Evalute Rydberg constant by putting the value of the fundamental constants in its expression

Explanation

To evaluate the Rydberg constant (R) using the expression and fundamental constants, we can follow these steps:

### Step 1: Write the expression for the Rydberg constant
The Rydberg constant (R) is given by the formula:

\[ R = \frac{m_e e^4}{8 h^3 c \epsilon_0^2} \]

where:
- \( m_e \) = mass of the electron
- \( e \) = charge of the electron
- \( h \) = Planck's constant
- \( c \) = speed of light
- \( \epsilon_0 \) = permittivity of free space

### Step 2: Substitute the values of the fundamental constants
We will substitute the known values of the fundamental constants into the expression:

- \( m_e = 9.1 \times 10^{-31} \) kg (mass of electron)
- \( e = 1.6 \times 10^{-19} \) C (charge of electron)
- \( h = 6.63 \times 10^{-34} \) J·s (Planck's constant)
- \( c = 3 \times 10^8 \) m/s (speed of light)
- \( \epsilon_0 = 8.85 \times 10^{-12} \) C²/(N·m²) (permittivity of free space)

### Step 3: Plug in the values into the formula
Now, substituting these values into the Rydberg constant formula:

\[ R = \frac{(9.1 \times 10^{-31}) (1.6 \times 10^{-19})^4}{8 (6.63 \times 10^{-34})^3 (3 \times 10^8) (8.85 \times 10^{-12})^2} \]

### Step 4: Calculate the numerator
Calculate the numerator:

\[ Numerator = 9.1 \times 10^{-31} \times (1.6 \times 10^{-19})^4 \]

Calculating \( (1.6 \times 10^{-19})^4 \):

\[ (1.6 \times 10^{-19})^4 = 6.5536 \times 10^{-76} \]

Thus,

\[ Numerator = 9.1 \times 10^{-31} \times 6.5536 \times 10^{-76} = 5.964 \times 10^{-106} \]

### Step 5: Calculate the denominator
Calculate the denominator:

\[ Denominator = 8 \times (6.63 \times 10^{-34})^3 \times (3 \times 10^8) \times (8.85 \times 10^{-12})^2 \]

Calculating \( (6.63 \times 10^{-34})^3 \):

\[ (6.63 \times 10^{-34})^3 = 2.917 \times 10^{-101} \]

Calculating \( (8.85 \times 10^{-12})^2 \):

\[ (8.85 \times 10^{-12})^2 = 7.832 \times 10^{-23} \]

Now substituting these values into the denominator:

\[ Denominator = 8 \times 2.917 \times 10^{-101} \times 3 \times 10^8 \times 7.832 \times 10^{-23} \]

Calculating this gives:

\[ Denominator = 8 \times 2.917 \times 3 \times 7.832 \times 10^{-116} = 6.866 \times 10^{-115} \]

### Step 6: Calculate Rydberg constant
Now, we can calculate R:

\[ R = \frac{5.964 \times 10^{-106}}{6.866 \times 10^{-115}} \approx 1.097 \times 10^{7} \text{ m}^{-1} \]

### Final Result
Thus, the evaluated Rydberg constant is:

\[ R \approx 1.097 \times 10^{7} \text{ m}^{-1} \]

---