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Chapter 7: Problem 52

Calculate the velocities of electrons with de Broglie wavelengths of \(1.0 \times 10^{2} \mathrm{~nm}\) and \(1.0 \mathrm{~nm}\), respectively.

Short Answer

Expert verified
The velocity of the electron with a de Broglie wavelength of \(1.0\times 10^{2} \mathrm{~nm}\) is approximately \(7.28 \times 10^5 \mathrm{m/s}\), and the velocity of the electron with a de Broglie wavelength of \(1.0 \mathrm{~nm}\) is approximately \(7.28 \times 10^7 \mathrm{m/s}\).

Step by step solution

01

Convert nanometers to meters

To make the calculation easier, let's first convert the given de Broglie wavelengths from nanometers to meters. 1 nm = \(1 \times 10^{-9}\) m So, for the given wavelengths: - \(1.0\times 10^{2} \mathrm{~nm} = 1.0\times 10^{2} \times 10^{-9} \mathrm{m} = 1.0\times 10^{-7} \mathrm{m} \) - \(1.0 \mathrm{~nm} = 1.0\times 10^{-9} \mathrm{m}\)
02

Rearrange the de Broglie wavelength formula

In this step, we need to rearrenge the de Broglie wavelength formula to solve for the velocity of the electron (\(v\)): \[ v = \frac{h}{m\lambda} \]
03

Compute the velocities

Now we have the formula to calculate the velocity, we just need to insert the known values of mass, wavelength, and Planck's constant into the equation to get the velocities for both cases. For \(\lambda = 1.0 \times 10^{-7} \mathrm{m}\), \[ v_1 = \frac{(6.63 \times 10^{-34} \mathrm{Js})}{(9.11 \times 10^{-31} \mathrm{kg})(1.0 \times 10^{-7} \mathrm{m})} \] For \(\lambda = 1.0 \times 10^{-9} \mathrm{m}\), \[ v_2 = \frac{(6.63 \times 10^{-34} \mathrm{Js})}{(9.11 \times 10^{-31} \mathrm{kg})(1.0 \times 10^{-9} \mathrm{m})} \]
04

Calculate the velocities

With the given values, we can now calculate the velocities: - For \(v_1\): \[ v_1 \approx \frac{6.63 \times 10^{-34}}{9.11 \times 10^{-31} \times 1.0 \times 10^{-7}} \approx 7.281 \times 10^5 \mathrm{m/s} \] - For \(v_2\): \[ v_2 \approx \frac{6.63 \times 10^{-34}}{9.11 \times 10^{-31} \times 1.0 \times 10^{-9}} \approx 7.281 \times 10^7 \mathrm{m/s} \] The velocity of the electron with a de Broglie wavelength of \(1.0\times 10^{2} \mathrm{~nm}\) is approximately \(7.28 \times 10^5 \mathrm{m/s}\), and the velocity of the electron with a de Broglie wavelength of \(1.0 \mathrm{~nm}\) is approximately \(7.28 \times 10^7 \mathrm{m/s}\).

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Most popular questions from this chapter

Assume that we are in another universe with different physical laws. Electrons in this universe are described by four quantum numbers with meanings similar to those we use. We will call these quantum numbers \(p, q, r\), and \(s .\) The rules for these quantum numbers are as follows: \(p=1,2,3,4,5, \ldots\) \(q\) takes on positive odd integers and \(q \leq p\) \(r\) takes on all even integer values from \(-q\) to \(+q\). (Zero is considered an even number.) \(s=+\frac{1}{2}\) or \(-\frac{1}{2}\) a. Sketch what the first four periods of the periodic table will look like in this universe. b. Wh?t are the atomic numbers of the first four elements you would expect to be least reactive? c. Give an example, using elements in the forst four rows, of ionic compounds with the formulas \(\mathrm{XY}, \mathrm{XY}_{2}, \mathrm{X}_{2} \mathrm{Y}, \mathrm{XY}_{3}\), and \(\mathrm{X}_{2} \mathrm{Y}_{3}\) d. How many electrons can have \(p=4, q=3 ?\) e. How many electrons can have \(p=3, q=0, r=0 ?\) f. How many electrons can have \(p=6\) ?

Consider an electron for a hydrogen atom in an excited state. The maximum wavelength of electromagnetic radiation that can completely remove (ionize) the electron from the \(\mathrm{H}\) atom is \(1460 \mathrm{~nm} .\) What is the initial excited state for the electron \((n=?) ?\)

For each of the following pairs of elements \((\mathrm{C}\) and \(\mathrm{N}) \quad(\mathrm{Ar}\) and \(\mathrm{Br})\) pick the atom with a. more favorable (exothermic) electron affinity. b. higher ionization energy. c. larger size.

An ionic compound of potassium and oxygen has the empirical formula KO. Would you expect this compound to be potassium(II) oxide or potassium peroxide? Explain.

How many orbitals in an atom can have the designation \(5 p, 3 d_{z^{2}}\), \(4 d, n=5, n=4 ?\)
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