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

First, we find the frequency of the emitted photon: $$f=\frac{3\times {10}^8}{150\times {10}^{-9}}=2\times {10}^{15}\mathrm{Hz}$$ Next, calculate the energy of one photon: $$E=hf=(6.626\times {10}^{-34})(2\times {10}^{15})=1.33\times {10}^{-18}\mathrm{J}$$ Now, find the total number of emitted photons: $$\text{number of photons}=\frac{1.98\times {10}^5}{1.33\times {10}^{-18}}\approx 1.49\times {10}^{23}$$ Finally, since each carbon atom emits one photon, the number of carbon atoms in the sample is approximately $1.49\times {10}^{23}$.

Step-by-step solution

Calculate the energy of one photon emitted by carbon

We will use Planck's equation to find the energy of one photon emitted by carbon: $$E=hf$$ where $E$ is the energy of the photon, $h\approx 6.626\times {10}^{-34}\mathrm{J}.\mathrm{s}$ is Planck's constant, and $f$ is the frequency of the photon. Since the wavelength ($\lambda$) of the emitted photon is given, we can find the frequency using the speed of light ($c\approx 3\times {10}^8\mathrm{m}/\mathrm{s}$): $$f=\frac{c}{\lambda }$$

Find the frequency of the photon emitted

We are given the wavelength as $150\mathrm{nm}$. Let's convert it to meters and find the frequency: $$\lambda =150\times {10}^{-9}\mathrm{m}$$ Using the speed of light, we can find the frequency: $$f=\frac{3\times {10}^8}{150\times {10}^{-9}}$$

Calculate the energy of one photon

Now that we have the frequency, we can find the energy of one photon emitted by carbon: $$E=hf=(6.626\times {10}^{-34})\left(\frac{3\times {10}^8}{150\times {10}^{-9}}\right)$$

Find the total number of emitted photons

We are given the total amount of energy emitted as $1.98\times {10}^5\mathrm{J}$. To find the total number of emitted photons, we will divide the total energy by the energy of one photon: $$\text{number of photons}=\frac{1.98\times {10}^5}{E}$$

Calculate the number of carbon atoms

Finally, since we are assuming that each carbon atom emits one photon, the number of carbon atoms is equal to the number of emitted photons. So the number of carbon atoms in the sample is: $$\text{number of carbon atoms}=\text{number of photons}$$ Calculate the values in each step to find the number of carbon atoms in the sample.

Key concepts

Planck's constant

Planck's constant, denoted by the symbol $h$, is a fundamental constant that plays a crucial role in quantum mechanics. Its value is approximately $6.626\times {10}^{-34}\text{ J.s}$. This tiny constant relates the energy of a photon to the frequency of its electromagnetic wave.

Planck's equation, $E=hf$, expresses this relationship, where $E$ is energy, $h$ is Planck's constant, and $f$ is frequency.

• It shows that energy is quantized, meaning it can only take specific discrete values.
• Higher frequency waves have more energy per photon.

Understanding Planck's constant helps us grasp why and how atoms can absorb or emit electromagnetic radiation in quantized amounts, pivotal for calculating photonic interactions at an atomic level.

Speed of light

The speed of light, represented by $c$, is one of the constants defining our understanding of physics. It is roughly $3\times {10}^8\text{ m/s}$.

This constant is key to linking wavelength and frequency of light through the equation $c=\lambda f$, where $\lambda$ is the wavelength and $f$ the frequency. This equation allows us to compute the frequency when the wavelength is given, or vice versa.

• The speed of light is a universal constant and is the maximum speed at which all energy, matter, and information in the universe can travel.
• Light travels fastest in a vacuum and slows down when passing through materials due to interaction with atoms.

This comprehension of the speed of light sets the stage for further insight into how electromagnetic scattering and absorption operate at the atom level.

Energy calculation

Energy calculation for photons is an essential process in understanding electromagnetic wave interactions. The equation $E=hf$ is central here, where $E$ stands for energy, $h$ is Planck’s constant, and $f$ signifies frequency.

To find the energy of a single photon:

• First, determine its frequency using the wavelength and the speed of light $f=\frac{c}{\lambda }$.
• Substitute this frequency into Planck's equation to find $E$.

This relationship shows how light’s energy is tied directly to its frequency, linking mathematical models to physical processes such as photon emission and absorption by atoms. This calculation aids in determining how many photons are present in a given energy amount, crucial for applications like the exercise on calculating carbon atoms.

Wavelength and Frequency

Wavelength and frequency are vital properties describing electromagnetic waves, including light. Wavelength, $\lambda$, refers to the distance between consecutive wave peaks and is typically measured in meters. Frequency, $f$, is the number of wave cycles passing a point per second, measured in hertz (Hz).

These quantities are inversely related through the speed of light: $c=\lambda f$.

• Shorter wavelengths correspond to higher frequencies and vice versa.
• This inverse relationship helps determine wave properties from measurable quantities.

In photon emission, understanding this link helps convert between wavelength and frequency for energy calculations. These concepts are especially useful in spectroscopy, where analyzing light from substances reveals details about their atomic and molecular structures.