Question

Alveoli are tiny sacs of air in the lungs. Their average diameter is \(5.0 \times 10^{-5} \mathrm{~m} .\) Calculate the uncertainty in the velocity of an oxygen molecule \(\left(5.3 \times 10^{-26} \mathrm{~kg}\right)\) trapped within a sac. (Hint: The maximum uncertainty in the position of the molecule is given by the diameter of the sac.)

Short answer

The uncertainty in the velocity of the oxygen molecule is approximately \(\Delta v = 9.95 \times 10^{-3} \text{ m/s}\).

Step-by-step solution

Identify the Known Values

We are given the diameter of the alveoli \(5.0 \times 10^{-5} \text{ m}\) as the uncertainty in the position \(\Delta x\). The mass of the oxygen molecule is \(5.3 \times 10^{-26} \text{ kg}\).

Write Heisenberg's Uncertainty Principle Formula

The formula for Heisenberg's Uncertainty Principle is: \[ \Delta x \cdot \Delta p \geq \frac{h}{4 \pi} \]Where \(\Delta x\) is the position uncertainty, \(\Delta p\) is the momentum uncertainty, and \(h\) is Planck's constant \(6.626 \times 10^{-34} \text{ J·s}\).

Solve for Momentum Uncertainty \(\Delta p\)

Rearrange the formula to solve for \(\Delta p\): \[ \Delta p \geq \frac{h}{4 \pi \cdot \Delta x} \]Substitute the known values: \[ \Delta p \geq \frac{6.626 \times 10^{-34}}{4 \pi \cdot 5.0 \times 10^{-5}} \]Calculate \(\Delta p\).

Convert Momentum Uncertainty to Velocity Uncertainty

The momentum \(p\) is given by \(p = m \cdot v\), where \(m\) is the mass and \(v\) is the velocity. Thus, the uncertainty in momentum \(\Delta p\) is related to the uncertainty in velocity \(\Delta v\) as:\[ \Delta p = m \cdot \Delta v \]Rearranging gives:\[ \Delta v = \frac{\Delta p}{m} \]Substitute \(\Delta p\) from Step 3 and the oxygen molecule's mass to find \(\Delta v\).

Calculate the Result

Using the expression from Step 4, compute \(\Delta v\):\[ \Delta v = \frac{6.626 \times 10^{-34}}{4 \pi \cdot 5.0 \times 10^{-5} \times 5.3 \times 10^{-26}} \]Calculating this expression yields the uncertainty in velocity \(\Delta v\).

Key concepts

Alveoli

Alveoli are tiny, balloon-like structures inside our lungs that play a crucial role in the respiratory system. These small sacs are the endpoint of the respiratory tree and are structured to maximize the exchange of gases like oxygen and carbon dioxide. They have thin walls and are surrounded by numerous capillaries, allowing for efficient gas exchange.

When we breathe in, oxygen fills the alveoli and diffuses across its thin membrane into the blood capillaries. Meanwhile, carbon dioxide from the blood diffuses into the alveoli to be exhaled. The high surface area of many alveoli allows for a large amount of gas exchange simultaneously, making it essential for our breathing process.

• Alveoli increase the surface area for gas exchange.
• They help in oxygenating blood and removing carbon dioxide.
• Efficient function is crucial for respiratory health.

Oxygen Molecule

Oxygen molecules are vital for the existence of most life forms on Earth. An oxygen molecule is made of two oxygen atoms bonded together, forming the chemical formula O₂. This diatomic molecule is what we commonly refer to as the oxygen we breathe.

Inside the alveoli, oxygen molecules are responsible for the life-sustaining function of respiration. They move across the alveolar membrane into the blood, where they bind to hemoglobin in red blood cells. This process is crucial for delivering oxygen to all parts of the body.

• Oxygen molecules consist of two oxygen atoms.
• They are essential for cellular respiration.
• Oxygen is transported from the alveoli to the bloodstream.

Momentum Uncertainty

Momentum uncertainty, as described by Heisenberg's Uncertainty Principle, highlights the limitation in simultaneously knowing the exact position and momentum of a particle. In the context of quantum mechanics, there is always a trade-off between the precision of measuring a particle's position and the momentum.

In our exercise, the oxygen molecule in an alveolus has an inherent uncertainty in its momentum because of the uncertainty in its position. The principle is expressed mathematically by the equation: \[ \Delta x \cdot \Delta p \geq \frac{h}{4 \pi} \] where \( \Delta x \) is the uncertainty in position and \( \Delta p \) is the uncertainty in momentum. Understanding momentum uncertainty is pivotal in quantum mechanics since it sets the fundamental limits on our measurements.

• The more accurately you know a particle's position, the less accurately you can know its momentum, and vice versa.
• This principle is a foundation of quantum mechanics.
• It explains why we can never have perfect precision in measurements at the quantum level.

Velocity Uncertainty

Velocity uncertainty arises when considering the uncertainty in a particle's momentum, especially when mass is constant. By Heisenberg's formula, as we identify the momentum uncertainty in a system, we also determine the velocity uncertainty. In simpler terms, if the position of an oxygen molecule inside an alveolus is known with some uncertainty, its velocity cannot be precisely determined.

The relationship between momentum and velocity is given by \( p = m \cdot v \), where \( m \) is mass, and \( v \) is velocity. Thus, an uncertainty in momentum \( \Delta p \) translates to an uncertainty in velocity \( \Delta v \) using the equation: \[ \Delta v = \frac{\Delta p}{m} \] This indicates that smaller masses like oxygen molecules experience greater uncertainties in velocity due to Heisenberg's principle. Understanding velocity uncertainty helps us comprehend the behavior of particles at a microscopic level influenced by quantum effects.

• This concept helps in predicting how fast a particle like an oxygen molecule is moving, within a range of uncertainty.
• Is indicative of the inability to precisely measure both position and velocity simultaneously.
• Influences experiments involving small particles and quantum behavior.