To calculate the Rydberg constant for hydrogen using the given ionization energy, we can follow these steps:
### Step 1: Understand the Relationship
The Rydberg formula for the hydrogen atom is given by:
\[
\frac{1}{\lambda} = R \left( \frac{1}{n_1^2} - \frac{1}{n_2^2} \right)
\]
where:
- \( R \) is the Rydberg constant,
- \( n_1 \) is the principal quantum number of the lower energy level,
- \( n_2 \) is the principal quantum number of the higher energy level.
### Step 2: Set the Quantum Numbers
For ionization, the electron is removed from the atom, which corresponds to \( n_2 \) approaching infinity (\( n_2 = \infty \)). The ground state of hydrogen corresponds to \( n_1 = 1 \).
### Step 3: Substitute the Values
Substituting these values into the Rydberg formula gives:
\[
\frac{1}{\lambda} = R \left( \frac{1}{1^2} - \frac{1}{\infty^2} \right) = R \left( 1 - 0 \right) = R
\]
Thus, we have:
\[
\frac{1}{\lambda} = R
\]
### Step 4: Relate Energy to Wavelength
The energy \( E \) of the ionization process can be expressed in terms of wavelength \( \lambda \):
\[
E = \frac{hc}{\lambda}
\]
Where:
- \( h \) is Planck's constant (\( 6.626 \times 10^{-34} \, \text{J s} \)),
- \( c \) is the speed of light (\( 3 \times 10^8 \, \text{m/s} \)).
### Step 5: Convert Ionization Energy to Joules
The ionization energy given is \( 13.6 \, \text{eV} \). To convert this to joules, we use the conversion factor:
\[
1 \, \text{eV} = 1.6 \times 10^{-19} \, \text{J}
\]
Thus,
\[
E = 13.6 \, \text{eV} \times 1.6 \times 10^{-19} \, \text{J/eV} = 2.176 \times 10^{-18} \, \text{J}
\]
### Step 6: Substitute Energy into the Equation
Now we can relate this energy to the wavelength:
\[
E = \frac{hc}{\lambda} \implies \lambda = \frac{hc}{E}
\]
Substituting the values for \( h \) and \( c \):
\[
\lambda = \frac{(6.626 \times 10^{-34} \, \text{J s})(3 \times 10^8 \, \text{m/s})}{2.176 \times 10^{-18} \, \text{J}}
\]
Calculating this gives:
\[
\lambda = \frac{1.9878 \times 10^{-25}}{2.176 \times 10^{-18}} \approx 9.13 \times 10^{-8} \, \text{m}
\]
### Step 7: Calculate Rydberg Constant
Now, substituting \( \lambda \) back into the equation for \( R \):
\[
R = \frac{1}{\lambda} = \frac{1}{9.13 \times 10^{-8}} \approx 1.095 \times 10^7 \, \text{m}^{-1}
\]
### Final Answer
Thus, the Rydberg constant for hydrogen is approximately:
\[
R \approx 1.097 \times 10^7 \, \text{m}^{-1}
\]
