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 in 5 hours, we will follow these steps:

### Step 1: Calculate the total energy emitted by the lamp.
The energy emitted by the lamp can be calculated using the formula:
\[ \text{Energy} = \text{Power} \times \text{Time} \]
Given:
- Power = 50 W
- Time = 5 hours

First, convert the time from hours to seconds:
\[ 5 \text{ hours} = 5 \times 3600 \text{ seconds} = 18000 \text{ seconds} \]

Now, calculate the energy:
\[ \text{Energy} = 50 \text{ W} \times 18000 \text{ s} = 900000 \text{ J} = 9 \times 10^5 \text{ J} \]

### Step 2: Use the energy to find the number of photons emitted.
The energy of a single photon can be calculated using the formula:
\[ E = \frac{n \cdot h \cdot c}{\lambda} \]
Where:
- \(E\) is the total energy,
- \(n\) is the number of photons,
- \(h\) is Planck's constant (\(6.626 \times 10^{-34} \text{ J s}\)),
- \(c\) is the speed of light (\(3 \times 10^8 \text{ m/s}\)),
- \(\lambda\) is the wavelength in meters.

Given:
- Wavelength = 5893 Å = \(5893 \times 10^{-10} \text{ m}\) (since \(1 \text{ Å} = 10^{-10} \text{ m}\))

Now, rearranging the formula to find \(n\):
\[ n = \frac{E \cdot \lambda}{h \cdot c} \]

### Step 3: Substitute the values into the equation.
Substituting the known values:
\[ n = \frac{(9 \times 10^5 \text{ J}) \cdot (5893 \times 10^{-10} \text{ m})}{(6.626 \times 10^{-34} \text{ J s}) \cdot (3 \times 10^8 \text{ m/s})} \]

### Step 4: Calculate the number of photons.
Calculating the denominator:
\[ h \cdot c = (6.626 \times 10^{-34}) \cdot (3 \times 10^8) = 1.9878 \times 10^{-25} \text{ J m} \]

Now, substituting back into the equation for \(n\):
\[ n = \frac{(9 \times 10^5) \cdot (5893 \times 10^{-10})}{1.9878 \times 10^{-25}} \]

Calculating the numerator:
\[ 9 \times 10^5 \cdot 5893 \times 10^{-10} = 5.3037 \times 10^{-4} \text{ J m} \]

Now substituting into the equation:
\[ n = \frac{5.3037 \times 10^{-4}}{1.9878 \times 10^{-25}} \approx 2.67 \times 10^{24} \]

### Final Answer:
The number of photons emitted in 5 hours by a 50 W lamp is approximately:
\[ n \approx 2.67 \times 10^{24} \text{ photons} \]