Question

What would the minimum work function for a metal have to be for visible light (380-750 nm) to eject photoelectrons?

Short answer

The minimum work function must be less than 3.26 eV.

Step-by-step solution

Understand the Problem

The problem is asking for the minimum work function needed for photoelectrons to be ejected by visible light. This involves the photoelectric effect, where light must have enough energy, given by the photon's energy, to overcome the work function of the metal.

Identify Key Formulas

The energy of a photon is given by the formula $E=\frac{hc}{\lambda }$, where $h$ is Planck's constant ($6.626\times {10}^{-34}\text{ J s}$), $c$ is the speed of light ($3\times {10}^8\text{ m/s}$), and $\lambda$ is the wavelength of light.

Calculate the Maximum Photon Energy

For the shortest wavelength of visible light (380 nm), the energy of the photon is maximum. Convert 380 nm to meters: $380\text{ nm}=380\times {10}^{-9}\text{ m}$. Substitute into the equation: $E=\frac{(6.626\times {10}^{-34})(3\times {10}^8)}{380\times {10}^{-9}}$. Calculate to find $E$.

Solve for Minimum Work Function

The minimum work function must be equal to or less than the maximum photon energy calculated in Step 3. Convert the energy from Joules to electronvolts (1 eV = $1.602\times {10}^{-19}$ J), to express the work function in eV. Compute the value.

Key concepts

Work Function

The work function is a fundamental concept in the context of the photoelectric effect. It represents the minimum energy required to dislodge an electron from the surface of a material, typically a metal. In simple terms, it's like the energy "threshold" that must be crossed for electrons to escape from the material when light hits it.
The work function is crucial because only photons with energy equal to or greater than this value can successfully eject electrons. For metals that interact with visible light, determining the minimum work function is essential for applications like photoelectric sensors and solar cells.

• This energy is usually measured in electronvolts (eV), a more convenient unit than Joules when working on the atomic scale.
• When photons from visible light strike a metal, their energy must surpass the work function for electrons to be emitted.
• The photoelectric effect reveals how light can be thought of as particles (photons) with distinct energy levels based on their wavelength.

Photon Energy

Photon energy is a key factor in the photoelectric effect. It dictates whether or not an electron will be ejected from the surface of a metal. The energy of a photon is dependent on its wavelength and can be calculated using the formula:$$E=\frac{hc}{\lambda }$$Here, $E$ represents the photon’s energy, $h$ is Planck's constant, $c$ is the speed of light, and $\lambda$ is the wavelength of the photon.
Understanding photon energy helps us predict which wavelengths are capable of ejecting electrons. Shorter wavelengths have higher frequencies and therefore more energy. Consequently, they have a better chance of overcoming the work function of a metal.

• Photon energy is directly proportional to frequency and inversely proportional to wavelength.
• The higher the energy of a photon, the more likely it is to eject electrons from a metal surface.
• In visible light, photons from the violet end (shorter wavelength) possess more energy than those from the red end.

Visible Light Spectrum

The visible light spectrum is the portion of the electromagnetic spectrum that the human eye can see. It ranges from approximately 380 nm to 750 nm in wavelength. Each color within this spectrum corresponds to a different wavelength, with violet having the shortest wavelength and red having the longest.
When considering the photoelectric effect, it's important to understand how the visible light spectrum relates to photon energy. Shorter wavelengths (like violet) carry higher energy photons than longer wavelengths (like red).

• The visible spectrum usually appears as a continuous range of colors, from violet to red.
• Only photons within specific energy ranges can overcome the work function of certain metals.
• This is why, in our problem, we calculate the minimum work function by considering the shortest wavelength (380 nm) since it has the highest energy potential.

Planck's Constant

Planck's constant, denoted as $h$, is a fundamental quantity in quantum mechanics that plays a key role in the calculation of photon energy. Its value is approximately $6.626\times {10}^{-34}\text{ Js}$. Planck's constant is pivotal because it bridges wave and particle models of light, showing light as quantized packets of energy called photons.
In the formula for photon energy $E=\frac{hc}{\lambda }$, Planck's constant is multiplied by the speed of light $c$ and divided by wavelength $\lambda$ to find the energy of a photon.

• Planck’s constant is a fundamental constant used in various quantum mechanical equations, not just in calculations involving photons.
• This constant underpins the quantization of energy, limiting energy exchanges to discrete amounts.
• By understanding this constant, we gain deeper insights into the dual nature of electromagnetic radiation.