Problem 56 - Chemistry Matter and Change

Question

Platinum What minimum frequency of light is needed to eject a photoelectron from atoms of platinum, which require at least \(9.08 \times 10^{-19} \mathrm{J} /\) photon?

Step-by-Step Solution

Answer

The minimum frequency of light needed to eject a photoelectron from platinum atoms, which require at least \(9.08 \times 10^{-19} \mathrm{J} / \mathrm{photon}\), is \(1.37 \times 10^{15} \mathrm{s}^{-1}\).

Step 1: Understand the photoelectric effect

The photoelectric effect demonstrates that photons (particles of light) can impart energy to electrons. When light with a frequency above a certain threshold strikes the surface of a metal, it can cause the ejection of electrons, called photoelectrons. The energy required to eject a photoelectron from the metal surface is called the work function. For this exercise, the work function of platinum is given as \(9.08 \times 10^{-19} \mathrm{J}\) per photon.

Step 2: Applying Planck's equation

Planck's equation relates the energy of a photon to its frequency as follows: \[ E = h\nu \] where \(E\) is the energy of the photon, \(h\) is Planck's constant (\(h = 6.63 \times 10^{-34} \) J/s), and \(\nu\) is the frequency of the light. Given the work function of platinum (\(9.08 \times 10^{-19}\) J), we can determine the minimum frequency of light required to eject electrons using the equation: \[ \nu = \frac{E}{h} \]

Step 4: Calculate the minimum frequency

Now, by plugging the given values into the formula, we can find the minimum required frequency of light: \[ \nu = \frac{9.08 \times 10^{-19} \mathrm{J}}{6.63 \times 10^{-34}\mathrm{J/s}} \] \[ \nu = 1.37 \times 10^{15} \mathrm{s}^{-1} \] Therefore, the minimum frequency of light necessary to eject a photoelectron from platinum atoms is \(1.37 \times 10^{15} \mathrm{s}^{-1}\).

Key Concepts

Work Function

The work function is a fundamental concept in understanding the photoelectric effect. It is essentially the minimum energy needed to release an electron from the surface of a material. For different materials, this energy requirement, measured in joules per photon, varies significantly. In the context of platinum, the work function is given as \(9.08 \times 10^{-19} \, \text{J/photon}\).
This value implies that any photon with energy less than this amount won’t be able to dislodge an electron. Think of the work function as a barrier that light must overcome to facilitate electron ejection. Understanding the work function helps in determining the threshold frequency below which no photoelectrons are emitted. This principle is paramount in fields like material science and electronics, where controlling electron flow is crucial.

Planck's Equation

Planck's equation forms the backbone of quantum mechanics related to light and electromagnetic waves. It connects the frequency of light \(u\) to its energy \(E\) through the formula:\[E = hu\]Where \(h\) is Planck's constant with a value of \(6.63 \times 10^{-34} \text{ J/s}\).
This equation was pivotal in proposing that energy is quantized, existing in discrete packets known as photons.
Applying Planck's equation allows us to calculate how much energy a given frequency of light provides. Conversely, knowing the energy and requiring the frequency, as in our exercise, can allow us to use this equation to deduce the frequency necessary to overcome a known work function.

Photon Energy

Photon energy exemplifies the distinct nature of light particles and is key to many technological innovations, like photovoltaic cells. Photon energy describes the amount of energy carried by a single photon, and it plays a critical role in determining whether a photon can trigger the photoelectric emission of an electron.
Using Planck's equation \(E = hu\), we can determine the energy of any photon given its frequency. This energy must be compared to the work function to ascertain whether it is sufficient to discharge an electron from a material.

• Higher frequency photons possess greater energy.
• If photon energy surpasses the work function, electrons will be emitted.

Understanding photon energy helps in numerous applications, including the development of sensitive light-detection technologies.

Minimum Frequency of Light

The minimum frequency of light, in the realm of the photoelectric effect, is the lowest frequency that can still provide enough energy to surpass a material's work function, ejecting electrons.
Using the work function \(E\) and Planck's constant \(h\), we calculate this frequency \(u\) using the relation:\[u = \frac{E}{h}\]For platinum, with a work function of \(9.08 \times 10^{-19} \text{ J}\), the calculated minimum frequency is \(1.37 \times 10^{15} \text{ s}^{-1}\).
This frequency, also termed 'threshold frequency,' signifies the cutoff point: below this frequency, photons lack the energy to free electrons, and above it, emission is possible. Knowledge of the minimum frequency is essential for designing systems that leverage electron emissions, such as photoelectric sensors and solar cells.