To find the ionization energy of a hydrogen atom in different units (Ergs, Joules, and electron volts), we can follow these steps:
### Step 1: Understand Ionization Energy
Ionization energy is the energy required to remove the most loosely bound electron from an atom. For the hydrogen atom, this energy can be calculated using the formula for the energy levels of hydrogen-like atoms.
### Step 2: Calculate Energy Levels
The energy of the electron in the nth energy level of a hydrogen atom is given by the formula:
\[
E_n = -\frac{13.6 \, \text{eV}}{n^2}
\]
For the ground state (n=1):
\[
E_1 = -\frac{13.6 \, \text{eV}}{1^2} = -13.6 \, \text{eV}
\]
### Step 3: Determine Final Energy
When the electron is completely removed (ionized), it is at an infinite distance from the nucleus, and its energy is considered to be:
\[
E_\infty = 0 \, \text{eV}
\]
### Step 4: Calculate Ionization Energy
The ionization energy (IE) can be calculated as:
\[
IE = E_\infty - E_1 = 0 - (-13.6 \, \text{eV}) = 13.6 \, \text{eV}
\]
### Step 5: Convert Electron Volts to Ergs
To convert the ionization energy from electron volts to Ergs, we use the conversion factor:
\[
1 \, \text{eV} = 1.602 \times 10^{-12} \, \text{Ergs}
\]
Thus, the ionization energy in Ergs is:
\[
IE \, (\text{in Ergs}) = 13.6 \, \text{eV} \times 1.602 \times 10^{-12} \, \text{Ergs/eV} = 21.8 \times 10^{-12} \, \text{Ergs}
\]
### Step 6: Convert Electron Volts to Joules
To convert the ionization energy from electron volts to Joules, we use the conversion factor:
\[
1 \, \text{eV} = 1.602 \times 10^{-19} \, \text{Joules}
\]
Thus, the ionization energy in Joules is:
\[
IE \, (\text{in Joules}) = 13.6 \, \text{eV} \times 1.602 \times 10^{-19} \, \text{J/eV} = 2.179 \times 10^{-18} \, \text{J}
\]
This can also be expressed as:
\[
IE \, (\text{in Joules}) = 21.8 \times 10^{-19} \, \text{J}
\]
### Final Results
- Ionization Energy in Ergs: \( 21.8 \times 10^{-12} \, \text{Ergs} \)
- Ionization Energy in Joules: \( 21.8 \times 10^{-19} \, \text{J} \)
- Ionization Energy in Electron Volts: \( 13.6 \, \text{eV} \)
### Summary
The ionization energy for a hydrogen atom is:
- **In Ergs**: \( 21.8 \times 10^{-12} \, \text{Ergs} \)
- **In Joules**: \( 21.8 \times 10^{-19} \, \text{J} \)
- **In Electron Volts**: \( 13.6 \, \text{eV} \)
