Problem 26 - Chemistry and Chemical Reactivity

Question

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

Step-by-Step Solution

Answer

The wavelength is approximately \(1.41 \times 10^{-33}\) meters.

Step 1: Convert Bullet's Mass to Kilograms

First, convert the mass of the bullet from grams to kilograms since the standard unit in physics for mass is kilograms. The mass is given as \(1.50\, \text{g}\). To convert grams to kilograms, divide by 1000. So, \(1.50\, \text{g} = \frac{1.50}{1000} = 1.50 \times 10^{-3} \text{ kg}\).

Step 2: Convert Bullet's Velocity to Meters per Second

The velocity is given in miles per hour (mph). To use it in calculations, convert this to meters per second (m/s). The conversion factor is: \(1\, \text{mile} = 1609.34\, \text{meters}\) Thus, \(7.00 \times 10^2 \text{ mph} = 700 \times \frac{1609.34}{3600} \approx 312.7\, \text{m/s}\).

Step 3: Use de Broglie Wavelength Formula

The de Broglie wavelength formula is given by \(\lambda = \frac{h}{mv}\), where:- \(\lambda\) is the wavelength,- \(h\) is Planck's constant \(\approx 6.626 \times 10^{-34} \text{ m}^2 \text{ kg/s}\),- \(m\) is the mass of the object,- \(v\) is the velocity of the object.Now substitute the values into the de Broglie wavelength formula:\[\lambda = \frac{6.626 \times 10^{-34}}{(1.50 \times 10^{-3}) \times 312.7}\]

Step 4: Calculate the Wavelength

Compute the wavelength using the values from the formula:\[\lambda = \frac{6.626 \times 10^{-34}}{0.46905} \approx 1.41 \times 10^{-33}\, \text{meters} \].

Key Concepts

Mass Conversion

When dealing with scientific calculations, it's crucial to ensure all units are in the standard format used in physics and most sciences. Mass is typically measured in kilograms (kg). If you have mass in grams (g), the conversion is simple. You divide the number of grams by 1000 to find the value in kilograms.
For example, converting a mass of 1.50 g to kilograms:

• Determine the mass in grams: 1.50 g
• Convert grams to kilograms by dividing by 1000: \(1.50 \div 1000 = 1.50 \times 10^{-3} \text{ kg} \)

This conversion is essential for maintaining consistency in calculations, especially when calculating de Broglie wavelengths.

Velocity Conversion

Velocity often needs conversion because it can be measured in different units, like miles per hour (mph) or meters per second (m/s). In physics, meters per second is the standard unit.
To convert mph to m/s, use the conversion factor where 1 mile is equal to 1609.34 meters, and there are 3600 seconds in an hour.

• Start with the given velocity: 700 mph
• Convert miles to meters: 700 miles \( \times 1609.34 \text{ meters/mile} \)
• Convert hours to seconds: divide by 3600 seconds/hour
• Final calculation: \(700 \times \frac{1609.34}{3600} \approx 312.7 \text{ m/s} \)

Converting to m/s is vital for using formulas like the de Broglie wavelength equation, where SI units are standard.

Planck's Constant

Planck's constant is a fundamental constant in quantum mechanics, symbolized by \(h\). It plays a pivotal role in the de Broglie wavelength equation, among others. Its value is approximately \(6.626 \times 10^{-34} \text{ m}^2 \text{ kg/s} \).
This constant is critical when dealing with calculations involving energy and momentum of particles.
In the de Broglie wavelength formula, \(\lambda = \frac{h}{mv}\), Planck's constant provides a bridge between a particle's momentum (\(mv\)) and its associated wave properties.

• \(h \approx 6.626 \times 10^{-34} \text{ m}^2 \text{ kg/s} \)
• Allows calculation of wavelength from mass and velocity
• Highlights particle-wave duality in quantum physics

Understanding Planck's constant helps you appreciate the quantum mechanics underlying many physical processes.