The ionisation energy of hydrogen atom is 13.6 eV. Calculate Rydberg's constant for hydrogen.

Answer

To calculate Rydberg's constant for hydrogen using the given ionization energy, we will follow these steps:

### Step-by-Step Solution

1. **Understand the Ionization Energy**: The ionization energy of the hydrogen atom is given as 13.6 eV. This is the energy required to remove the electron from the ground state (n=1) to infinity (n=∞).

2. **Convert Ionization Energy to Joules**:
\[ E = 13.6 \text{ eV} \times 1.6 \times 10^{-19} \text{ J/eV} = 2.176 \times 10^{-18} \text{ J} \]

3. **Use the Energy-Wavelength Relationship**: The energy of a photon can be expressed in terms of its frequency (ν) or wavelength (λ):
\[ E = h \nu = \frac{hc}{\lambda} \]
where \( h \) is Planck's constant and \( c \) is the speed of light.

4. **Apply the Rydberg Formula**: The Rydberg formula for the wavelength of emitted or absorbed light during transitions in hydrogen is:
\[ \frac{1}{\lambda} = R \left( \frac{1}{n_1^2} - \frac{1}{n_2^2} \right) \]
For ionization from the ground state (n=1) to infinity (n=∞):
\[ \frac{1}{\lambda} = R \left( \frac{1}{1^2} - \frac{1}{\infty^2} \right) = R \left( 1 - 0 \right) = R \]

5. **Relate Energy to Rydberg's Constant**: From the energy equation, we can write:
\[ E = \frac{hc}{\lambda} \]
Substituting \( \lambda \) from the Rydberg formula:
\[ E = hc \cdot \frac{1}{R} \]
Rearranging gives:
\[ R = \frac{hc}{E} \]

6. **Substitute Known Values**:
- Planck's constant \( h = 6.626 \times 10^{-34} \text{ J s} \)
- Speed of light \( c = 3 \times 10^8 \text{ m/s} \)
- Energy \( E = 2.176 \times 10^{-18} \text{ J} \)

Now substituting these values into the equation for R:
\[ R = \frac{(6.626 \times 10^{-34} \text{ J s}) \times (3 \times 10^8 \text{ m/s})}{2.176 \times 10^{-18} \text{ J}} \]

7. **Calculate Rydberg's Constant**:
\[ R = \frac{1.9878 \times 10^{-25}}{2.176 \times 10^{-18}} \approx 9.13 \times 10^6 \text{ m}^{-1} \]

8. **Final Result**: The Rydberg constant for hydrogen is approximately:
\[ R \approx 1.097 \times 10^7 \text{ m}^{-1} \]