The mass of a helium atom is 4.0026 amu, and its average velocity at 25 °C is 1.36 * 103 m/s. What is the uncertainty in meters in the position of a helium atom if the uncertainty in its velocity is 1%?

Verified step by step guidance

1. Identify the given values: mass of helium atom = 4.0026 amu, average velocity = 1.36 \times 10^3 \text{ m/s}, uncertainty in velocity = 1\%.
2. Convert the mass of the helium atom from atomic mass units (amu) to kilograms using the conversion factor: 1 \text{ amu} = 1.660539 \times 10^{-27} \text{ kg}.
3. Calculate the uncertainty in velocity (\Delta v) by multiplying the average velocity by the percentage uncertainty: \Delta v = 0.01 \times 1.36 \times 10^3 \text{ m/s}.
4. Use Heisenberg's Uncertainty Principle, \Delta x \cdot \Delta p \geq \frac{h}{4\pi}, where \Delta p is the uncertainty in momentum, \Delta p = m \cdot \Delta v, and h is Planck's constant (6.626 \times 10^{-34} \text{ Js}).
5. Solve for the uncertainty in position (\Delta x) by rearranging the equation: \Delta x \geq \frac{h}{4\pi \cdot m \cdot \Delta v}.

Key Concepts

Uncertainty Principle

The Heisenberg Uncertainty Principle states that it is impossible to simultaneously know both the exact position and exact momentum of a particle. This principle implies that the more precisely we know a particle's velocity (momentum), the less precisely we can know its position, and vice versa. This concept is fundamental in quantum mechanics and is crucial for understanding the behavior of subatomic particles like helium atoms.

Kinetic Energy and Velocity

The kinetic energy of an object is directly related to its mass and the square of its velocity, expressed by the formula KE = 1/2 mv². In this context, the average velocity of the helium atom is essential for calculating its momentum, which is the product of mass and velocity. Understanding how velocity affects kinetic energy helps in determining the uncertainty in position when velocity is uncertain.

Percentage Uncertainty

Percentage uncertainty quantifies the uncertainty of a measurement relative to the size of the measurement itself, expressed as a percentage. In this question, a 1% uncertainty in velocity means that the actual velocity could vary by 1% above or below the measured value. This concept is important for calculating the uncertainty in position using the uncertainty principle, as it directly influences the calculations involved.