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 follow these steps:

### Step 1: Understand the relationship between power, energy, and photons
The power (P) of the lamp is given as 100 W. Power is defined as the energy emitted per unit time. Therefore, the energy emitted in one second is equal to the power of the lamp.

\[ E = P \times t \]
Since we are considering one second, we have:
\[ E = 100 \, \text{J} \]

### Step 2: Calculate the energy of one photon
The energy (E_photon) of a single photon can be calculated using the formula:
\[ E_{\text{photon}} = \frac{hc}{\lambda} \]
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)

Now substituting the values:
\[ E_{\text{photon}} = \frac{(6.625 \times 10^{-34} \, \text{J s}) \times (3 \times 10^8 \, \text{m/s})}{5.89 \times 10^{-9} \, \text{m}} \]

### Step 3: Calculate the energy of one photon
Calculating the numerator:
\[ E_{\text{photon}} = \frac{(6.625 \times 3) \times 10^{-34 + 8}}{5.89 \times 10^{-9}} = \frac{19.875 \times 10^{-26}}{5.89 \times 10^{-9}} \]
Now, performing the division:
\[ E_{\text{photon}} = 3.375 \times 10^{-18} \, \text{J} \]

### Step 4: Calculate the number of photons emitted per second
Now, we can find the number of photons emitted per second (N) using the formula:
\[ N = \frac{E}{E_{\text{photon}}} \]
Substituting the values:
\[ N = \frac{100 \, \text{J}}{3.375 \times 10^{-18} \, \text{J}} \]

### Step 5: Perform the calculation
\[ N \approx 2.96 \times 10^{19} \]

### Final Answer
The number of photons emitted by a 100 W sodium lamp in one second is approximately:
\[ N \approx 2.96 \times 10^{19} \]