Problem 12 - Chemistry and Chemical Reactivity

Question

You are an engineer designing a switch that works by the photoelectric effect. The metal you wish to use in your device requires \(6.7 \times 10^{-19} \mathrm{J} /\) atom to remove an electron. Will the switch work if the light falling on the metal has a wavelength of \(540 \mathrm{nm}\) or greater? Why or why not?

Step-by-Step Solution

Answer

No, the switch will not work because the photon energy is insufficient to remove an electron.

Step 1: Calculate the Energy of the Incoming Photons

The energy of a photon is given by the formula \(E = \frac{hc}{\lambda}\), where \(h\) is Planck's constant \( (6.626 \times 10^{-34} \text{Js} )\), \(c\) is the speed of light \( (3.0 \times 10^{8} \text{m/s})\), and \(\lambda\) is the wavelength of the light. Substituting in \(\lambda = 540 \times 10^{-9} \text{m}\), we find the energy of the photon.

Step 2: Perform the Calculation for Photon Energy

Substitute the given values into the photon energy formula: \[E = \frac{(6.626 \times 10^{-34} \text{Js})(3.0 \times 10^{8} \text{m/s})}{540 \times 10^{-9} \text{m}} = 3.68 \times 10^{-19} \text{J} \text{ per photon}.\]

Step 3: Compare Photon Energy with Required Energy

The energy required to remove an electron from the metal is \(6.7 \times 10^{-19} \text{J}\). Compare this with the energy of a photon \(3.68 \times 10^{-19} \text{J}\). Since the photon's energy is less than the required energy, a photon with \(540 \text{ nm}\) wavelength cannot remove an electron.

Step 4: Draw Conclusion Based on Comparison

Since the energy of light with a wavelength of \(540 \text{ nm}\) or greater is less than the energy required to remove an electron, the switch will not work under these conditions.

Key Concepts

Photon Energy Calculation

The energy of a photon is crucial in understanding the photoelectric effect. It refers to the energy carried by a single photon, which can influence how electrons interact with surfaces. To calculate this energy, you use the equation:

• \(E = \frac{hc}{\lambda}\)

In this formula, \(h\) stands for Planck's constant, \(c\) is the speed of light, and \(\lambda\) represents the wavelength of the light. By substituting the given values into this equation, you determine the photon's energy, which ultimately helps in assessing whether an electron can be ejected from a material when it absorbs a photon.

Planck's Constant

Planck's constant, represented as \(h\), plays a key role in connecting energy and frequency in the quantum world. It is a fundamental constant with the value \(6.626 \times 10^{-34} \text{Js}\).
In the context of the photoelectric effect, it serves as a proportional factor in the equation that calculates photon energy. It essentially allows us to convert between the energy experienced on microscopic scales and the more macroscopically observant qualities like light frequency or wavelength.
Being aware of Planck's constant helps us understand how tiny packets of energy in photons can have significant implications on materials and devices like switches that utilize the photoelectric effect.

Speed of Light

The speed of light, denoted as \(c\), is a universal constant in physics. It travels at \(3.0 \times 10^{8} \text{ m/s}\) in a vacuum and plays a vital role in many physical equations.
In the calculation of photon energy, it functions as part of the equation that combines with Planck's constant and the wavelength of light. Its exact value is imperative to make sure calculations remain accurate when determining important energy thresholds, like those needed for the photoelectric effect to occur.

Wavelength and Energy Relationship

Understanding the relationship between wavelength and energy is essential to the photoelectric effect. Wavelength, represented by \(\lambda\), refers to the distance between consecutive peaks of a wave.
For light, a shorter wavelength means higher energy, while a longer wavelength corresponds to lower energy.

• This relationship is inversely proportional, which means as the wavelength decreases, the energy increases.
• The formula connecting them is expressed as: \(E = \frac{hc}{\lambda}\).

If a photon's wavelength is too long, it won't possess the energy necessary to eject an electron, which is why shorter wavelengths (or higher-energy photons) are often needed in processes like the photoelectric effect. Therefore, understanding and calculating these elements become crucial when designing devices that rely on specific energy thresholds, like the photoelectric switch in our exercise.