The number of photons of light of `barv =4.5×10^6m^(−1)` necessary to provide 2 J of energy are:

Options

A. `4xx10^(17)`
B. `2.2xx10^(18)`
C. `4xx10^(18)`
D. `3xx10^(17)`

Explanation

To solve the problem of finding the number of photons of light needed to provide 2 J of energy with a wave number of \( \bar{\nu} = 4.5 \times 10^6 \, \text{m}^{-1} \), we can follow these steps:

### Step-by-Step Solution

**Step 1: Understand the relationship between energy, number of photons, and wave number.**

The energy \( E \) of \( n \) photons can be expressed as:
\[ E = n \cdot E_{\text{photon}} \]
where \( E_{\text{photon}} \) is the energy of a single photon.

**Step 2: Calculate the energy of a single photon in terms of wave number.**

The energy of a single photon can be expressed using the wave number \( \bar{\nu} \):
\[ E_{\text{photon}} = h \cdot c \cdot \bar{\nu} \]
where:
- \( h \) is Planck's constant (\( 6.626 \times 10^{-34} \, \text{J s} \)),
- \( c \) is the speed of light (\( 3 \times 10^8 \, \text{m/s} \)),
- \( \bar{\nu} \) is the wave number.

**Step 3: Substitute the values into the equation for energy of a single photon.**

Substituting the values:
\[ E_{\text{photon}} = (6.626 \times 10^{-34} \, \text{J s}) \cdot (3 \times 10^8 \, \text{m/s}) \cdot (4.5 \times 10^6 \, \text{m}^{-1}) \]

**Step 4: Calculate \( E_{\text{photon}} \).**

Calculating this gives:
\[ E_{\text{photon}} = 6.626 \times 10^{-34} \times 3 \times 10^8 \times 4.5 \times 10^6 \]
\[ E_{\text{photon}} = 8.93 \times 10^{-19} \, \text{J} \]

**Step 5: Use the total energy to find the number of photons.**

Now, we know the total energy \( E = 2 \, \text{J} \). We can find the number of photons \( n \) using:
\[ n = \frac{E}{E_{\text{photon}}} \]
Substituting the values:
\[ n = \frac{2 \, \text{J}}{8.93 \times 10^{-19} \, \text{J}} \approx 2.24 \times 10^{18} \]

### Final Answer
The number of photons necessary to provide 2 J of energy is approximately:
\[ n \approx 2.24 \times 10^{18} \]