Number of photons emitted by 100 W sodium lamp in one second is (Given `lamda= 5.89 xx 10^(-9) m , h = 6.625 ×× 10 ^(-34)J s` )

Options

A. `2.9xx10^(20)`
B. `1.9xx10^(18)`
C. `2.9xx10^(18)`
D. `1.9xx10^(20)`

Answer

To find the number of photons emitted by a 100 W sodium lamp in one second, we can use the relationship between power, energy, and the properties of photons.

### Step-by-step Solution:

1. **Understand the relationship between power, energy, and time**:
\[ \text{Power} = \frac{\text{Energy}}{\text{Time}} \]
Given that the time is 1 second, we can express energy as:
\[ \text{Energy} = \text{Power} \times \text{Time} = 100 \, \text{W} \times 1 \, \text{s} = 100 \, \text{J} \]

2. **Relate energy to the number of photons**:
The energy of one photon can be calculated using the formula:
\[ E = \frac{hc}{\lambda} \]
where:
- \( h = 6.625 \times 10^{-34} \, \text{J s} \) (Planck's constant)
- \( c = 3 \times 10^8 \, \text{m/s} \) (speed of light)
- \( \lambda = 5.89 \times 10^{-9} \, \text{m} \) (wavelength)

3. **Calculate the energy of one photon**:
Plugging in the values:
\[ E = \frac{(6.625 \times 10^{-34} \, \text{J s}) \times (3 \times 10^8 \, \text{m/s})}{5.89 \times 10^{-9} \, \text{m}} \]

First calculate the numerator:
\[ (6.625 \times 10^{-34}) \times (3 \times 10^8) = 1.9875 \times 10^{-25} \, \text{J m} \]

Now divide by the wavelength:
\[ E = \frac{1.9875 \times 10^{-25}}{5.89 \times 10^{-9}} \approx 3.375 \times 10^{-17} \, \text{J} \]

4. **Calculate the number of photons emitted**:
The number of photons \( n \) can be calculated by dividing the total energy by the energy of one photon:
\[ n = \frac{\text{Total Energy}}{\text{Energy per photon}} = \frac{100 \, \text{J}}{3.375 \times 10^{-17} \, \text{J}} \]

Performing the division:
\[ n \approx 2.96 \times 10^{18} \]

5. **Final Result**:
Rounding off, the number of photons emitted by the 100 W sodium lamp in one second is approximately:
\[ n \approx 2.9 \times 10^{18} \]