Problem 52 - Chemistry: The Central Science in SI Units, Global Edition

Question

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?

Step-by-Step Solution

Answer

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 1: 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 \(1.055 \times 10^{-34} \mathrm{J\cdot s}\).

Step 2: 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: \(p_{neutron} = m_{neutron} \times v\) \(p_{neutron} = (1.67 \times 10^{-27} \: kg) \times (3.00 \times 10^{5} \: m/s)\) \(p_{neutron} = 5.01 \times 10^{-22} \: kg\: m/s\)

Step 3: 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: \(\Delta p_{neutron} = m_{neutron} \times \Delta v\) \(\Delta p_{neutron} = (1.67 \times 10^{-27} \: kg) \times (0.01 \times 10^{5} \: m/s)\) \(\Delta p_{neutron} = 1.67 \times 10^{-23} \: kg\: m/s\)

Step 4: 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: \(\Delta x_{neutron} \geq \frac{\hbar}{2 \Delta p_{neutron}}\) \(\Delta x_{neutron} \geq \frac{1.055 \times 10^{-34} \: J \cdot s}{2(1.67 \times 10^{-23} \: kg\: m/s)}\) \(\Delta x_{neutron} \geq 3.16 \times 10^{-12} \: m\)

Step 5: 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.

Key Concepts

Electron Position Uncertainty

The electron is a tiny particle with a very small mass, making its position difficult to pin down precisely. In quantum mechanics, the degree of uncertainty in an electron's position can be calculated using Heisenberg's Uncertainty Principle. This principle states that the more precisely we know the momentum of a particle, the less precisely we can know its position, and vice versa.

For an electron moving at a speed of \(3.00 \times 10^{5} \: m/s\), uncertainty in its position is calculated by taking into account the mass of the electron (\(9.11 \times 10^{-31} \: kg\)) and the uncertainty in its velocity. The uncertainty in momentum (Δp) is given by multiplying the electron's mass by the uncertainty in its speed. Once Δp is known, we can find the position uncertainty (Δx) by rearranging the Heisenberg equation:

• \(\Delta x_{electron} \geq \frac{\hbar}{2 \Delta p_{electron}}\)
• The electron's position uncertainty turns out to be about ◇\(5.77 \times 10^{-9} \: m\).

Therefore, the position of an electron is not very certain due to its small mass and high velocity, which gives it a larger position uncertainty.

Neutron Position Uncertainty

Neutrons are considerably heavier than electrons, and this greater mass affects their position uncertainty. Just like with the electron, the Heisenberg Uncertainty Principle applies to neutrons too. However, due to their larger mass, we can generally know the position of a neutron with greater precision than that of an electron.

For a neutron moving at the same speed as the electron, we calculate position uncertainty in a similar manner. This time, the neutron's mass is utilized: \(1.67 \times 10^{-27} \: kg\). The uncertainty in its momentum (Δp) can be found in the same way we calculated for the electron. Once this is determined, we derive the position uncertainty:

• \(\Delta x_{neutron} \geq \frac{\hbar}{2 \Delta p_{neutron}}\)
• The neutron's position uncertainty is approximately \(3.16 \times 10^{-12} \: m\), which is notably smaller compared to the electron.

This means that even with the same velocity, the neutron's position can be determined with much better precision.

Momentum Calculation

Momentum, a crucial concept in physics, is the product of mass and velocity. In this exercise, we calculated the momentum for both the electron and the neutron to examine their respective uncertainties.

For calculating momentum (p), the formula we use is:

• \(p = m \times v\)

where:

• \(m\) is the mass of the particle,
• \(v\) is the velocity of the particle.

Given the masses of the electron \(9.11 \times 10^{-31} \: kg\) and the neutron \(1.67 \times 10^{-27} \: kg\), combined with a velocity of \(3.00 \times 10^{5} \: m/s\), we performed the following calculations:

• For the electron, \(p_{electron} = 2.733 \times 10^{-25} \: kg \cdot m/s\).
• For the neutron, \(p_{neutron} = 5.01 \times 10^{-22} \: kg \cdot m/s\).

This momentum understanding allowed us to further calculate their respective position uncertainties using the Heisenberg Uncertainty Principle. Knowing momentum is key to connecting how precisely we can predict a particle's position and velocity.