Solution:
Hint: As we go up in the energy diagram of hydrogen, the energy difference decreases.
In the energy diagram of hydrogen, the energy difference between levels decreases as we move up. Here, the maximum energy difference is between $\text{n}=1$ to $\text{n}=2$.
Here option two cannot be the answer because we have to check the only transition occur due to emission, not absorption.
Emission of energy occurs when an electron transitions from a higher energy level to a lower energy level. We are looking for the transition that emits the maximum energy, which corresponds to the largest energy difference between the initial and final states.
The energy of an electron in a hydrogen atom is given by the formula:
$E_n = -\frac{13.6}{\text{n}^2} \text{ eV}$
The energy difference for a transition from an initial state $\text{n}_i$ to a final state $\text{n}_f$ is:
$\Delta E = E_{n_f} - E_{n_i} = -13.6 \left( \frac{1}{\text{n}_f^2} - \frac{1}{\text{n}_i^2} \right) \text{ eV}$
For emission, $\Delta E$ will be negative, indicating energy is released. To find the maximum energy emitted, we need to find the transition with the largest absolute value of $\Delta E$.
Let's analyze the given options for emission (higher $\text{n}$ to lower $\text{n}$):
1. $2 \rightarrow 1$:
$\Delta E = -13.6 \left( \frac{1}{1^2} - \frac{1}{2^2} \right) = -13.6 \left( 1 - \frac{1}{4} \right) = -13.6 \left( \frac{3}{4} \right) = -10.2 \text{ eV}$
2. $1 \rightarrow 4$: (This is absorption, not emission, so it's not the answer)
3. $4 \rightarrow 3$:
$\Delta E = -13.6 \left( \frac{1}{3^2} - \frac{1}{4^2} \right) = -13.6 \left( \frac{1}{9} - \frac{1}{16} \right) = -13.6 \left( \frac{16 - 9}{144} \right) = -13.6 \left( \frac{7}{144} \right) \approx -0.66 \text{ eV}$
4. $3 \rightarrow 2$:
$\Delta E = -13.6 \left( \frac{1}{2^2} - \frac{1}{3^2} \right) = -13.6 \left( \frac{1}{4} - \frac{1}{9} \right) = -13.6 \left( \frac{9 - 4}{36} \right) = -13.6 \left( \frac{5}{36} \right) \approx -1.89 \text{ eV}$
Comparing the absolute values of the energy emitted:
$| -10.2 \text{ eV} |$ for $2 \rightarrow 1$
$| -0.66 \text{ eV} |$ for $4 \rightarrow 3$
$| -1.89 \text{ eV} |$ for $3 \rightarrow 2$
The largest energy emitted is for the transition $2 \rightarrow 1$, with an energy of $10.2 \text{ eV}$. This is consistent with the hint that the energy difference decreases as we move up in energy levels (i.e., the largest energy difference is between the lowest energy levels).
The final answer is $\boxed{1}$