Question

A 250 watt bulb emits monochromatic light wavelength 198.78 nm. How many photons are emitted by the bulb per second?

Hint:

Look for mathematical simplifications before doing raw arithmetic. Values like $198.78$ in a physics/chemistry problem are almost always chosen specifically to cancel cleanly with combinations of fundamental constants like $h$ and $c$ ($6.626 \times 3 = 19.878$).

Answer 1

Step 1: Understanding the Concept:
Power (Watts) is defined as energy emitted per unit time (Joules/second). The total energy emitted by the bulb in one second is made up of the sum of the discrete energies of all individual photons emitted.

Step 2: Key Formula or Approach:
1. Relate Power to total energy per second: $E_{\text{total}} = \text{Power} \times \text{time}$
2. Calculate energy of a single photon: $E_{\text{photon}} = \frac{hc}{\lambda}$
3. Equate total energy to number of photons times energy per photon: $E_{\text{total}} = n \times E_{\text{photon}}$
Combine to find $n$ (photons per second): $n = \frac{P}{E_{\text{photon}}} = \frac{P \lambda}{hc}$

Step 3: Detailed Explanation:
Given values:
- Power ($P$) = 250 W = 250 J/s. Therefore, energy emitted in 1 second ($E_{\text{total}}$) = 250 J.
- Wavelength ($\lambda$) = $198.78 \text{ nm} = 198.78 \times 10^{-9} \text{ m}$
- Planck's constant ($h$) $\approx 6.626 \times 10^{-34} \text{ J s}$
- Speed of light ($c$) $\approx 3 \times 10^8 \text{ m/s}$
First, calculate the energy of one single photon:
$E_{\text{photon}} = \frac{hc}{\lambda}$
$E_{\text{photon}} = \frac{(6.626 \times 10^{-34} \text{ J s}) \times (3 \times 10^8 \text{ m/s})}{198.78 \times 10^{-9} \text{ m}}$
$E_{\text{photon}} = \frac{19.878 \times 10^{-26}}{198.78 \times 10^{-9}} \text{ J}$
Notice the convenient numbers designed for easy calculation without a calculator:
$E_{\text{photon}} = \left(\frac{19.878}{198.78}\right) \times 10^{-26 - (-9)}$
$E_{\text{photon}} = 0.1 \times 10^{-17} \text{ J}$
$E_{\text{photon}} = 10^{-18} \text{ J}$
Now, calculate the number of photons ($n$) emitted per second:
$n = \frac{E_{\text{total}}}{E_{\text{photon}}}$
$n = \frac{250 \text{ J}}{10^{-18} \text{ J/photon}}$
$n = 250 \times 10^{18} \text{ photons}$
$n = 2.5 \times 10^{20} \text{ photons}$

Step 4: Final Answer:
The number of photons emitted per second is $2.5 \times 10^{20}$.