If the wavelength of light emitted by a sodium vapour lamp is 5893 A then the numebr of photons emitted in 5 hour by a 50 W lamp will be

Options

A. `7.4 xx10^(20)` photons
B. `4.46 xx10^(22)` photons
C. `2.7 xx10^(24)` photons
D. `5.4 xx10^(23)` photons

Answer

To solve the problem of finding the number of photons emitted by a sodium vapor lamp with a wavelength of 5893 Å in 5 hours by a 50 W lamp, we will follow these steps:

### Step 1: Convert Time to Seconds
First, we need to convert the time from hours to seconds.

\[ \text{Time in seconds} = 5 \text{ hours} \times 60 \text{ minutes/hour} \times 60 \text{ seconds/minute} = 18000 \text{ seconds} \]

### Step 2: Calculate Total Energy
Next, we calculate the total energy emitted by the lamp using the formula:

\[ \text{Energy} = \text{Power} \times \text{Time} \]

Given that the power of the lamp is 50 W:

\[ \text{Energy} = 50 \text{ W} \times 18000 \text{ s} = 900000 \text{ J} = 9 \times 10^5 \text{ J} \]

### Step 3: Calculate Energy of One Photon
Now, we need to calculate the energy of one photon emitted by the lamp. The energy of a photon can be calculated using the formula:

\[ E = \frac{hc}{\lambda} \]

Where:
- \( h \) (Planck's constant) = \( 6.626 \times 10^{-34} \text{ J s} \)
- \( c \) (speed of light) = \( 3 \times 10^8 \text{ m/s} \)
- \( \lambda \) (wavelength) = \( 5893 \text{ Å} = 5893 \times 10^{-10} \text{ m} \)

Substituting the values:

\[ E = \frac{(6.626 \times 10^{-34} \text{ J s}) \times (3 \times 10^8 \text{ m/s})}{5893 \times 10^{-10} \text{ m}} \]

Calculating the energy:

\[ E \approx \frac{1.9878 \times 10^{-25}}{5893 \times 10^{-10}} \approx 3.37 \times 10^{-19} \text{ J} \]

### Step 4: Calculate the Number of Photons
Finally, we can find the number of photons emitted by dividing the total energy by the energy of one photon:

\[ \text{Number of photons} = \frac{\text{Total Energy}}{\text{Energy of one photon}} = \frac{9 \times 10^5 \text{ J}}{3.37 \times 10^{-19} \text{ J}} \]

Calculating this gives:

\[ \text{Number of photons} \approx 2.67 \times 10^{24} \]

### Final Answer
Thus, the number of photons emitted in 5 hours by a 50 W sodium vapor lamp is approximately:

\[ \text{Number of photons} \approx 2.7 \times 10^{24} \]