The energy change is greatest for a hydrogen atom when its state changes from

Options

A. `n = 2 to n = 1`
B. `n = 3 to n = 2`
C. `n = 4 to n = 3`
D. `n = 5 to n = 4`

Explanation

To determine the greatest energy change for a hydrogen atom when its state changes, we can follow these steps:

### Step 1: Understand the Energy Levels
The energy of a hydrogen atom in a given energy level \( n \) is given by the formula:
\[ E_n = -\frac{13.6 \, \text{eV}}{n^2} \]
where \( n \) is the principal quantum number (1, 2, 3, ...).

### Step 2: Calculate the Energy for Each Level
We will calculate the energy for the first five energy levels (n = 1 to n = 5):

- For \( n = 1 \):
\[ E_1 = -\frac{13.6}{1^2} = -13.6 \, \text{eV} \]

- For \( n = 2 \):
\[ E_2 = -\frac{13.6}{2^2} = -\frac{13.6}{4} = -3.4 \, \text{eV} \]

- For \( n = 3 \):
\[ E_3 = -\frac{13.6}{3^2} = -\frac{13.6}{9} \approx -1.51 \, \text{eV} \]

- For \( n = 4 \):
\[ E_4 = -\frac{13.6}{4^2} = -\frac{13.6}{16} \approx -0.85 \, \text{eV} \]

- For \( n = 5 \):
\[ E_5 = -\frac{13.6}{5^2} = -\frac{13.6}{25} \approx -0.54 \, \text{eV} \]

### Step 3: Calculate the Energy Changes
Now we will calculate the energy change for transitions between different energy levels:

1. From \( n = 2 \) to \( n = 1 \):
\[ \Delta E_{2 \to 1} = E_1 - E_2 = (-13.6) - (-3.4) = -10.2 \, \text{eV} \]

2. From \( n = 3 \) to \( n = 2 \):
\[ \Delta E_{3 \to 2} = E_2 - E_3 = (-3.4) - (-1.51) = -1.89 \, \text{eV} \]

3. From \( n = 4 \) to \( n = 3 \):
\[ \Delta E_{4 \to 3} = E_3 - E_4 = (-1.51) - (-0.85) = -0.66 \, \text{eV} \]

4. From \( n = 5 \) to \( n = 4 \):
\[ \Delta E_{5 \to 4} = E_4 - E_5 = (-0.85) - (-0.54) = -0.31 \, \text{eV} \]

### Step 4: Compare Energy Changes
Now we compare the absolute values of the energy changes calculated:

- \( |\Delta E_{2 \to 1}| = 10.2 \, \text{eV} \)
- \( |\Delta E_{3 \to 2}| = 1.89 \, \text{eV} \)
- \( |\Delta E_{4 \to 3}| = 0.66 \, \text{eV} \)
- \( |\Delta E_{5 \to 4}| = 0.31 \, \text{eV} \)

### Conclusion
The greatest energy change occurs when the hydrogen atom transitions from \( n = 2 \) to \( n = 1 \), with an energy change of \( 10.2 \, \text{eV} \).

### Final Answer
The energy change is greatest for a hydrogen atom when its state changes from \( n = 2 \) to \( n = 1 \).

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