convert to joules: $\Delta E = 12.75 \times 1.602\times10^{-19} = 2.043\times10^{-18}\ \text{J}$.

(This lies in the ultraviolet range.)

Converting to wavelength

$\lambda = \frac{c}{f} = \frac{3.00\times10^{8}}{3.08\times10^{15}} \approx 9.73\times10^{-8}\ \text{m} = 97.3\ \text{nm}$ …

Method: Energy-Level Transition (Bohr Model)

This problem uses the Bohr model of the hydrogen atom, where the energy of an electron in level $n$ is given by:

The photon energy absorbed equals the difference between the final and initial energy levels.

Step 1: Write the energy of each level

For the ground state ($n = 1$):

For the excited state ($n = 4$):

Step 2: Find the photon energy absorbed

The photon must supply exactly the energy gap:

Step 3: Convert photon energy to joules

For calculations with wavelength and frequency, use SI units:

Step 4: Find the frequency

Use Planck's relation $E = hf$:

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Common Mistakes on This Photon Energy Problem

Students often lose marks on this seemingly straightforward question. Here are the most frequent errors and how to avoid each.

Mistake 1: Using the wrong energy formula

Many students plug $E = hf$ directly without connecting it to the hydrogen energy levels. They forget that the photon's energy must exactly equal the difference between two atomic energy levels.

How to avoid: Always start by writing the energy of the $n$-th level in hydrogen:

The photon energy is the gap between the final and initial states:

Never use $E_n$ alone — the photon doesn't "carry" the full energy of a level; it carries only the transition energy.

Mistake 2: Sign errors in the energy difference

Students sometimes write $\Delta E = E_1 - E_4$ (reversed order) and get a negative value. Then they either panic or take the absolute value without understanding why.

How to avoid: The atom absorbs energy, so the final state has higher (less negative) energy. The correct order is:

Since $E_4 > E_1$ (both negative, but $E_4$ is closer to zero), $\Delta E$ comes out positive — as it must for absorption.

Mistake 3: Forgetting to convert eV to joules

The Rydberg constant is often given in eV, but $h$ and $c$ in the wavelength formula use SI units. Students plug eV directly into $E = hc/\lambda$ and get nonsense.

How to avoid: After finding $\Delta E$ in eV, convert to joules:

Then use:

Where $h = 6.626 \times 10^{-34} \text{ J·s}$ and $c = 3.00 \times 10^8 \text{ m/s}$.

Mistake 4: Confusing wavelength and frequency formulas

Students mix up $f = c/\lambda$ and $\lambda = c/f$, or use $E = hc/\lambda$ but forget which variable to solve for.

How to avoid: Write the two relations clearly:

Solve step by step: first find $f$ from $E$, then find $\lambda$ from $f$. Don't try to jump to both at once.

Mistake 5: Arithmetic errors with fractions

When computing $E_4 - E_1$, students often mishandle the fractions:

$\Delta E = (-0.85) - (-13.6) = -0.85 + 13.6 = 12.75 \text{ eV}$ …