The energy of an electromagnetic radiation is `19.875 xx 10^(-13)` ergs. What is the wave number in `cm^(-1)` ? (`h = 6.625 xx 10^(-27)` erg.sec, `c = 3 xx 10^(10) cm.sec^(-1)`)

Options

A. 1000
B. `10^(6)`
C. 100
D. 10000

Answer

To find the wave number (ν̅) in cm⁻¹ given the energy of electromagnetic radiation, we can follow these steps:

### Step 1: Write down the formula
The relationship between energy (E), Planck's constant (h), speed of light (c), and wavelength (λ) can be expressed as:
\[ E = \frac{hc}{\lambda} \]
We also know that the wave number (ν̅) is defined as:
\[ \nu̅ = \frac{1}{\lambda} \]
From these, we can rearrange the energy formula to express wave number:
\[ \nu̅ = \frac{E}{hc} \]

### Step 2: Substitute the values
Given:
- Energy, \( E = 19.875 \times 10^{-13} \) ergs
- Planck's constant, \( h = 6.625 \times 10^{-27} \) erg·sec
- Speed of light, \( c = 3 \times 10^{10} \) cm/sec

Now we substitute these values into the wave number formula:
\[ \nu̅ = \frac{19.875 \times 10^{-13}}{(6.625 \times 10^{-27}) \times (3 \times 10^{10})} \]

### Step 3: Calculate the denominator
First, calculate \( hc \):
\[ hc = (6.625 \times 10^{-27}) \times (3 \times 10^{10}) \]
\[ hc = 19.875 \times 10^{-17} \text{ erg·cm} \]

### Step 4: Calculate the wave number
Now substitute \( hc \) back into the wave number equation:
\[ \nu̅ = \frac{19.875 \times 10^{-13}}{19.875 \times 10^{-17}} \]
\[ \nu̅ = \frac{19.875}{19.875} \times 10^{4} \]
\[ \nu̅ = 1 \times 10^{4} \text{ cm}^{-1} \]

### Step 5: Final result
Thus, the wave number is:
\[ \nu̅ = 10,000 \text{ cm}^{-1} \]

### Conclusion
The final answer is:
**10,000 cm⁻¹**