Problem 154 - A Complete Resource Book in Chemistry for JEE Main

Question

Uncertainty in position of a particle of \(25 \mathrm{~g}\) in space is \(10^{-5} \mathrm{~m} .\) Hence uncertainty in velocity \(\left(\mathrm{ms}^{-1}\right)\) is (Planck constant \(\mathrm{h}=6.6 \times 10^{-34} \mathrm{Js}\) ) \(\quad\) [2002] (a) \(2.1 \times 10^{-28}\) (b) \(2.1 \times 10^{-34}\) (c) \(0.5 \times 10^{-34}\) (d) \(5.0 \times 10^{-24}\)

Step-by-Step Solution

Answer

The uncertainty in velocity is approximately \(2.1 \times 10^{-28}\text{ ms}^{-1}\) (Option a).

Step 1: Understand Heisenberg's Uncertainty Principle

Heisenberg's Uncertainty Principle states that the uncertainty in position \( \Delta x \) and the uncertainty in momentum \( \Delta p \) of a particle are related by the inequality \( \Delta x \Delta p \geq \frac{h}{4\pi} \), where \( h \) is the Planck's constant.

Step 2: Calculate Uncertainty in Momentum

The uncertainty in momentum \( \Delta p \) can be expressed as \( \Delta p = m \Delta v \), where \( m \) is mass and \( \Delta v \) is the uncertainty in velocity. Rearranging the inequality from Step 1, we get \( \Delta p \geq \frac{h}{4\pi \Delta x} \).

Step 3: Substitute Values

Substitute the given values: mass \( m = 25 \text{ g} = 0.025 \text{ kg} \), uncertainty in position \( \Delta x = 10^{-5} \text{ m} \), and \( h = 6.6 \times 10^{-34} \text{ Js} \) into the equation \( \Delta p \geq \frac{h}{4\pi \Delta x} \).

Step 4: Solve for Uncertainty in Velocity

Calculate \( \Delta v = \frac{\Delta p}{m} \). Using the inequality \( \Delta p \geq \frac{6.6 \times 10^{-34}}{4\pi \times 10^{-5}} \), find \( \Delta v \geq \frac{\frac{6.6 \times 10^{-34}}{4\pi \times 10^{-5}}}{0.025} \).

Step 5: Calculate the Numerical Value

Compute the values to get \( \Delta v \geq \frac{6.6 \times 10^{-34}}{4\pi \times 10^{-5} \times 0.025} \approx 2.1 \times 10^{-28} \text{ ms}^{-1} \).

Key Concepts

Uncertainty in Position

In a world governed by quantum mechanics, the concept of uncertainty in position is a fundamental idea put forth by Werner Heisenberg. It suggests that there is an inherent limit to how precisely we can know both the position and momentum of a particle even if we have ideal measuring tools. This limitation is not due to technical imperfections but is an intrinsic property of nature. When we talk about uncertainty in position, denoted as \( \Delta x \), we are referring to the range of positions where a particle might be found. This range increases when we try to measure a particle's velocity or momentum with greater accuracy. This mysterious yet fascinating element of nature is pivotal because it reshapes our understanding of determinism and prediction in physics.

• Uncertainty in position implies a spread or range in possible positions.
• Greater certainty in position leads to increased uncertainty in momentum.
• This uncertainty is not an error but a natural aspect of microscopic particles.

Uncertainty in Momentum

Hand in hand with uncertainty in position comes the concept of uncertainty in momentum. In physics, a particle's momentum is a product of its mass and velocity. Uncertainty in momentum \( \Delta p \) indicates how much variation or spread there can be in a particle's momentum. This is not just a quirky principle but a deep-rooted characteristic emerging from the quantum world. Because position and momentum are intertwined, trying to measure one's value accurately increases the uncertainty in the other. This principle is pivotal in dictating the behavior of subatomic particles where precision measurement becomes a dance of trade-offs.

• Momentum uncertainty involves impreciseness in velocity or weight changes.
• Increased precision in position affects our ability to measure momentum accurately.
• Momentum and position measurement is a trade-off, a balance mandated by quantum mechanics.

Planck's Constant

Planck's constant, symbolized as \( h \), is a fundamental figure in quantum mechanics and central to Heisenberg's Uncertainty Principle. With a value of \( 6.6 \times 10^{-34} \text{ Js} \), this constant is pivotal in the realms of the very small - particles at the atomic and subatomic scale. Its existence allows us to calculate the lower bounds of uncertainty in both position and momentum. Planck's constant arises in the equation \( \Delta x \Delta p \geq \frac{h}{4\pi} \), forming the bedrock of intensity limits between position and momentum.

• Planck's constant refers to the quantization of certain physical properties.
• It defines the scale where quantum effects can't be ignored.
• In calculations, it sets the finite boundary on the accuracy of measurements in quantum mechanics.

Each factor intertwined elucidates the nature of the universe on a quantum level, challenging our classical perception of certainty and determination.