The energy of stable states of the hydrogen atom is given by `E_(n)=-2.18xx10^(-8)//n^(2)[J]` where n denotes the principal quantum number.
Calculate the energy differences between `n=2` (first excited state) and `n=1` (ground state) and between `n=7` and `n=1`.

Answer

To solve the problem, we will calculate the energy differences between the specified states of the hydrogen atom using the given formula for energy levels.

### Step 1: Write down the energy formula
The energy of stable states of the hydrogen atom is given by:
\[ E_n = -\frac{2.18 \times 10^{-8}}{n^2} \, \text{J} \]

### Step 2: Calculate the energy for \( n = 1 \) (ground state)
Substituting \( n = 1 \) into the formula:
\[ E_1 = -\frac{2.18 \times 10^{-8}}{1^2} = -2.18 \times 10^{-8} \, \text{J} \]

### Step 3: Calculate the energy for \( n = 2 \) (first excited state)
Substituting \( n = 2 \) into the formula:
\[ E_2 = -\frac{2.18 \times 10^{-8}}{2^2} = -\frac{2.18 \times 10^{-8}}{4} = -5.45 \times 10^{-9} \, \text{J} \]

### Step 4: Calculate the energy difference between \( n = 2 \) and \( n = 1 \)
The energy difference \( \Delta E_{2 \to 1} \) is given by:
\[ \Delta E_{2 \to 1} = E_1 - E_2 \]
Substituting the values:
\[ \Delta E_{2 \to 1} = (-2.18 \times 10^{-8}) - (-5.45 \times 10^{-9}) \]
\[ = -2.18 \times 10^{-8} + 5.45 \times 10^{-9} \]
\[ = -1.635 \times 10^{-8} \, \text{J} \]

### Step 5: Calculate the energy for \( n = 7 \)
Substituting \( n = 7 \) into the formula:
\[ E_7 = -\frac{2.18 \times 10^{-8}}{7^2} = -\frac{2.18 \times 10^{-8}}{49} = -4.45 \times 10^{-10} \, \text{J} \]

### Step 6: Calculate the energy difference between \( n = 7 \) and \( n = 1 \)
The energy difference \( \Delta E_{7 \to 1} \) is given by:
\[ \Delta E_{7 \to 1} = E_1 - E_7 \]
Substituting the values:
\[ \Delta E_{7 \to 1} = (-2.18 \times 10^{-8}) - (-4.45 \times 10^{-10}) \]
\[ = -2.18 \times 10^{-8} + 4.45 \times 10^{-10} \]
\[ = -2.1355 \times 10^{-8} \, \text{J} \]

### Final Answers
1. The energy difference between \( n = 2 \) and \( n = 1 \) is:
\[ \Delta E_{2 \to 1} = 1.635 \times 10^{-8} \, \text{J} \]
2. The energy difference between \( n = 7 \) and \( n = 1 \) is:
\[ \Delta E_{7 \to 1} = 2.1355 \times 10^{-8} \, \text{J} \]