To solve the problem of determining the number of photons emitted per second by a 20 W light source emitting monochromatic light of wavelength 600 nm in terms of Avogadro's constant, we can follow these steps:
### Step 1: Understand the relationship between power, energy, and number of photons
The power (P) of the light source can be expressed in terms of the energy of the photons emitted. The energy (E) of a single photon can be calculated using the formula:
\[ E = \frac{hc}{\lambda} \]
where:
- \( h \) 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 of light (in meters)
### Step 2: Convert the wavelength from nanometers to meters
The wavelength given is 600 nm. To convert this to meters:
\[ \lambda = 600 \, \text{nm} = 600 \times 10^{-9} \, \text{m} = 6 \times 10^{-7} \, \text{m} \]
### Step 3: Calculate the energy of a single photon
Now, substituting the values into the energy formula:
\[ E = \frac{(6.626 \times 10^{-34} \, \text{J s})(3 \times 10^8 \, \text{m/s})}{6 \times 10^{-7} \, \text{m}} \]
Calculating this gives:
\[ E = \frac{1.9878 \times 10^{-25} \, \text{J m}}{6 \times 10^{-7} \, \text{m}} \]
\[ E \approx 3.313 \times 10^{-19} \, \text{J} \]
### Step 4: Calculate the number of photons emitted per second
The number of photons (n) emitted per second can be calculated using the formula:
\[ n = \frac{P}{E} \]
where \( P = 20 \, \text{W} \) (the power of the light source).
Substituting the values:
\[ n = \frac{20 \, \text{W}}{3.313 \times 10^{-19} \, \text{J}} \]
\[ n \approx 6.03 \times 10^{19} \, \text{photons/s} \]
### Step 5: Express the number of photons in terms of Avogadro's constant
Avogadro's constant \( N_A \) is approximately \( 6.022 \times 10^{23} \, \text{mol}^{-1} \). To express the number of photons in terms of \( N_A \):
\[ n \text{ (in terms of } N_A) = \frac{6.03 \times 10^{19}}{6.022 \times 10^{23}} N_A \]
Calculating this gives:
\[ n \approx 10^{-4} N_A \]
### Final Answer
Thus, the number of photons emitted per second in terms of Avogadro's constant is approximately:
\[ n \approx 10^{-4} N_A \]
