Question

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.

Short answer

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

Step-by-step solution

Find the energy of 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: \(E = (6.626 \times 10^{-34} \text{Js})(2 \times 10^{15} \text{Hz}) = 1.3252 \times 10^{-18} \text{J}\)

Calculate the number of photons

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.

Present the final answer

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

Key concepts

Planck's Equation

Planck's Equation is a fundamental concept in understanding the energy properties of photons. It provides a direct relationship between a photon's energy and its frequency. The equation is expressed as \(E = hf\). Here, \(E\) signifies the energy of the photon, \(h\) represents Planck's constant, which is approximately \(6.626 \times 10^{-34} \text{Js}\), and \(f\) denotes the frequency of the photon. This equation forms the backbone of quantum mechanics, illustrating that energy is quantized and exchanged in discrete amounts called quanta. Understanding this equation is crucial when calculating the energy associated with photons of different frequencies or wavelengths.

Wavelength-to-Frequency Conversion

In many scenarios, we need to switch between the wavelength and frequency descriptions of light, since some information is easier to obtain in either form. To do this, we utilize the equation \(f = \frac{c}{\lambda}\). Here, \(f\) is the frequency, \(c\) is the speed of light with a constant value of \(3 \times 10^{8} \text{m/s}\), and \(\lambda\) is the wavelength of the light.

• Wavelength is typically given in nanometers (nm) in many problems.
• Convert it to meters by multiplying by \(10^{-9}\).
• This formula makes it possible to obtain the frequency necessary to use in Planck’s Equation.

By converting the wavelength to frequency, we further bridge the gap between the macroscopic scale of wave phenomena and the microscopic nature of quantum mechanics.

Carbon Atom Emission

Carbon atoms, like all atoms, have discrete energy levels. When they absorb sufficient energy, electrons get excited to higher energy levels. Upon returning to their original state, these electrons emit energy in the form of photons. In the given problem, each carbon atom emits a single photon of specific energy when returning to its ground state.

• This is a fundamental aspect of atomic emission spectra.
• The emitted photon's energy corresponds to the energy difference between the two levels the electron transitioned between.

Understanding this emission process helps in calculating how many photons, and thus how many atoms, are involved in the emission when a sample of carbon releases a total specified amount of energy.

Photon Frequency

The frequency of a photon is central to determining its energy. As derived previously using the equation \(f = \frac{c}{\lambda}\), the frequency provides a link between a photon's inherent energy and its observable properties such as color or type of radiation (like UV, visible, etc.).In the problem's context, when dealing with a photon emitted by a carbon atom, this frequency helps to quantify the energy that each photon contains, using Planck's Equation. This particular exercise demonstrates a unique scenario where calculations shift parameters from wavelength to frequency to energy, all bound tightly by these fundamental equations.

Total Energy Emission

Total energy emission is the cumulative energy that is released by all the photons emitted by a sample or body. In contexts like the one provided, it involves understanding how individual photon energies combine to account for the total emitted energy. This is critical for computing the number of photons involved.According to the text's problem, we use the formula:\[\text{Number of photons} = \frac{\text{Total energy emitted}}{\text{Energy of one photon}}\]This approach ensures precision by considering the energy of one photon, calculated using frequency in Planck's Equation, to evaluate the total energy emitted. With such knowledge, we can quantify attributes like the amount or number of individual emitters (like atoms), based on their collective energy output.