Calculate the value of Rydberg's constant.

Options

A. `1.097×10^7 m^(-1)`
B. `8.85×10^(-12) m^(-1)`
C. `1.6×10^(-19) m^(-1)`
D. `6.63×10^(-34)m^(-1)`

Answer

To calculate the value of Rydberg's constant (R), we can use the formula:

\[ R = \frac{2 \pi^2 m e^4}{h^3 c} \]

where:
- \( m \) is the mass of the electron,
- \( e \) is the charge of the electron,
- \( h \) is Planck's constant,
- \( c \) is the speed of light.

### Step 1: Identify the constants needed for the calculation
- Mass of the electron, \( m = 9.1 \times 10^{-31} \) kg
- Charge of the electron, \( e = 1.6 \times 10^{-19} \) C
- Speed of light, \( c = 3 \times 10^8 \) m/s
- Planck's constant, \( h = 6.63 \times 10^{-34} \) J·s
- Value of \( \pi \approx 3.14 \)

### Step 2: Substitute the values into the formula
Substituting the values into the formula, we have:

\[ R = \frac{2 \times (3.14)^2 \times (9.1 \times 10^{-31}) \times (1.6 \times 10^{-19})^4}{(6.63 \times 10^{-34})^3 \times (3 \times 10^8)} \]

### Step 3: Calculate the numerator
First, calculate the numerator:

1. Calculate \( (3.14)^2 \):
\[ (3.14)^2 \approx 9.8596 \]

2. Calculate \( (1.6 \times 10^{-19})^4 \):
\[ (1.6)^4 = 6.5536 \quad \text{and} \quad (10^{-19})^4 = 10^{-76} \Rightarrow (1.6 \times 10^{-19})^4 = 6.5536 \times 10^{-76} \]

3. Now, calculate the entire numerator:
\[ 2 \times 9.8596 \times (9.1 \times 10^{-31}) \times (6.5536 \times 10^{-76}) \]

Performing the multiplication:
\[ \approx 2 \times 9.8596 \times 9.1 \times 6.5536 \times 10^{-31-76} \approx 2 \times 9.8596 \times 9.1 \times 6.5536 \times 10^{-107} \]

This gives us a numerical value for the numerator.

### Step 4: Calculate the denominator
Now calculate the denominator:

1. Calculate \( (6.63 \times 10^{-34})^3 \):
\[ (6.63)^3 \approx 291.61 \quad \text{and} \quad (10^{-34})^3 = 10^{-102} \Rightarrow (6.63 \times 10^{-34})^3 \approx 291.61 \times 10^{-102} \]

2. Now multiply by \( 3 \times 10^8 \):
\[ 291.61 \times 10^{-102} \times 3 \times 10^8 = 874.83 \times 10^{-94} \]

### Step 5: Final calculation
Now we can finalize the calculation of \( R \):

\[ R = \frac{\text{Numerator}}{\text{Denominator}} \approx \frac{\text{Value from Step 3}}{874.83 \times 10^{-94}} \]

After performing the division, we find:

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

### Conclusion
The value of Rydberg's constant is approximately \( 1.097 \times 10^7 \, \text{m}^{-1} \).

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