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 needed is approximately \(1.37 \times 10^{15} \text{ Hz}\).

Step 1: Understand the Concept

This problem involves the photoelectric effect, where light of a certain minimum frequency can eject electrons from a material's surface. The energy of the photon must be equal to or greater than the work function of the material to eject an electron.

Step 2: Identify the Given Data

We are given that the energy required to eject a photoelectron from a platinum atom is \(9.08 \times 10^{-19} \text{ J/photon}\). This value is the work function \( W \) of platinum.

Step 3: Use the Energy of a Photon Formula

The energy of a photon \( E \) is given by the formula \( E = h \cdot f \), where \( h \) is Planck's constant \(6.626 \times 10^{-34} \text{ J s}\), and \( f \) is the frequency of the photon.

Step 4: Set up the Equation for Minimum Frequency

Since the energy must be at least the work function to eject a photoelectron, we set \( h \cdot f = W \) to find the minimum frequency \( f \).

Step 5: Solve for Frequency

Rearrange the equation to solve for \( f \): \( f = \frac{W}{h} = \frac{9.08 \times 10^{-19}}{6.626 \times 10^{-34}} \).

Step 6: Calculate the Frequency

Perform the division: \( f = \frac{9.08 \times 10^{-19}}{6.626 \times 10^{-34}} \approx 1.37 \times 10^{15} \text{ Hz}\).

Step 7: Conclusion

The minimum frequency of light needed to eject a photoelectron from a platinum atom is approximately \(1.37 \times 10^{15} \text{ Hz}\).

Key Concepts

Photon Energy

Photon energy is the energy carried by a single photon, which is a particle of light. This energy is crucial for understanding phenomena like the photoelectric effect. The amount of energy a photon has depends on its frequency. Higher frequency photons have higher energy, while lower frequency photons have less energy.

To calculate the energy of a photon, we use the formula:

• \( E = h \cdot f \)

Here, \( E \) is the photon energy, \( h \) is Planck's constant, and \( f \) represents the frequency of the photon. This formula shows that energy is directly proportional to frequency, meaning as frequency increases, so does the energy of the photon.

Photons with sufficient energy can cause electrons to be ejected from certain materials, which is the basis for devices like solar panels and photo detectors. Understanding photon energy allows us to manipulate light in practical applications, making it a fundamental topic in both physics and engineering.

Work Function

The work function is a critical concept when exploring the interaction between light and matter. It represents the minimum energy necessary to remove an electron from the surface of a material. This value is specific to each material and affects how the material behaves when exposed to light.

In the context of the photoelectric effect, if the energy of an incoming photon is at least equal to the work function, an electron can be released from the surface of the material. Hence, the work function is a crucial threshold. For platinum, which was discussed in the exercise, this threshold is given as \(9.08 \times 10^{-19} \text{ J} \) per photon.

Recognizing this concept helps us understand why certain materials exhibit photoelectric properties while others do not. It's all about whether the photon's energy can meet or exceed the work function necessary to free electrons. This concept not only supports solar technology but is also used in vacuum tubes and semiconductor devices.

Planck's Constant

Planck's constant \( (h) \) is one of the fundamental constants in physics. It plays a pivotal role in quantifying the relationship between the energy of a photon and its frequency, making it essential for the study of quantum mechanics. The value of Planck's constant is approximately \( 6.626 \times 10^{-34} \text{ Js} \).

This constant is the scaling factor in the photon energy formula:

• \( E = h \cdot f \)

By determining how much energy corresponds to a given frequency, Planck's constant connects the quantum scale phenomena of photons to larger scale concepts observable in everyday physics.

The introduction of Planck's constant was a massive leap forward in understanding atomic and subatomic scales, revolutionizing how we think about energy and matter. It underpins many equations and theories relating to quantum mechanics, including Schrödinger's equation, which models the behavior of particles at quantum levels.