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Chapter 7: Problem 148

The mathematical equation for studying the photoelectric effect is \(h \nu=W+\frac{1}{2} m_{e} u^{2}\) where \(v\) is the frequency of light shining on the metal, \(W\) is the work function, and \(m_{e}\) and \(u\) are the mass and speed of the ejected electron. In an experiment, a student found that a maximum wavelength of \(351 \mathrm{nm}\) is needed to just dislodge electrons from a zinc metal surface. Calculate the speed (in \(\mathrm{m} / \mathrm{s})\) of an ejected electron when she employed light with a wavelength of \(313 \mathrm{nm}\)

Short Answer

Expert verified
Substitute all the values into each step following the method given and solve it step by step to get the final speed of an electron.

Step by step solution

01

Calculate Frequency of Given Wavelength

Using the equation \(v = c / \lambda \), where \(\lambda = 313\) nm, convert \(313 \) nm to meters by multiplying by \(1x10^{-9}\) then substitute \(c = 3x10^8 m/s\) and \(\lambda = 313x10(-9)\) m to find \(v\).
02

Calculate Energy of Given Wavelength

Now, we calculate the energy using Planck's equation \(E = h \times v\), by substituting \(h = 6.63x10^{-34} Js\) and \(v\) calculated in Step 1.
03

Calculate Work Function of Zinc

To calculate the work function of zinc, first calculate the frequency of the light that corresponds to the maximum wavelength using the same method as in step 1. Instead of \(313 nm\), here we substitute \(\lambda = 351 nm\) to find \(v_{max}\). Now, calculate the energy \(E_{max}\) using Planck's equation by substituting \(h = 6.63x10^{-34} Js\) and \(v_{max}\). This energy \(E_{max}\) is the work function \(W\).
04

Calculate Kinetic Energy

The energy transferred to the electron is the difference between the energy of the light and the work function of zinc. So, \(KE = E - E_{max}\).
05

Calculate Speed of Electron

According to kinetic energy theory, \(KE = 0.5 \times m \times u^2\), where \(m\) is the mass of the electron \(9.1 x 10^{-31}\) kg. Solve this equation for \(u\) using the value of \(KE\) calculated in Step 4.

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