Carbon absorbs energy at a wavelength of \(150 . \mathrm{nm.}\) The total amount of energy emitted by a carbon sample is \(1.98 \times 10^{5} \mathrm{J}\) Calculate the number of carbon atoms present in the sample, assuming that each atom emits one photon.

We can use the Planck's equation, which relates the energy of a photon to its wavelength, to calculate the energy of one photon. The equation is: \(E = hf\), where \(E\) is the energy of the photon, \(h\) is the Planck's constant (\(6.626 \times 10^{-34} \text{Js}\)), and \(f\) is the frequency of the photon. We are given the wavelength, so we need to convert it to frequency using the equation: \(f = \frac{c}{\lambda}\), where \(f\) is the frequency of the photon, \(c\) is the speed of light (\(3 \times 10^8 \text{m/s}\)), and \(\lambda\) is the wavelength of the photon. First, convert the wavelength to meters: \(\lambda = 150 \times 10^{-9} \text{m}\) Now, we can find the frequency: \(f = \frac{3 \times 10^8 \text{m/s}}{150 \times 10^{-9} \text{m}} = 2 \times 10^{15} \text{Hz}\) Finally, we can find the energy of one photon: \(E = (6.626 \times 10^{-34} \text{Js})(2 \times 10^{15} \text{Hz}) = 1.3252 \times 10^{-18} \text{J}\)

We are given the total energy emitted by the carbon sample, which is \(1.98 \times 10^5 \text{J}\). To calculate the number of photons, we can use the energy of one photon that we calculated in Step 1: Number of photons = \(\frac{\text{Total energy}}{\text{Energy of one photon}}\) Number of photons = \(\frac{1.98 \times 10^5 \text{J}}{1.3252 \times 10^{-18} \text{J}} = 1.4939 \times 10^{23}\) Since we have made the assumption that each carbon atom emits one photon, this number is also the number of carbon atoms present in the sample.

The number of carbon atoms present in the sample is approximately \(1.4939 \times 10^{23}\).