A 50 watt bulb emits monochromatic red light of wavelength of 795 nm. The number of photons emitted per second by the bulb is \( x \times 10^{20} \). The value of \( x \) is _________. (Nearest integer)

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Accepted Answer

Detailed Step-by-Step Solution

This problem uses Planck's Quantum Theory, relating the total power emitted to the energy of individual photons.

Step 1: Understand Power in terms of Energy

Power (\(P\)) is defined as the total energy emitted per unit time (Joules per second).

P = 50 \text{ W} = 50 \text{ J/s}

This means the total energy emitted by the bulb in one second is \( E_{\text{total}} = 50 \text{ J} \).

Step 2: Relate Total Energy to Number of Photons

If \( n \) is the number of photons emitted per second, the total energy is the sum of the energies of these \( n \) photons:

E_{\text{total}} = n \times E_{\text{photon}}

The energy of a single photon is given by \( E_{\text{photon}} = \frac{hc}{\lambda} \). Substituting this in:

P = n \times \frac{hc}{\lambda}

Rearranging the formula to solve for \( n \) (number of photons):

n = \frac{P \times \lambda}{hc}

Step 3: Substitute the Values and Calculate

\( P = 50 \text{ J/s} \)
\( \lambda = 795 \text{ nm} = 795 \times 10^{-9} \text{ m} \)
\( h = 6.63 \times 10^{-34} \text{ J s} \)
\( c = 3.0 \times 10^8 \text{ m/s} \)

n = \frac{50 \times 795 \times 10^{-9}}{6.63 \times 10^{-34} \times 3.0 \times 10^8}

Calculate the numerator and denominator separately:

\text{Numerator} = 39750 \times 10^{-9} \) \( \text{Denominator} = 19.89 \times 10^{-26}

Now, divide them:

n = \frac{39750}{19.89} \times 10^{-9 - (-26)} \) \( n \approx 1998.49 \times 10^{17}

Convert to scientific notation to match the question format (\( \times 10^{20} \)):

n = 1.99849 \times 10^{20} \approx 2 \times 10^{20}

Conclusion: We are given that \( n = x \times 10^{20} \). Comparing this with our result \( 2 \times 10^{20} \), the value of \( x \) to the nearest integer is 2.

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Number of photons emitted per second by a 50W bulb | Chemca Solution

A 50 watt bulb emits monochromatic red light of wavelength of 795 nm. The number of photons emitted per second by the bulb is \( x \times 10^{20} \). The value of \( x \) is _________. (Nearest integer)

Detailed Step-by-Step Solution

Step 1: Understand Power in terms of Energy

Step 2: Relate Total Energy to Number of Photons

Step 3: Substitute the Values and Calculate

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