Problem 21 - Chemistry

Question

The work function of potassium is \(3.68 \times 10^{-19} \mathrm{~J}\). (a) What is the minimum frequency of light needed to eject electrons from the metal? (b) Calculate the kinetic energy of the ejected electrons when light of frequency equal to \(8.62 \times 10^{14} \mathrm{~s}^{-1}\) is used for irradiation.

Step-by-Step Solution

Answer

(a) The minimum frequency of light needed to eject electrons from potassium is \(5.55 \times 10^{14} s^{-1}\). (b) The kinetic energy of the ejected electrons when light of frequency equal to \(8.62 \times 10^{14} s^{-1}\) is used for irradiation is \(1.86 \times 10^{-19} J\).

Step 1: Minimum Frequency Calculation

The energy of a photon is given by \(E = hf\), where \(E\) is the energy of the photon, \(h\) is Planck's constant (\(6.63 \times 10^{-34} J \cdot s\)), and \(f\) is the frequency we are trying to find. To obtain the frequency, we rearrange the above formula as \(f = E/h\) and substitute the given work function value (\(3.68 \times 10^{-19} J\)) as the energy: \(f = (3.68 \times 10^{-19} J) / (6.63 \times 10^{-34} J \cdot s) = 5.55 \times 10^{14} s^{-1}\).

Step 2: Kinetic Energy Calculation

Now, we have to find out the kinetic energy of the electrons ejected by the light with frequency \(8.62 \times 10^{14} s^{-1}\). The kinetic energy is given by \(KE = hf - \Phi\) where \(\Phi\) is the work function and \(h\) is again Planck's constant. Thus, substituting the given values, we have \(KE = (6.63 \times 10^{-34} J \cdot s)(8.62 \times 10^{14} s^{-1}) - 3.68 \times 10^{-19} J = 1.86 \times 10^{-19} J\).

Key Concepts

Work Function

The concept of the work function is fundamental when discussing the photoelectric effect. It represents the minimum amount of energy required to remove an electron from the surface of a metal. This value is unique to each material and an essential threshold in determining if a photon can cause electron ejection. For instance, potassium has a work function of approximately 3.68 eV. If the energy of an incoming photon is less than this work function, it will not be sufficient to eject electrons from the potassium surface, indicating the material-specific nature of the phenomenon.

In the context of our example, the very first step to solving for the minimum frequency of light that can eject electrons is tied to the work function, signifying its importance in calculations related to the photoelectric effect.

Photon Energy

Photon energy is the amount of energy carried by a single photon, which is directly proportional to its frequency and inversely proportional to its wavelength. The relation is embodied by the equation
\( E = hf \),
where \( E \) is the energy of the photon, \( h \) is Planck's constant, and \( f \) is the frequency of the light. When photons strike a metal surface, their energy can be transferred to the electrons, and if this energy is greater than the work function of the metal, electrons are ejected. With the given frequency of light in our textbook exercise, calculating the energy of photons allows for the determination of the kinetic energy of ejected electrons, thus bridging the concepts of photon energy and electron behavior upon light interaction.

Planck's Constant

Planck's constant is a fundamental physical constant denoted by \( h \) and has a value of approximately \( 6.63 \times 10^{-34} J \cdot s \). This constant is a key element in quantum mechanics and plays a pivotal role in the equation relating the energy of a photon to its frequency. Planck's constant establishes the quantization of energy in the realm of atomic and subatomic particles, and forms the backbone of understanding phenomena such as the photoelectric effect. Within the exercise, Planck's constant is used to compute both the minimum frequency needed to eject electrons from potassium using its work function, and the kinetic energy of those ejected electrons when irradiated with a specific frequency of light.

Kinetic Energy of Electrons

The kinetic energy of electrons is the energy that an electron possesses due to its motion, which in the context of the photoelectric effect, is the result of being ejected from a metal's surface. Any surplus energy from the photon that's not used to overcome the work function is converted into kinetic energy of the ejected electrons. The equation used to measure this is
\( KE = hf - \bf{\Phi} \),
where \( KE \) is the kinetic energy, \( hf \) is the photon energy, and \( \Phi \) is the work function. From our example, after the light with a certain frequency hits the potassium surface, we calculate the kinetic energy of the emitted electrons. This reveals how the initial photon energy is partitioned between the work needed to remove an electron and the energy with which it departs.

Frequency Threshold

The frequency threshold, or threshold frequency, refers to the minimum frequency that a photon must have to eject an electron from a metal surface, overcoming the work function. It sets a 'cut-off' point, below which no electrons will be emitted regardless of the intensity of the light. This concept exemplifies the quantum notion that energy comes in discrete packets and that only those photons with sufficient energy per packet (or frequency) can cause the photoelectric effect. In our potassium example, the calculated threshold frequency shows the minimum required input to initiate electron ejection, emphasizing the practical applications of these fundamental principles to understanding and predicting the behavior of electrons under various light exposures.