Problem 26 - Chemistry and Chemical Reactivity

Question

A rifle bullet (mass \(=1.50 \mathrm{g}\) ) has a velocity of \(7.00 \times\) \(10^{2}\) mph. What is the wavelength associated with this bullet?

Step-by-Step Solution

Answer

The wavelength associated with the bullet is approximately \(1.407 \times 10^{-33}\) meters.

Step 1: Convert units

First, we need to convert the mass of the bullet from grams to kilograms and the velocity from miles per hour to meters per second. Mass: 1.50 grams = 0.00150 kilogramsVelocity: Convert the velocity from mph to m/s: \[ 7.00 \times 10^{2} \text{ mph} \times \frac{1609.34 \text{ m}}{1 \text{ mile}} \times \frac{1 \text{ hour}}{3600 \text{ seconds}} \approx 313.984 \text{ m/s} \]

Step 2: Use de Broglie wavelength formula

The de Broglie wavelength formula is given by:\[ \lambda = \frac{h}{mv} \]where:- \(h\) is Planck's constant \(6.626 \times 10^{-34} \text{ m}^2 \text{ kg} / \text{s}\).- \(m\) is the mass of the object.- \(v\) is the velocity of the object.Plug in the values we converted earlier into this formula.

Step 3: Calculate the wavelength

Substitute the known values into the de Broglie equation:\[ \lambda = \frac{6.626 \times 10^{-34}}{0.00150 \times 313.984} \]Perform the calculation to find \(\lambda\):\[ \lambda \approx \frac{6.626 \times 10^{-34}}{0.470976} \approx 1.407 \times 10^{-33} \text{ meters} \]

Key Concepts

Understanding Quantum Mechanics

Quantum mechanics is a fundamental theory in physics that describes the complex behaviors of matter and energy at very small scales, such as atoms and subatomic particles. It challenges classical mechanics, which works well at larger scales but fails to accurately predict outcomes in the micro world. This field of science introduces concepts like wave-particle duality, where particles such as electrons exhibit both wave and particle characteristics. The idea is crucial to understanding phenomena like the de Broglie wavelength, where even a particle with mass, like a bullet, can have a wavelength associated with it. This realization revolutionized how physicists understand the structure and behavior of our universe at microscopic levels.

The Significance of Planck's Constant

Planck's constant (\( h \)), named after physicist Max Planck, is a fundamental constant that connects the energy of a photon with its frequency. Its value is approximately \( 6.626 \times 10^{-34} \text{ m}^2 \text{ kg} / \text{s} \), a very small number indicating the quantum scale of action in the physical world.

• Planck's constant is integral to the de Broglie equation: \( \lambda = \frac{h}{mv} \), which suggests wavelength \( \lambda \) is inversely proportional to mass and velocity.
• It plays a central role in quantum mechanics, where it sets the scale of quantum effects and is a key player in the uncertainty principle, formulated by Werner Heisenberg.

Understanding this constant allows us to grasp the scale at which quantum effects become significant. In everyday life, the impact of these quantum scales is negligible, which is why classical mechanics suffices for most human-scale phenomena.

The Importance of Unit Conversion

Unit conversion is an essential step in physics that ensures measurements are in consistent units. This process involves changing a measurement from one unit to another, such as converting velocity from miles per hour (mph) to meters per second (m/s), to fit the metrics used in calculations.

• Accurate unit conversion prevents errors in computations and allows standardization, making results comparable and understandable globally.
• In the given physics problem, the bullet's velocity was converted to \( 313.984 \text{ m/s} \) using the conversion factors \( 1 \text{ mph} = 0.44704 \text{ m/s} \).

Practicing conversions regularly helps build a better intuition for measurements, which is vital for solving physics problems efficiently. Ensuring all parts of an equation have matching units is one of the keys to accurate and meaningful scientific calculations.