The number of photons of light of `barv =2.5×10^6m^(−1)` necessary to provide 2 J of energy are:

Options

A. `6xx10^(17)`
B. `4xx10^(17)`
C. `4xx10^(18)`
D. `3.6xx10^(17)`

Explanation

To solve the problem of finding the number of photons necessary to provide 2 J of energy when the wave number (ν̄) is given as \(2.5 \times 10^6 \, \text{m}^{-1}\), we can follow these steps:

### Step-by-Step Solution:

1. **Understand the relationship between energy and photons**:
The energy (E) of n photons can be expressed as:
\[ E = n \cdot E_{\text{photon}} \]
where \(E_{\text{photon}} = h \cdot \nu\) and \(h\) is Planck's constant, and \(\nu\) is the frequency of the light.

2. **Relate wave number to wavelength**:
The wave number (ν̄) is related to the wavelength (λ) by the equation:
\[ \nū = \frac{1}{\lambda} \]
Therefore, we can express the wavelength as:
\[ \lambda = \frac{1}{\nū} \]

3. **Calculate the frequency**:
The frequency (ν) can be calculated using the speed of light (c) and the wavelength (λ):
\[ \nu = \frac{c}{\lambda} = c \cdot \nū \]

4. **Substituting values**:
We know:
- \(c = 3 \times 10^8 \, \text{m/s}\)
- \(h = 6.626 \times 10^{-34} \, \text{J s}\)
- \(\nū = 2.5 \times 10^6 \, \text{m}^{-1}\)

Therefore, the energy of one photon is:
\[ E_{\text{photon}} = h \cdot \nu = h \cdot (c \cdot \nū) \]

5. **Calculate the energy of one photon**:
\[ E_{\text{photon}} = 6.626 \times 10^{-34} \, \text{J s} \cdot (3 \times 10^8 \, \text{m/s} \cdot 2.5 \times 10^6 \, \text{m}^{-1}) \]
\[ E_{\text{photon}} = 6.626 \times 10^{-34} \cdot 7.5 \times 10^{14} \, \text{J} \]
\[ E_{\text{photon}} \approx 4.9695 \times 10^{-19} \, \text{J} \]

6. **Calculate the number of photons (n)**:
Now, using the total energy (E = 2 J):
\[ n = \frac{E}{E_{\text{photon}}} = \frac{2 \, \text{J}}{4.9695 \times 10^{-19} \, \text{J}} \]
\[ n \approx 4.02 \times 10^{18} \]

### Final Answer:
The number of photons necessary to provide 2 J of energy is approximately \(4.02 \times 10^{18}\).