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

Options

A. 2 x `10^18`
B. 1.4 x `10^18`
C. 2 x `10^14`
D. `1.2xx10^(-13)`

Explanation

To find the number of photons of light necessary to provide 1 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:
\[ E = n \cdot h \cdot c \cdot \bar{\nu} \]
Where:
- \( E \) is the energy (1 J in this case),
- \( n \) is the number of photons,
- \( h \) 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 (\( 3.5 \times 10^6 \, \text{m}^{-1} \)).

### Step 2: Rearranging the formula to solve for \( n \)
We can rearrange the equation to solve for \( n \):
\[ n = \frac{E}{h \cdot c \cdot \bar{\nu}} \]

### Step 3: Substitute the known values into the equation
Substituting the values into the equation:
\[ n = \frac{1 \, \text{J}}{(6.626 \times 10^{-34} \, \text{J s}) \cdot (3 \times 10^8 \, \text{m/s}) \cdot (3.5 \times 10^6 \, \text{m}^{-1})} \]

### Step 4: Calculate the denominator
Calculating the denominator:
\[ h \cdot c \cdot \bar{\nu} = (6.626 \times 10^{-34}) \cdot (3 \times 10^8) \cdot (3.5 \times 10^6) \]
Calculating step-by-step:
1. \( 6.626 \times 10^{-34} \times 3 \times 10^8 = 1.9878 \times 10^{-25} \)
2. \( 1.9878 \times 10^{-25} \times 3.5 \times 10^6 = 6.9553 \times 10^{-19} \)

### Step 5: Calculate \( n \)
Now substituting back into the equation for \( n \):
\[ n = \frac{1}{6.9553 \times 10^{-19}} \approx 1.438 \times 10^{18} \]

### Step 6: Final answer
Thus, the number of photons necessary to provide 1 J of energy is approximately:
\[ n \approx 1.4 \times 10^{18} \]

### Summary
The number of photons of light of wave number \( \bar{\nu} = 3.5 \times 10^6 \, \text{m}^{-1} \) necessary to provide 1 J of energy is approximately \( 1.4 \times 10^{18} \).

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