Question

How much energy is emitted by Avogadro number of atoms, if they each emit a light wave of \(400 \mathrm{~nm}\) wavelength?

Short answer

The total energy emitted by Avogadro's number of atoms is \(E_\text{total} = ((6.63 \times 10^{-34}\,\text{J.s}) \times f) \times (6.022 \times 10^{23}\,\text{atoms/moles})\), where \(f = \frac{3\times10^8\,\text{m/s}}{400 \times 10^{-9}\,\text{m}}\).

Step-by-step solution

Calculate the frequency of the light wave

Using the relationship between frequency and wavelength, we can calculate the frequency of the light wave with a wavelength of 400 nm: \(f = \frac{c}{\lambda}\) To utilize this equation, we need to convert the wavelength from nanometers to meters: \(400\,\text{nm} = 400 \times 10^{-9}\,\text{m}\) Now, we can calculate the frequency: \(f = \frac{3\times10^8\,\text{m/s}}{400 \times 10^{-9}\,\text{m}}\)

Calculate the energy of one photon

Now that we have the frequency, we can find the energy of one photon using the Planck's equation: \(E = h \times f\) where \(h = 6.63 \times 10^{-34}\,\text{J.s}\) is the Planck's constant. \(E = (6.63 \times 10^{-34}\,\text{J.s}) \times f\)

Calculate the total energy emitted by Avogadro's number of atoms

Finally, to find the total energy emitted by Avogadro's number of atoms, we need to multiply the energy of one photon by Avogadro's number (\(N_A = 6.022 \times 10^{23}\,\text{atoms/moles}\)): \(E_\text{total} = E \times N_A\) Now, substitute the values of E and \(N_A\) to get the final answer: \(E_\text{total} = ((6.63 \times 10^{-34}\,\text{J.s}) \times f) \times (6.022 \times 10^{23}\,\text{atoms/moles})\)

Key concepts

Avogadro's Number

Avogadro's Number is a fundamental constant in chemistry and physics. It represents the number of atoms, molecules, or other particles in one mole of a substance. This number, approximately \(6.022 \times 10^{23}\), is especially useful when dealing with quantities in chemical reactions and various physical transformations.When a problem involves Avogadro's Number, such as calculating the total energy emitted by multiple atoms or molecules, it's crucial to remember that this number provides you with a way to convert between the macroscopic scale of substances that we can observe and the microscopic scale of atoms and molecules.

• Link to Moles: Avogadro's Number bridges the gap between an atomic scale and a human-scale quantity, by defining the relationship between moles and particles.
• Use in Calculations: It is often used to scale up calculations from one particle to many, as seen when calculating total energy emissions from atoms.
• Constant in Equations: Often noted as \(N_A\), it appears in equations dealing with quantities of particles in chemical reactions.

Wavelength and Frequency

Understanding the relationship between wavelength and frequency is vital when analyzing light waves. Both are key properties of waves and are inversely related. This means that as the wavelength of a wave increases, its frequency decreases, and vice versa.The fundamental equation that connects these two properties is:\[ c = \lambda \times f \]Here:

• \(c\) is the speed of light in a vacuum, \(3 \times 10^8\,\text{m/s}\).
• \(\lambda\) is the wavelength, the distance between successive crests of a wave.
• \(f\) is the frequency, the number of wave cycles that pass a fixed point per second.

For practical calculations, wavelengths are often converted from nanometers (nm) to meters (m). By knowing the wavelength, you can easily determine the frequency using the speed of light. This conversion process is vital in understanding how different wavelengths, such as visible light at \(400\) nm, translate to specific frequencies which in turn affect energy calculations.

Planck's Constant

Planck's Constant is an essential constant in quantum mechanics that relates the energy of a photon to its frequency. It is denoted as \(h\) and has a value of \(6.63 \times 10^{-34}\,\text{J.s}\). This constant is crucial when dealing with energy quantization in the study of photons.The equation that represents this relationship is:\[ E = h \times f \]Where:

• \(E\) is the energy of a photon.
• \(h\) represents Planck's Constant.
• \(f\) is the frequency of the photon.

Planck's Constant is found at the heart of calculations involving photon energy, particularly in fields like quantum mechanics, photonics, and spectroscopy. Understanding this constant helps explain how energy is quantized and how light interacts with matter at microscopic levels. This knowledge is not only theoretical but has practical applications in technology such as lasers, LED lighting, and even solar panels.