To calculate the energy required to remove an electron from a hydrogen atom in its ground state (also known as the ionization energy of hydrogen), we can use the formula derived from the Bohr model of the hydrogen atom.
The energy of an electron in the nth level of a hydrogen atom is given by:
\[ E_n = -\frac{ke^2}{2a_0 n^2} \]
Where:
- \(E_n\) is the energy of the electron at level \(n\),
- \(k\) is Coulomb's constant (\(8.9875 \times 10^9 , \text{N m}^2/\text{C}^2\)),
- \(e\) is the elementary charge (\(1.602 \times 10^{-19} , \text{C}\)),
- \(a_0\) is the Bohr radius (\(5.29177 \times 10^{-11} , \text{m}\)),
- \(n\) is the principal quantum number (1 for the ground state).
For hydrogen, in its ground state (\(n=1\)), the energy can also be calculated using a simplified formula:
\[ E_1 = -13.6 , \text{eV} \]
This value reflects the energy associated with the first energy level of the hydrogen atom.
To find the energy required to remove the electron from the ground state (ionization energy), we take the absolute value of this energy:
\[ \text{Ionization Energy} = |E_1| = 13.6 , \text{eV} \]
Thus, the energy required to remove the electron from a hydrogen atom in its ground state is 13.6 eV.