Question

Titanium metal requires light with a maximum wavelength of \(286 \mathrm{nm}\) to emit electrons. (a) What is the minimum energy of the photons necessary to emit electrons from titanium via the photoelectric effect? (b) What is the frequency of this radiation? (c) Is it possible to eject electrons from titanium metal using infrared light? (d) If titanium is irradiated with light of wavelength \(276 \mathrm{nm}\), what is the maximum possible kinetic energy of the emitted electrons?

Short answer

(a) The minimum energy of the photons necessary to emit electrons from titanium via the photoelectric effect is \(E = 6.63 \times 10^{-34} \mathrm{Js} \times \frac{3.0 \times 10^8 \mathrm{m/s}}{286 \times 10^{-9} \mathrm{m}}\). (b) The frequency of this radiation is \(\nu = \frac{3.0 \times 10^8 \mathrm{m/s}}{286 \times 10^{-9} \mathrm{m}}\). (c) It is not possible to eject electrons from titanium metal using infrared light as the maximum wavelength of infrared light (700 nm) is greater than the given wavelength of 286 nm. (d) To find the maximum possible kinetic energy of emitted electrons, first calculate the energy of photons at wavelength \(276 \mathrm{nm}\), \(E_2 = 6.63 \times 10^{-34} \mathrm{Js} \times \frac{3.0 \times 10^8 \mathrm{m/s}}{276 \times 10^{-9} \mathrm{m}}\). Then, the maximum possible kinetic energy is \(K = E_2 - E\).

Step-by-step solution

Calculate the minimum energy of the photons to emit electrons

We need to find the minimum energy of the photons to emit electrons from titanium via the photoelectric effect. We can use the formula \(E = h \nu\), where \(E\) is the energy, \(h\) is Planck's constant (\(6.63 \times 10^{-34} \mathrm{Js}\)), and \(\nu\) is the frequency. First, we need to find the frequency. The frequency can be calculated using the formula \(\nu = \frac{c}{\lambda}\), where \(c\) is the speed of light (\(3.0 \times 10^8 \mathrm{m/s}\)), and \(\lambda\) is the wavelength. Given the maximum wavelength \(\lambda = 286 \times 10^{-9} \mathrm{m}\), let's calculate the frequency: \[\nu = \frac{3.0 \times 10^8 \mathrm{m/s}}{286 \times 10^{-9} \mathrm{m}}\] Now, we will use this frequency to calculate the minimum energy of the photons: \[E = h \nu\]

Calculate the frequency of the radiation

We have already calculated the frequency of the radiation in step 1. So, no additional calculations are needed.

Determine whether it is possible to eject electrons using infrared light

To figure out if it's possible to eject electrons using infrared light, we need to compare the maximum wavelength of infrared light with the given wavelength of 286 nm. If the maximum wavelength of infrared light is less than 286 nm, then it is not possible to eject electrons using infrared light. The range of wavelengths for infrared light is approximately 700 nm to 1 mm. Since 286 nm is shorter than the minimum wavelength of infrared light, it is not possible to eject electrons from titanium metal using infrared light.

Calculate the maximum possible kinetic energy of the emitted electrons

Given the wavelength of 276 nm, we first need to find the energy of photons using this wavelength. \[\lambda_2 = 276 \times 10^{-9} \mathrm{m}\] \[E_2 = h \frac{c}{\lambda_2}\] The maximum possible kinetic energy of the emitted electrons can be calculated using the formula: \[K = E_2 - E\] Where \(K\) is the kinetic energy, \(E_2\) is the energy of photons at wavelength \(276 \mathrm{nm}\), and \(E\) is the minimum energy of photons. Calculate the maximum possible kinetic energy of the emitted electrons using the calculated values for \(E_2\) and \(E\).

Key concepts

Planck's Constant

Planck's constant, denoted as \(h\), is a fundamental constant in quantum mechanics. Its value is approximately \(6.63 \times 10^{-34} \, \mathrm{Js}\). It plays a crucial role in the quantization of energy and is primarily used to describe the properties of photons, which are particles of light.

In the context of the photoelectric effect, Planck's constant is part of the equation \(E = h u\), where \(E\) is the energy of a photon and \(u\) is its frequency. This equation implies that energy is quantized and only comes in integer multiples of these energy quanta which are determined by Planck's constant. It also highlights the direct proportionality between energy and frequency, meaning as the frequency of light increases, its energy does as well.

Understanding Planck's constant helps students appreciate how energy transition occurs at a microscopic level, distinguishing classical physics from quantum mechanics.

Photon Energy

Photon energy refers to the energy carried by a photon, which is a fundamental particle representing a quantum of light. The equation \(E = h u\) describes how a photon's energy depends on its frequency. To find the energy of a photon, we can use Planck's constant \(h = 6.63 \times 10^{-34} \, \mathrm{Js}\) and the frequency of the light \(u\).

The photon energy is significant because it dictates whether particular events, like the emission of electrons from a metal surface, can occur. For example, in the given exercise about titanium, only photons with a certain minimum energy threshold will cause electron emission. This threshold energy corresponds to the work function, a property of the material.

Calculation of photon energy is straightforward:

• Find the frequency using \(u = \frac{c}{\lambda}\), where \(c\) is the speed of light and \(\lambda\) is the wavelength.
• Use the frequency in \(E = h u\) to find the energy.

By knowing the energy content in photons hitting a surface, one can predict and analyze reactions like electron ejections, essential in fields like photovoltaics and electronics.

Infrared Light

Infrared light is a type of electromagnetic radiation with wavelengths longer than visible light, ranging roughly from 700 nm to 1 mm. It falls between visible light and microwave radiation on the electromagnetic spectrum, and while it is invisible to the human eye, we feel it as heat.

In the context of the photoelectric effect, infrared light’s longer wavelength means it has lower energy than visible light. One of the key aspects to understand here is the inverse relationship between wavelength and frequency, where a longer wavelength corresponds to a lower frequency and thus lower energy, as per the formula \(E = h u\).

For the case with titanium, where its work function requires a photon energy higher than what typical infrared light can provide, it implies:

• Infrared light cannot eject electrons from titanium due to insufficient energy.
• The energy tied to infrared light is generally absorbed as heat rather than causing electronic transitions.

Understanding infrared light is crucial for applications in thermal imaging, night-vision, and in understanding heat absorption and emission in various materials.