The ionisation energy of hydrogen atom is `1.312xx10^(6) J "mol"^(-1)`. Calculate the energy required to excite an electron in a hydrogen atom from the ground state to the first excited state.

Explanation

To calculate the energy required to excite an electron in a hydrogen atom from the ground state to the first excited state, we can follow these steps:

### Step 1: Understand the Ionization Energy
The ionization energy (IE) of hydrogen is given as \(1.312 \times 10^6 \, \text{J mol}^{-1}\). This energy is the energy required to remove an electron from the ground state (n=1) to infinity (n=∞).

### Step 2: Convert Ionization Energy to Joules per Atom
To find the energy per atom, we need to divide the ionization energy by Avogadro's number (\(N_A = 6.022 \times 10^{23} \, \text{mol}^{-1}\)):
\[ \text{Energy per atom} = \frac{1.312 \times 10^6 \, \text{J mol}^{-1}}{6.022 \times 10^{23} \, \text{mol}^{-1}} \approx 2.18 \times 10^{-18} \, \text{J atom}^{-1} \]

### Step 3: Calculate the Energy of the Ground State (E1)
The energy of the ground state (E1) can be represented as:
\[ E_1 = -\frac{Z^2 \cdot 13.6 \, \text{eV}}{n^2} \]
For hydrogen, \(Z = 1\) and \(n = 1\):
\[ E_1 = -\frac{1^2 \cdot 13.6 \, \text{eV}}{1^2} = -13.6 \, \text{eV} \]
To convert this to joules:
\[ E_1 = -13.6 \times 1.6 \times 10^{-19} \, \text{J} \approx -2.18 \times 10^{-18} \, \text{J} \]

### Step 4: Calculate the Energy of the First Excited State (E2)
Using the same formula for the first excited state (n=2):
\[ E_2 = -\frac{Z^2 \cdot 13.6 \, \text{eV}}{n^2} = -\frac{1^2 \cdot 13.6 \, \text{eV}}{2^2} = -\frac{13.6 \, \text{eV}}{4} = -3.4 \, \text{eV} \]
Converting this to joules:
\[ E_2 = -3.4 \times 1.6 \times 10^{-19} \, \text{J} \approx -5.44 \times 10^{-19} \, \text{J} \]

### Step 5: Calculate the Energy Required for Excitation
The energy required to excite the electron from the ground state (E1) to the first excited state (E2) is given by:
\[ \Delta E = E_2 - E_1 \]
Substituting the values:
\[ \Delta E = (-5.44 \times 10^{-19} \, \text{J}) - (-2.18 \times 10^{-18} \, \text{J}) \]
\[ \Delta E = -5.44 \times 10^{-19} + 2.18 \times 10^{-18} = 1.635 \times 10^{-18} \, \text{J} \]

### Final Answer
The energy required to excite an electron in a hydrogen atom from the ground state to the first excited state is approximately:
\[ \Delta E \approx 1.635 \times 10^{-18} \, \text{J} \]

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