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Chapter 7: Problem 163

It takes 476 kJ to remove 1 mole of electrons from the atoms at the surface of a solid metal. What is the maximum wavelength of light that can remove a single electron from an atom at the surface of this solid metal?

Short Answer

Expert verified
The maximum wavelength of light that can remove a single electron from an atom at the surface of this solid metal is \(\lambda = \frac{2.998 \times 10^{8} \text{ m/s} \times 6.626 \times 10^{-34} \text{Js}}{\frac{476000\text{ J/mol}}{6.022\times 10^{23}\text{ electrons/mol}}}\), which can be calculated to get the value in meters.

Step by step solution

01

Convert energy in kJ/mol to J/mol

Given, the energy required to remove 1 mole of electrons is 476 kJ. We need to convert it to J/mol. 1 kJ = 1000 J So, 476 kJ = 476 * 1000 J = 476000 J/mol
02

Calculate the energy required to remove one electron

The energy required to remove 1 mole of electrons is 476000 J/mol. We need to find the energy required to remove one electron. Number of electrons in a mole = Avogadro's number = \(6.022 \times 10^{23}\) electrons/mol Energy per electron = \(\frac{476000\text{ J/mol}}{6.022\times 10^{23}\text{ electrons/mol}}\)
03

Calculate the frequency of light

From the energy per electron, we can find the frequency of light using Planck's equation: \(E = h\nu \) Where E is the energy per electron, h is Planck's constant (\(6.626 \times 10^{-34} \text{ Js} \)), and \(\nu\) is the frequency. Energy per electron = \(\frac{476000\text{ J/mol}}{6.022\times 10^{23}\text{ electrons/mol}}\) \(\nu = \frac{E}{h}\) \( \nu = \frac{\frac{476000\text{ J/mol}}{6.022\times 10^{23}\text{ electrons/mol}}}{6.626 \times 10^{-34} \text{Js}}\)
04

Calculate the speed of light

Since the speed of light (c) is equal to the product of frequency (\(\nu\)) and wavelength (\(\lambda\)), we can find the wavelengths by dividing the speed of light by the frequency. \(c = 2.998 \times 10^{8} \text{m/s}\) \(\lambda = \frac{c}{\nu}\) \(\lambda = \frac{2.998 \times 10^{8} \text{m/s}}{\frac{\frac{476000\text{ J/mol}}{6.022\times 10^{23}\text{ electrons/mol}}}{6.626 \times 10^{-34} \text{Js}}}\)
05

Solve for the maximum wavelength

Now, we can put all the values in the given formula and find the maximum wavelength. \(\lambda = \frac{2.998 \times 10^{8} \text{ m/s} \times 6.626 \times 10^{-34} \text{Js}}{\frac{476000\text{ J/mol}}{6.022\times 10^{23}\text{ electrons/mol}}}\) After solving the above expression, we get the maximum wavelength (in meters) that can remove a single electron from an atom at the surface of this solid metal.

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