Calculate the first excitation energy of electron in the hydrogen atom.

Explanation

To calculate the first excitation energy of the electron in the hydrogen atom, we will follow these steps:

### Step 1: Understand the Energy Levels
In a hydrogen atom, the energy levels are defined by the principal quantum number \( n \). The first energy level corresponds to \( n = 1 \) and the second energy level corresponds to \( n = 2 \). The first excitation energy occurs when the electron transitions from \( n = 1 \) to \( n = 2 \).

### Step 2: Use the Energy Formula
The energy of an electron in a hydrogen atom at a given energy level \( n \) is given by the formula:
\[ E_n = -\frac{1.312 \times 10^6}{n^2} \text{ joules per mole} \]
where \( E_n \) is the energy at level \( n \).

### Step 3: Calculate \( E_1 \) and \( E_2 \)
1. For \( n = 1 \):
\[ E_1 = -\frac{1.312 \times 10^6}{1^2} = -1.312 \times 10^6 \text{ joules per mole} \]

2. For \( n = 2 \):
\[ E_2 = -\frac{1.312 \times 10^6}{2^2} = -\frac{1.312 \times 10^6}{4} = -3.28 \times 10^5 \text{ joules per mole} \]

### Step 4: Calculate the First Excitation Energy
The first excitation energy (\( \Delta E \)) is the difference between the energy at \( n = 2 \) and \( n = 1 \):
\[ \Delta E = E_2 - E_1 \]
Substituting the values we calculated:
\[ \Delta E = \left(-3.28 \times 10^5\right) - \left(-1.312 \times 10^6\right) \]
This simplifies to:
\[ \Delta E = -3.28 \times 10^5 + 1.312 \times 10^6 \]
\[ \Delta E = 1.312 \times 10^6 - 3.28 \times 10^5 = 9.84 \times 10^5 \text{ joules per mole} \]

### Final Answer
The first excitation energy of the electron in the hydrogen atom is:
\[ \Delta E = 9.84 \times 10^5 \text{ joules per mole} \]

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