A source of monochromatic radiation of wavelength 400 nm provides 1000 J of energy in 10 seconds. When this radiation falls on the surface of sodium, \(x \times 10^{20}\) electrons are ejected per second. Assume that wavelength 400 nm is sufficient for ejection of electron from the surface of sodium metal. The value of \(x\) is ________. (Nearest integer) (\(h = 6.626 \times 10^{-34} \text{ Js}\))
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For a quick estimate, the energy of a 400 nm photon is roughly 3.1 eV. Multiplying by \(1.6 \times 10^{-19}\) gives the energy in Joules.
Step 1: Understanding the Concept:
In the photoelectric effect, one photon ejections one electron if the photon's energy is higher than the metal's work function.
Total energy provided per second (power) is the product of the number of photons per second and the energy of a single photon. Step 2: Key Formula or Approach:
1. Energy of one photon: \(E = \frac{hc}{\lambda}\).
2. Number of photons per second (\(n\)) = \(\frac{\text{Total Power}}{\text{Energy of one photon}}\). Step 3: Detailed Explanation:
1. Calculate Power (\(P\)):
Energy = \(1000 \text{ J}\), Time = \(10 \text{ s}\).
Power \(P = \frac{1000}{10} = 100 \text{ J/s}\).
2. Calculate Energy of one photon (\(E\)):
\(\lambda = 400 \text{ nm} = 400 \times 10^{-9} \text{ m}\).
\[ E = \frac{6.626 \times 10^{-34} \times 3 \times 10^8}{400 \times 10^{-9}} \]
\[ E = 4.9695 \times 10^{-19} \text{ J} \]
3. Calculate Number of electrons per second (\(n\)):
\[ n = \frac{100}{4.9695 \times 10^{-19}} \approx 20.12 \times 10^{19} \]
\[ n \approx 2.012 \times 10^{20} \]
So, \(x \approx 2\). Step 4: Final Answer:
The value of \(x\) is 2.
