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Chapter 6: Problem 52

Calculate the uncertainty in the position of (a) an electron moving at a speed of \((3.00 \pm 0.01) \times 10^{5} \mathrm{~m} / \mathrm{s},(\mathbf{b})\) a neutron moving at this same speed. (The masses of an electron and a neutron are given in the table of fundamental constants in the inside cover of the text.) (c) Based on your answers to parts (a) and (b), which can we know with greater precision, the position of the electron or of the neutron?

Short Answer

Expert verified
The position uncertainties for the electron and the neutron are calculated using the Heisenberg Uncertainty Principle. The position uncertainty for the electron is \(5.77 \times 10^{-9} \: m\) (5.77 nm) and for the neutron is \(3.16 \times 10^{-12} \: m\) (3.16 pm). Since the uncertainty for the neutron is lower than that of the electron, we can know the position of the neutron with greater precision.

Step by step solution

01

Understanding the Heisenberg Uncertainty Principle formula

The Heisenberg Uncertainty Principle states that the product of the uncertainties in position (Δx) and momentum (Δp) for a particle must be greater than or equal to a constant value, which is ħ/2: \[\Delta x \cdot \Delta p \geq \frac{\hbar}{2}\] Here, \(\hbar\) is the reduced Planck constant, which equals \(1.055 \times 10^{-34} \mathrm{J\cdot s}\).
02

Calculate the momenta of electron and neutron

First, we need to calculate the momentum (p) for the electron and neutron with the given speed value, multiplying it by their respective masses. The mass of an electron is \(9.11 \times 10^{-31} \: kg\), and the mass of a neutron is \(1.67 \times 10^{-27} \: kg\). For the electron: \(p_{electron} = m_{electron} \times v\) \(p_{electron} = (9.11 \times 10^{-31} \: kg) \times (3.00 \times 10^{5} \: m/s)\) \(p_{electron} = 2.733 \times 10^{-25} \: kg\: m/s\) For the neutron: \(p_{neutron} = m_{neutron} \times v\) \(p_{neutron} = (1.67 \times 10^{-27} \: kg) \times (3.00 \times 10^{5} \: m/s)\) \(p_{neutron} = 5.01 \times 10^{-22} \: kg\: m/s\)
03

Calculate the uncertainty in the momenta of electron and neutron

For both the electron and neutron, the uncertainty in momentum (Δp) can be calculated using the mass and the given uncertainty in speed (± 0.01 \(\times\) 10^5 m/s). For the electron: \(\Delta p_{electron} = m_{electron} \times \Delta v\) \(\Delta p_{electron} = (9.11 \times 10^{-31} \: kg) \times (0.01 \times 10^5 \: m/s)\) \(\Delta p_{electron} = 9.11 \times 10^{-27} \: kg\: m/s\) For the neutron: \(\Delta p_{neutron} = m_{neutron} \times \Delta v\) \(\Delta p_{neutron} = (1.67 \times 10^{-27} \: kg) \times (0.01 \times 10^{5} \: m/s)\) \(\Delta p_{neutron} = 1.67 \times 10^{-23} \: kg\: m/s\)
04

Calculate the position uncertainties for electron and neutron

Now, we will use the Heisenberg Uncertainty Principle formula to calculate the position uncertainty (Δx) for both the electron and the neutron: For the electron: \(\Delta x_{electron} \geq \frac{\hbar}{2 \Delta p_{electron}}\) \(\Delta x_{electron} \geq \frac{1.055 \times 10^{-34} \: J \cdot s}{2(9.11 \times 10^{-27} \: kg\: m/s)}\) \(\Delta x_{electron} \geq 5.77 \times 10^{-9} \: m\) For the neutron: \(\Delta x_{neutron} \geq \frac{\hbar}{2 \Delta p_{neutron}}\) \(\Delta x_{neutron} \geq \frac{1.055 \times 10^{-34} \: J \cdot s}{2(1.67 \times 10^{-23} \: kg\: m/s)}\) \(\Delta x_{neutron} \geq 3.16 \times 10^{-12} \: m\)
05

Compare the position uncertainties and conclude

Comparing the position uncertainties calculated for the electron and the neutron, we can observe \(Δx_{electron} = 5.77\) nm and \(Δx_{neutron} = 3.16\) pm. The position uncertainty of the neutron is smaller than that of the electron, so we can know the position of the neutron with greater precision.

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Most popular questions from this chapter

The visible emission lines observed by Balmer all involved $n_{\mathrm{f}}=2 .$ (a) Which of the following is the best explanation of why the lines with \(n_{\mathrm{f}}=3\) are not observed in the visible portion of the spectrum: (i) Transitions to \(n_{\mathrm{f}}=3\) are not allowed to happen, (ii) transitions to \(n_{\mathrm{f}}=3\) emit photons in the infrared portion of the spectrum, (iii) transitions to \(n_{\mathrm{f}}=3\) emit photons in the ultraviolet portion of the spectrum, or (iv) transitions to \(n_{\mathrm{f}}=3\) emit photons that are at exactly the same wavelengths as those to \(n_{\mathrm{f}}=2 .\) (b) Calculate the wavelengths of the first three lines in the Balmer series-those for which \(n_{1}=3,4\), and 5 -and identify these lines in the emission spectrum shown in Figure 6.11

(a) What is the frequency of radiation that has a wavelength of $10 \mu \mathrm{m}\(, about the size of a bacterium? \)(\mathbf{b})$ What is the wavelength of radiation that has a frequency of $5.50 \times 10^{14} \mathrm{~s}^{-1} ?$ (c) Would the radiations in part (a) or part \((b)\) be visible to the human eye? (d) What distance does electromagnetic radiation travel in \(50.0 \mu \mathrm{s} ?\)

The following electron configurations represent excited states. Identify the element and write its ground-state condensed electron configuration. (a) \(1 s^{2} 2 s^{2} 2 p^{4} 3 s^{1}\) (b) \([\mathrm{Ne}] 3 s^{1} 3 p^{4} 4 p^{1}\) (c) \([\mathrm{Ar}] 4 s^{2} 3 d^{6} 4 p^{1}\)

Carbon dioxide in the atmosphere absorbs energy in the $4.0-4.5 \mu \mathrm{m}\( range of the spectrum. (a) Calculate the frequency of the \)4.0 \mu \mathrm{m}\( radiation. \)(\mathbf{b})$ In what region of the electromagnetic spectrum does this radiation occur?

Bohr's model can be used for hydrogen-like ions-ions that have only one electron, such as \(\mathrm{He}^{+}\) and \(\mathrm{Li}^{2+} .\) (a) Why is the Bohr model applicable to \(\mathrm{Li}^{2+}\) ions but not to neutral Li atoms? (b) The ground-state energies of \(\mathrm{B}^{4+}, \mathrm{C}^{5+},\) and \(\mathrm{N}^{6+}\) are tabulated as follows: By examining these numbers, propose a relationship between the ground-state energy of hydrogen-like systems and the nuclear charge, \(Z\). (Hint: Divide by the ground-state energy of hydrogen $\left.-2.18 \times 10^{-18} \mathrm{~J}\right)$ (c) Use the relationship you derive in part (b) to predict the ground-state energy of the \(\mathrm{Be}^{3+}\) ion.
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