The number of photons of light of `barv =3.5×10^7m^(−1)` necessary to provide 3 J of energy are:

Options

A. `6xx10^(17)`
B. `4xx10^(17)`
C. `5xx10^(18)`
D. `3xx10^(18)`

Explanation

To find the number of photons of light required to provide 3 J of energy with a given wave number \( \bar{\nu} = 3.5 \times 10^7 \, \text{m}^{-1} \), we can follow these steps:

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

The energy \( E \) of a photon can be expressed in terms of the number of photons \( n \), Planck's constant \( h \), the speed of light \( c \), and the wave number \( \bar{\nu} \) as follows:

\[ E = n \cdot h \cdot c \cdot \bar{\nu} \]

### Step 2: Rearrange the formula to solve for the number of photons \( n \)

From the above equation, we can rearrange it to find the number of photons:

\[ n = \frac{E}{h \cdot c \cdot \bar{\nu}} \]

### Step 3: Substitute the known values

Now, we substitute the known values into the equation:

- Energy \( E = 3 \, \text{J} \)
- Planck's constant \( h = 6.626 \times 10^{-34} \, \text{J s} \)
- Speed of light \( c = 3.0 \times 10^8 \, \text{m/s} \)
- Wave number \( \bar{\nu} = 3.5 \times 10^7 \, \text{m}^{-1} \)

Substituting these values gives:

\[ n = \frac{3 \, \text{J}}{(6.626 \times 10^{-34} \, \text{J s}) \cdot (3.0 \times 10^8 \, \text{m/s}) \cdot (3.5 \times 10^7 \, \text{m}^{-1})} \]

### Step 4: Calculate the denominator

First, calculate the denominator:

\[ h \cdot c \cdot \bar{\nu} = (6.626 \times 10^{-34}) \cdot (3.0 \times 10^8) \cdot (3.5 \times 10^7) \]

Calculating this step-by-step:

1. \( 6.626 \times 10^{-34} \cdot 3.0 \times 10^8 = 1.9878 \times 10^{-25} \)
2. \( 1.9878 \times 10^{-25} \cdot 3.5 \times 10^7 = 6.9563 \times 10^{-18} \)

### Step 5: Calculate the number of photons \( n \)

Now substitute this back into the equation for \( n \):

\[ n = \frac{3}{6.9563 \times 10^{-18}} \approx 4.31 \times 10^{17} \]

### Step 6: Final answer

Thus, the number of photons required is approximately:

\[ n \approx 4.31 \times 10^{17} \]

### Summary

The number of photons of light necessary to provide 3 J of energy is approximately \( 4.31 \times 10^{17} \).

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