The number of photons of light of `barv =4.5×10^5m^(−1)` necessary to provide 1 J of energy are:

Options

A. `1.2xx10^(-13)`
B. `1.1xx10^(-19)`
C. `2.2xx10^(-18)`
D. `1.1xx10^(-13)`

Explanation

To solve the problem of finding the number of photons required to provide 1 J of energy with a wave number of \( \bar{\nu} = 4.5 \times 10^5 \, \text{m}^{-1} \), we can follow these steps:

### Step 1: Understand the relationship between energy, wavelength, and wave number
The wave number \( \bar{\nu} \) is related to the wavelength \( \lambda \) by the equation:
\[ \bar{\nu} = \frac{1}{\lambda} \]
From this, we can find the wavelength \( \lambda \):
\[ \lambda = \frac{1}{\bar{\nu}} = \frac{1}{4.5 \times 10^5} \, \text{m} \]

### Step 2: Calculate the wavelength
Calculating \( \lambda \):
\[ \lambda = \frac{1}{4.5 \times 10^5} \approx 2.22 \times 10^{-6} \, \text{m} \]

### Step 3: Use the energy of a photon formula
The energy \( E \) of a single photon can be calculated using the formula:
\[ E = \frac{hc}{\lambda} \]
where:
- \( h \) (Planck's constant) = \( 6.626 \times 10^{-34} \, \text{J s} \)
- \( c \) (speed of light) = \( 3.00 \times 10^8 \, \text{m/s} \)

### Step 4: Substitute the values into the energy formula
Substituting \( \lambda \) into the energy formula:
\[ E = \frac{(6.626 \times 10^{-34})(3.00 \times 10^8)}{2.22 \times 10^{-6}} \]

### Step 5: Calculate the energy of a single photon
Calculating \( E \):
\[ E \approx \frac{1.9878 \times 10^{-25}}{2.22 \times 10^{-6}} \approx 8.95 \times 10^{-20} \, \text{J} \]

### Step 6: Calculate the number of photons required
To find the number of photons \( n \) needed to provide 1 J of energy, we use:
\[ n = \frac{E_{\text{total}}}{E_{\text{photon}}} \]
where \( E_{\text{total}} = 1 \, \text{J} \):
\[ n = \frac{1}{8.95 \times 10^{-20}} \approx 1.12 \times 10^{19} \]

### Final Answer
The number of photons required is approximately:
\[ n \approx 1.1 \times 10^{19} \]