The number of photons of light of `barv =2.5×10^7m^(−1)` necessary to provide 2 J of energy are:

Options

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

Explanation

To solve the question of how many photons of light with a wave number \( \bar{\nu} = 2.5 \times 10^7 \, \text{m}^{-1} \) are necessary to provide 2 J of energy, we can follow these steps:

### Step 1: Understand the relationship between energy, number of photons, and wave number
The energy \( E \) provided by a number of photons \( n \) can be expressed as:
\[ E = n \cdot h \cdot c \cdot \bar{\nu} \]
where:
- \( E \) is the total energy (in joules),
- \( n \) is the number of photons,
- \( h \) is Planck's constant (\( 6.626 \times 10^{-34} \, \text{J s} \)),
- \( c \) is the speed of light (\( 3.00 \times 10^8 \, \text{m/s} \)),
- \( \bar{\nu} \) is the wave number (in \( \text{m}^{-1} \)).

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

### Step 3: Substitute the known values
Substituting the values into the equation:
- \( E = 2 \, \text{J} \)
- \( h = 6.626 \times 10^{-34} \, \text{J s} \)
- \( c = 3.00 \times 10^8 \, \text{m/s} \)
- \( \bar{\nu} = 2.5 \times 10^7 \, \text{m}^{-1} \)

The equation becomes:
\[ n = \frac{2}{(6.626 \times 10^{-34}) \cdot (3.00 \times 10^8) \cdot (2.5 \times 10^7)} \]

### Step 4: Calculate the denominator
Calculating the denominator:
\[ (6.626 \times 10^{-34}) \cdot (3.00 \times 10^8) \cdot (2.5 \times 10^7) = 4.9695 \times 10^{-18} \, \text{J} \]

### Step 5: Calculate the number of photons
Now substituting back into the equation for \( n \):
\[ n = \frac{2}{4.9695 \times 10^{-18}} \approx 4.028 \times 10^{17} \]

### Final Answer
Thus, the number of photons necessary to provide 2 J of energy is approximately:
\[ n \approx 4.028 \times 10^{17} \]