Question

The photoelectric threshold wavelength of a tungsten surface is 272 nm. Calculate the maximum kinetic energy of the electrons ejected from this tungsten surface by ultraviolet radiation of frequency 1.45 $\times$ 10${}^{15}$ Hz. Express the answer in electron volts.

Short answer

The maximum kinetic energy is 1.45 eV.

Step-by-step solution

Understand the Problem

We're given the threshold wavelength of tungsten and asked to calculate the maximum kinetic energy of electrons ejected when the surface is illuminated by UV light with a specified frequency. We'll use the photoelectric effect equations to find this energy.

Convert Threshold Wavelength to Threshold Frequency

We know that frequency $u$ and wavelength $\lambda$ are related by the speed of light $c$: $c=u\lambda$. Using this, the threshold frequency ($u_0$) can be calculated: $$u_0=\frac{c}{{\lambda }_0}$$ where ${\lambda }_0=272\text{ nm}=272\times {10}^{-9}\text{ m}$ and $c=3\times {10}^8\text{ m/s}$.

Calculate the Threshold Frequency

Using the formula from Step 2, calculate $u_0$: $$u_0=\frac{3\times {10}^8}{272\times {10}^{-9}}=1.10\times {10}^{15}\text{ Hz}$$

Find the Energy of Incident Photons

The energy of the incident photons $E$ is given by Planck's equation: $E=hu$. Here, $u=1.45\times {10}^{15}\text{ Hz}$, and $h=6.63\times {10}^{-34}\text{ J s}$.

Calculate the Energy of Incident Photons

Substitute the values into Planck's equation: $$E=6.63\times {10}^{-34}\times 1.45\times {10}^{15}=9.61\times {10}^{-19}\text{ J}$$

Calculate the Work Function

The work function $\varphi$ is the minimum energy required to eject an electron, calculated using $\varphi =h{u}_0$, where $u_0=1.10\times {10}^{15}\text{ Hz}$.

Determine the Work Function

Substitute into the equation: $$\varphi =6.63\times {10}^{-34}\times 1.10\times {10}^{15}=7.29\times {10}^{-19}\text{ J}$$

Calculate the Maximum Kinetic Energy

The maximum kinetic energy (KE) of the ejected electrons is given by $KE=E-\varphi$. Substitute the calculated values: $$KE=9.61\times {10}^{-19}-7.29\times {10}^{-19}=2.32\times {10}^{-19}\text{ J}$$

Convert Joules to Electron Volts

To convert the kinetic energy from joules to electron volts, use the conversion $1\text{ eV}=1.602\times {10}^{-19}\text{ J}$. Thus, $$KE=\frac{2.32\times {10}^{-19}}{1.602\times {10}^{-19}}\text{ eV}=1.45\text{ eV}$$

Final Answer

The maximum kinetic energy of the electrons ejected from the tungsten surface is $1.45\text{ eV}$.

Key concepts

Threshold Wavelength

In the realm of photoelectric effect, the threshold wavelength plays a pivotal role. It refers to the longest wavelength of light that is capable of ejecting electrons from a material's surface.

When a light of wavelength longer than this threshold strikes the substance, no electrons are ejected, no matter how intense the light may be. This happens because the energy of the photons is simply not enough to liberate an electron from the material's atomic structure.

To calculate the threshold frequency, we use the equation relating the speed of light ($c$) to frequency ($u$) and wavelength ($\lambda$): = u \lambda.

• Convert the given threshold wavelength to meters.
• Use the speed of light in meters per second ($3\times {10}^8$ m/s).
• Determine the threshold frequency ($u_0$), using the formula: $u_0=\frac{c}{{\lambda }_0}$.

This is crucial because the threshold frequency helps in determining the work function of the material, which is essential for understanding the photoelectric effect.

Kinetic Energy Calculation

The calculation of kinetic energy of electrons plays a central role in understanding the photoelectric effect. When light hits the surface of a material, it ejects electrons with specific kinetic energy.

The energy of each photon is given by Planck's equation $E=hu$, where $E$ is the energy, $h$ is Planck's constant ($6.63\times {10}^{-34}\text{ J s}$), and $u$ is the frequency of the light.

Once the incident energy is known, the next step is to calculate the work function, which we will discuss later. Subtract the work function from the photon energy to find the kinetic energy of the ejected electrons:

• Find the incident photon energy.
• Use the work function to get the effective energy of electrons.
• Calculate maximum kinetic energy using $KE=E-\varphi$.

Understanding this relationship is vital as it gives insights into how energy is transferred from light to electrons.

Work Function

The work function ($\varphi$) is a key concept when examining the photoelectric effect. It is defined as the minimum energy required to dislodge an electron from the surface of a material.

In most cases, the work function is tied to the intrinsic properties of the material itself. To calculate it, we multiply Planck’s constant ($h$) by the threshold frequency ($u_0$), which was derived from the threshold wavelength: $\varphi =h{u}_0$.

This value serves as a threshold, representing the barrier that photon energy must overcome to emit electrons.

• Calculate using the threshold frequency and Planck's constant.
• Represents the energy barrier for electron ejection.
• A crucial parameter in predicting electron emissions.

Understanding the work function helps us to predict how different materials will behave under light, assisting in the design and analysis of devices like solar panels and photo detectors.