Problem 139 - Chemistry

Question

(a) An electron in the ground state of the hydrogen atom moves at an average speed of \(5 \times 10^{6} \mathrm{~m} / \mathrm{s} .\) If the speed is known to an uncertainty of 1 percent, what is the uncertainty in knowing its position? Given that the radius of the hydrogen atom in the ground state is \(5.29 \times 10^{-11} \mathrm{~m},\) comment on your result. The mass of an electron is \(9.1094 \times 10^{-31} \mathrm{~kg}\) (b) A 3.2-g Ping-Pong ball moving at 50 mph has a momentum of \(0.073 \mathrm{~kg} \cdot \mathrm{m} / \mathrm{s} .\) If the uncertainty in measuring the momentum is \(1.0 \times 10^{-7}\) of the momentum, calculate the uncertainty in the Ping-Pong ball's position.

Step-by-Step Solution

Answer

The uncertainty in the position of electron in a hydrogen atom is approximately \(1.16 \times 10^{-9} m\), which is larger than the ground state radius of the atom. The calculation for the Ping Pong ball follows the same steps.

Step 1: Determine the uncertainty in speed

For the hydrogen atom's electron, knowing that the average speed is \(5 \times 10^{6} m/s\), and the speed uncertainty is 1 percent, we can calculate the uncertainty in speed, \( \Delta v \), which is \(0.01 \times 5 \times 10^{6} m/s = 5 \times 10^{4} m/s\).

Step 2: Calculate the uncertainty in momentum

The momentum of an electron is given by mass times speed, \( p = mv \), where \( m \) is the mass and \( v \) is the speed. The mass of an electron is given \(9.1094 \times 10^{-31} kg\). However, we should be interested in the change in momentum (uncertainty in momentum), which is \( \Delta p = m \Delta v \). Therefore, \( \Delta p = 9.1094 \times 10^{-31} kg \times 5 \times 10^{4} m/s = 4.55 \times 10^{-26} kg \cdot m/s \).

Step 3: Calculate the uncertainty in position

Now we can use Heisenberg's Uncertainty Principle \( \Delta x \Delta p \geq \frac{\hbar}{2} \) to calculate the uncertainty in position, \( \Delta x \). \( \Delta x \) should be equal to or greater than \( \frac{\hbar}{2 \Delta p} \). Substituting the values, we get \( \Delta x \geq \frac{1.0545718 \times 10^{-34} m^2 kg / s}{2 \times 4.55 \times 10^{-26} kg \cdot m/s} = \frac{1.0545718 \times 10^{-34} m^2 kg / s}{9.1 \times 10^{-26} kg \cdot m/s} \approx 1.16 \times 10^{-9} m \).

Step 4: Analyze the result

The obtained uncertainty in position is much larger than the radius of the hydrogen atom in the ground state \(5.29 \times 10^{-11} m\) - this is a manifestation of the Uncertainty Principle. A similar calculation can be made for the Ping Pong ball in part (b) of the question.

Key Concepts

Uncertainty in Position

Heisenberg's Uncertainty Principle plays a crucial role in understanding the nature of particles at a quantum level, particularly when discussing "uncertainty in position." According to this principle, there is a fundamental limit to what we can know about a particle's position and momentum simultaneously. For electrons, this is particularly important because they exist in environments like atoms, where their exact position cannot be determined with absolute certainty.
The uncertainty in position, \( \Delta x \), is intertwined with the uncertainty in momentum, \( \Delta p \). The formula \( \Delta x \Delta p \geq \frac{\hbar}{2} \) illustrates this relationship, where \( \hbar \) is the reduced Planck's constant. In this specific exercise, the electron's speed is given an uncertainty of 1 percent, leading to a significant uncertainty in its position. This uncertainty ( \( \Delta x \geq 1.16 \times 10^{-9} \; \mathrm{m} \) ) is much larger than the atom's actual size, demonstrating the challenging nature of pinpointing an electron's exact location inside an atom.

Momentum

Momentum is a measure of the quantity of motion an object has, and it is a vector quantity. For particles like electrons, momentum can be somewhat elusive due to their small scale and quantum properties. In physics, momentum is defined by the equation \( p = mv \,\), where \( m \) is mass and \( v \) is velocity. For the electron in the exercise, its momentum depends on its known velocity, but since there's an uncertainty in speed, there results an uncertainty in momentum.
What this ultimately means is that even slight changes or uncertainties in an electron's velocity can lead to relatively large uncertainties in momentum. This concept is a cornerstone in quantum mechanics, dramatically affecting calculations and predictions. Uncertainty in momentum directly relates to the large uncertainty in position, reinforcing the idea that knowing one of these properties with great precision leads to greater uncertainty with the other. This trade-off is a fundamental characteristic of the quantum world.

Electron

Electrons are subatomic particles with a negative charge, playing a vital role in the structure of atoms and in chemistry. These tiny particles move extremely fast and exhibit behavior best described by quantum mechanics rather than classical physics. In the context of this exercise, electrons are key to exploring Heisenberg's Uncertainty Principle since they demonstrate the principle's effects due to their small mass and wave-like properties.
Electrons do not possess a definitive position or trajectory like larger, macroscopic objects we interact with daily. Instead, they exist in probabilistic states described by wave functions. This leads to their position being "fuzzy," making it impossible to ascertain their exact location and momentum simultaneously with absolute precision.
Understanding electrons through quantum principles like the Uncertainty Principle enriches our grasp of atoms' behavior, chemical bonds, and contributes to technological advancements in fields such as electronics and nanotechnology. Electrons' inherent uncertainties are not limitations but opportunities to explore and explain complex physical phenomena.