The number of photons of light of `barv =3.5×10^6m^(−1)` necessary to provide 2 J of energy are:

Options

A. `6xx10^(17)`
B. `2.87xx10^(18)`
C. `6xx10^(-17)`
D. `2.87xx10^(-18)`

Explanation

To solve the problem of finding the number of photons of light required to provide 2 J of energy with a wave number of \( \bar{\nu} = 3.5 \times 10^6 \, \text{m}^{-1} \), we can follow these steps:

### Step 1: Understand the relationship between energy, number of photons, and wave number
The energy \( E \) provided by \( n \) photons can be expressed as:
\[ E = n \cdot h \cdot \nu \]
where:
- \( E \) is the total energy (in joules),
- \( n \) is the number of photons,
- \( h \) is Planck's constant (\( 6.626 \times 10^{-34} \, \text{J s} \)),
- \( \nu \) is the frequency of the light.

### Step 2: Relate wave number to frequency
The wave number \( \bar{\nu} \) is related to the wavelength \( \lambda \) by:
\[ \bar{\nu} = \frac{1}{\lambda} \]
The frequency \( \nu \) can be related to the wave number by:
\[ \nu = c \cdot \bar{\nu} \]
where \( c \) is the speed of light (\( 3 \times 10^8 \, \text{m/s} \)).

### Step 3: Substitute frequency into the energy equation
Substituting \( \nu \) into the energy equation gives:
\[ E = n \cdot h \cdot (c \cdot \bar{\nu}) \]
Rearranging this to find \( n \):
\[ n = \frac{E}{h \cdot c \cdot \bar{\nu}} \]

### Step 4: Substitute the known values
Now we can substitute the known values into the equation:
- \( E = 2 \, \text{J} \)
- \( h = 6.626 \times 10^{-34} \, \text{J s} \)
- \( c = 3 \times 10^8 \, \text{m/s} \)
- \( \bar{\nu} = 3.5 \times 10^6 \, \text{m}^{-1} \)

So,
\[ n = \frac{2}{(6.626 \times 10^{-34}) \cdot (3 \times 10^8) \cdot (3.5 \times 10^6)} \]

### Step 5: Calculate the value of \( n \)
Calculating the denominator:
\[ (6.626 \times 10^{-34}) \cdot (3 \times 10^8) \cdot (3.5 \times 10^6) = 6.626 \times 3 \times 3.5 \times 10^{-34 + 8 + 6} \]
Calculating the numerical part:
\[ 6.626 \times 3 \times 3.5 \approx 69.5 \]
Thus, the denominator becomes:
\[ 69.5 \times 10^{-20} = 6.95 \times 10^{-19} \]

Now substituting back into the equation for \( n \):
\[ n = \frac{2}{6.95 \times 10^{-19}} \approx 2.877 \times 10^{18} \]

### Final Answer
The number of photons required is approximately:
\[ n \approx 2.877 \times 10^{18} \]