Rydberg constant R is equal to

Options

A. `(me^(2))/(8 in_(0)^(2) ch^(3))`
B. `(me^(4))/(8 in_(0)^(2) ch^(2))`
C. `(m^(2) e^(4))/(8 in_(0)^(2) ch^(3))`
D. `(m^(4) e^(4))/(8 in_(0)^(2) ch^(3))`

Answer

To find the Rydberg constant \( R \), we can derive it from fundamental constants. The Rydberg constant is given by the formula:

\[ R = \frac{2\pi^2 k^2 m_e e^4}{h^2 c} \]

where:
- \( k \) is Coulomb's constant, which can be expressed as \( k = \frac{1}{4\pi \epsilon_0} \)
- \( m_e \) is the mass of the electron
- \( e \) is the elementary charge (the charge of the electron)
- \( h \) is Planck's constant
- \( c \) is the speed of light
- \( \epsilon_0 \) is the permittivity of free space

### Step-by-Step Solution:

1. **Substitute Coulomb's Constant**:
Replace \( k \) in the Rydberg constant formula:
\[ R = \frac{2\pi^2 \left(\frac{1}{4\pi \epsilon_0}\right)^2 m_e e^4}{h^2 c} \]

2. **Simplify the Expression**:
Simplifying the expression gives:
\[ R = \frac{2\pi^2 \cdot \frac{1}{16\pi^2 \epsilon_0^2} m_e e^4}{h^2 c} \]
This simplifies to:
\[ R = \frac{m_e e^4}{8 \epsilon_0^2 h^2 c} \]

3. **Final Expression**:
Thus, the final expression for the Rydberg constant \( R \) is:
\[ R = \frac{m_e e^4}{8 \epsilon_0^2 h^2 c} \]

### Conclusion:
The Rydberg constant \( R \) is equal to:
\[ R = \frac{m_e e^4}{8 \epsilon_0^2 h^2 c} \]