Problem 52 - Chemistry The Central Science In Si Units

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 of a neutron can be known with greater precision compared to an electron.

Step 1: Understanding the Uncertainty Principle

The Heisenberg Uncertainty Principle states that the uncertainty in position dx{ ext): \[ \Delta x \geq \frac{\hbar}{2 \cdot \Delta p} \] where
7do is the reduced Planck's constant and
7dort:xdap_ given by: \[ \Delta p = m \cdot \Delta v \] where
7dx{mp being mass and
7do is uncertainty in velocity.

Step 2: Calculate Uncertainty in Velocity

The uncertainty in the velocity for both the electron and the neutron is given as \( \Delta v = 0.01 \times 10^5 \text{ m/s} = 1 \times 10^3 \text{ m/s} \).

Step 3: Mass of Electron and Neutron

The mass of an electron \( m_{e} = 9.11 \times 10^{-31} \text{ kg} \) and the mass of a neutron \( m_{n} = 1.675 \times 10^{-27} \text{ kg} \).

Step 4: Calculate Uncertainty in Momentum for Electron

Calculate the uncertainty in momentum for the electron using \( \Delta p = m \cdot \Delta v \). So, for the electron: \( \Delta p_{e} = 9.11 \times 10^{-31} \text{ kg} \times 1 \times 10^3 \text{ m/s} = 9.11 \times 10^{-28} \text{ kg m/s} \).

Step 5: Calculate Position Uncertainty for Electron

Using \( \Delta x = \frac{\hbar}{2 \cdot \Delta p} \) with \( \hbar = 1.0545718 \times 10^{-34} \text{ J s} \): \( \Delta x_{e} = \frac{1.0545718 \times 10^{-34}}{2 \times 9.11 \times 10^{-28}} = 5.79 \times 10^{-8} \text{ m} \).

Step 6: Calculate Uncertainty in Momentum for Neutron

For the neutron
7do:xdap_:
7do:xdap_:ad
7dxmn \( \Delta p_{n} = m_{n} \cdot \Delta v \), so \( \Delta p_{n} = 1.675 \times 10^{-27} \text{ kg} \times 1 \times 10^3 \text{ m/s} = 1.675 \times 10^{-24} \text{ kg m/s} \).

Step 7: Calculate Position Uncertainty for Neutron

Use \( \Delta x = \frac{\hbar}{2 \cdot \Delta p} \): \( \Delta x_{n} = \frac{1.0545718 \times 10^{-34}}{2 \times 1.675 \times 10^{-24}} = 3.15 \times 10^{-11} \text{ m} \).

Step 8: Analyze and Compare Results

The position uncertainty for the electron is \( 5.79 \times 10^{-8} \text{ m} \), while for the neutron it is \( 3.15 \times 10^{-11} \text{ m} \). The electron's position has a larger uncertainty.

Key Concepts

The Electron: A Fundamental Particle

Electrons are one of the most important and well-known subatomic particles.
They carry a negative electric charge and are found in electron clouds surrounding the nucleus of an atom.
As part of the lepton family, electrons are fundamental particles and not composed of smaller sub-units.

• Electrons have a very small mass of approximately \(9.11 \times 10^{-31}\; \text{kg}\).
• Their movement and energy are crucial to understanding atomic structure and reactions.
• In the Heisenberg Uncertainty Principle, the electron's small mass implies a higher position uncertainty compared to heavier particles.

Despite their tiny mass, electrons play huge roles in chemical bonding, electricity, and magnetism.

Neutron: The Neutral Subatomic Particle

Neutrons are subatomic particles found within the nucleus of an atom, together with protons.
Unlike protons or electrons, neutrons carry no electric charge, making them neutral.
This neutrality plays a significant role in the stability of atomic nuclei.

• The mass of a neutron is about \(1.675 \times 10^{-27} \text{ kg}\), significantly heavier than an electron.
• Due to their larger mass, neutrons have lower position uncertainties when applying the Heisenberg Uncertainty Principle.
• Neutrons are crucial in nuclear reactions, such as fission and fusion.

Because of their substantial mass and neutrality, neutrons serve as a balancing force within the nucleus, aiding in the stability of atoms.

Momentum and Its Impact on Uncertainty

Momentum is a concept that describes the quantity of motion an object has, calculated as the product of its mass and velocity (\( p = m \times v \)).
In quantum mechanics, the Heisenberg Uncertainty Principle exploits this momentum, stated as:

• Greater mass or velocity leads to higher momentum.
• With increased momentum, position uncertainty tends to decrease.
• This concept helps in determining precision levels of different particles, such as electrons and neutrons.
• For an electron, lower mass resulted in a lower momentum and, consequently, higher uncertainty in position.

Understanding momentum is essential in physics as it allows us to calculate and predict the behavior of particles accurately.

Position Uncertainty and Its Implications

Position uncertainty is a central concept in the Heisenberg Uncertainty Principle.
It refers to the inability to measure both the position and momentum of a particle precisely at the same time.
Mathematically, the position uncertainty \( \Delta x \) is highly dependent on the particle's momentum uncertainty \( \Delta p \):

• The formula \( \Delta x \geq \frac{\hbar}{2 \cdot \Delta p} \) describes this inverse relationship.
• As momentum uncertainty increases, position uncertainty decreases, allowing for more precise location measurements.
• This concept is pivotal when determining the properties of subatomic particles such as electrons and neutrons.

Understanding position uncertainty helps us grasp the limitations innate to quantum measurements and methodologies.

The Reduced Planck's Constant in Quantum Mechanics

The reduced Planck's constant, denoted by \( \hbar \), is a fundamental constant in physics that plays a significant role in quantum mechanics.
It is derived from the Planck's constant \( h \) and is essential in formulations of the Heisenberg Uncertainty Principle:

• \( \hbar = \frac{h}{2\pi} \)
• The value of \( \hbar \) is \( 1.0545718 \times 10^{-34} \text{ J s} \).
• \( \hbar \) sets the scale at which quantum mechanical effects become significant, effectively bridging classical and quantum physics.
• It features prominently in equations involving wave-particle duality and quantum state probabilities.

The reduced Planck's constant is integral to understanding the quantum world's boundary conditions and behavior.