Evaluate Rydberg's constant.

Answer

To evaluate Rydberg's constant, we will follow these steps:

### Step 1: Understand the Energy Levels
According to Bohr's atomic model, the energy of an electron in the nth orbit of a hydrogen atom is given by the formula:
\[ E_n = -\frac{13.6 \, \text{eV}}{n^2} \]
where \( n \) is the principal quantum number.

### Step 2: Energy Change During Electron Transition
When an electron transitions from a higher energy level \( n_2 \) to a lower energy level \( n_1 \), the energy change \( \Delta E \) can be expressed as:
\[ \Delta E = E_{n_1} - E_{n_2} = \left(-\frac{13.6}{n_1^2}\right) - \left(-\frac{13.6}{n_2^2}\right) \]
This simplifies to:
\[ \Delta E = 13.6 \left(\frac{1}{n_1^2} - \frac{1}{n_2^2}\right) \, \text{eV} \]

### Step 3: Relate Energy Change to Wavelength
The energy of the emitted photon during this transition is also given by:
\[ E = \frac{hc}{\lambda} \]
where \( h \) is Planck's constant and \( c \) is the speed of light. Setting the two expressions for energy equal gives:
\[ \frac{hc}{\lambda} = 13.6 \left(\frac{1}{n_1^2} - \frac{1}{n_2^2}\right) \]

### Step 4: Rearranging for Wavelength
Rearranging the equation for \( \lambda \) gives:
\[ \frac{1}{\lambda} = \frac{13.6}{hc} \left(\frac{1}{n_1^2} - \frac{1}{n_2^2}\right) \]

### Step 5: Identify Rydberg's Constant
The Rydberg constant \( R_H \) is defined as:
\[ R_H = \frac{13.6}{hc} \]
Thus, we can express the equation as:
\[ \frac{1}{\lambda} = R_H \left(\frac{1}{n_1^2} - \frac{1}{n_2^2}\right) \]

### Step 6: Calculate Rydberg's Constant
To find \( R_H \), we need the values of \( h \) and \( c \):
- Planck's constant \( h \approx 6.626 \times 10^{-34} \, \text{Js} \)
- Speed of light \( c \approx 3 \times 10^8 \, \text{m/s} \)

Now substituting these values into the formula:
\[ R_H = \frac{13.6 \, \text{eV}}{(6.626 \times 10^{-34} \, \text{Js})(3 \times 10^8 \, \text{m/s})} \]

### Step 7: Convert Units
Since \( 1 \, \text{eV} = 1.6 \times 10^{-19} \, \text{J} \), we convert \( 13.6 \, \text{eV} \) to joules:
\[ 13.6 \, \text{eV} = 13.6 \times 1.6 \times 10^{-19} \, \text{J} \]

### Step 8: Final Calculation
Now we can calculate \( R_H \):
\[ R_H = \frac{13.6 \times 1.6 \times 10^{-19}}{(6.626 \times 10^{-34})(3 \times 10^8)} \]
After performing the calculations, we find:
\[ R_H \approx 1.097 \times 10^7 \, \text{m}^{-1} \]

### Conclusion
Thus, the value of Rydberg's constant is approximately:
\[ R_H \approx 1.097 \times 10^7 \, \text{m}^{-1} \]