The energy associated with the first orbit in the hydrogen atom is ` -2.18 xx 10^(-18) "J atom"^(-1)`. What is the energy associated with the fourth orbit ?

Answer

To find the energy associated with the fourth orbit of a hydrogen atom, we can use the formula derived from Bohr's theory of the hydrogen atom. The energy of an electron in a hydrogen atom is given by the formula:

\[ E_n = -\frac{R_H \cdot Z^2}{n^2} \]

Where:
- \(E_n\) is the energy of the nth orbit,
- \(R_H\) is the Rydberg constant (which is \(2.18 \times 10^{-18} \, \text{J}\) for hydrogen),
- \(Z\) is the atomic number (for hydrogen, \(Z = 1\)),
- \(n\) is the principal quantum number (the orbit number).

### Step-by-step Solution:

1. **Identify the given values**:
- Energy of the first orbit \(E_1 = -2.18 \times 10^{-18} \, \text{J}\)
- Atomic number for hydrogen \(Z = 1\)
- We need to find the energy for the fourth orbit \(n = 4\).

2. **Use the energy formula for the nth orbit**:
\[ E_n = -\frac{R_H \cdot Z^2}{n^2} \]

3. **Substituting the known values into the formula**:
\[ E_4 = -\frac{2.18 \times 10^{-18} \cdot 1^2}{4^2} \]

4. **Calculate \(4^2\)**:
\[ 4^2 = 16 \]

5. **Substitute \(16\) back into the equation**:
\[ E_4 = -\frac{2.18 \times 10^{-18}}{16} \]

6. **Perform the division**:
\[ E_4 = -0.13625 \times 10^{-18} \, \text{J} \]

7. **Express the answer in standard form**:
\[ E_4 = -1.36 \times 10^{-19} \, \text{J} \]

### Final Answer:
The energy associated with the fourth orbit of the hydrogen atom is:
\[ E_4 = -1.36 \times 10^{-19} \, \text{J} \]