Problem 25 - Chemistry and Chemical Reactivity

Question

Calculate the wavelength, in nanometers, associated with a \(46-\mathrm{g}\) golf ball moving at \(30 . \mathrm{m} / \mathrm{s}\) (about \(67 \mathrm{mph}) .\) At what speed must the ball travel to have a wavelength of \(5.6 \times 10^{-3} \mathrm{nm} ?\)

Step-by-Step Solution

Answer

The initial wavelength is \( 4.81 \times 10^{-25} \text{ nm} \). The ball must travel at \( 2.58 \times 10^{-22} \text{ m/s} \) for the given wavelength.

Step 1: Understand De Broglie Wavelength Formula

The wavelength \( \lambda \) of an object is given by De Broglie's formula: \[ \lambda = \frac{h}{mv} \] where \( h \) is Planck's constant \( 6.626 \times 10^{-34} \text{ m}^2 \text{ kg} / \text{s} \), \( m \) is the mass in kilograms, and \( v \) is the speed in meters per second.

Step 2: Convert Mass from Grams to Kilograms

Since mass needs to be in kilograms, convert the 46 g to kg by dividing by 1000. Therefore, the mass \( m = 0.046 \text{ kg} \).

Step 3: Calculate Wavelength at 30 m/s

Using the formula, substitute the values: \( \lambda = \frac{6.626 \times 10^{-34}}{0.046 \times 30} \). Calculate this expression to find the wavelength.

Step 4: Compute the Wavelength

After performing the calculation: \[ \lambda \approx 4.81 \times 10^{-34} \text{ meters} \]. Convert this to nanometers by multiplying by \( 10^9 \), resulting in \( \lambda \approx 4.81 \times 10^{-25} \text{ nm} \).

Step 5: Calculate Speed for a Given Wavelength

Given \( \lambda = 5.6 \times 10^{-3} \text{ nm} \), convert to meters: \( 5.6 \times 10^{-12} \text{ m} \). Rearrange De Broglie's formula to find speed: \( v = \frac{h}{m\lambda} \). Substitute the values: \( v = \frac{6.626 \times 10^{-34}}{0.046 \times 5.6 \times 10^{-12}} \).

Step 6: Compute the Speed

Calculate the speed: \[ v \approx 2.58 \times 10^{-22} \text{ m/s} \]. This is the speed necessary for the golf ball to have a wavelength of \( 5.6 \times 10^{-3} \text{ nm} \).

Key Concepts

Planck's Constant

Planck's Constant, a fundamental parameter in quantum mechanics, is represented as \( h \) and has a value of \( 6.626 \times 10^{-34} \text{ m}^2 \text{ kg} / \text{s} \). This constant is pivotal in calculating the wavelength of particles using the De Broglie wavelength formula. It describes the scale at which quantum effects become significant. In the formula \( \lambda = \frac{h}{mv} \), Planck's constant serves as the numerator and is crucial for determining the wavelength of an object with given mass \( m \) and velocity \( v \).

• Universality: It applies to all particles, from electrons to macroscopic objects like golf balls.
• Quantum Scale: Highlights the transition from classical to quantum physics, where wave-like behavior becomes observable.

Remember, though, that while Planck's constant is incredibly small, it plays a large role in bridging the concepts between mass, velocity, and observable wavelength.

Mass Conversion

In physics calculations involving De Broglie's wavelength, it's essential to have mass in the correct units, typically kilograms. Our exercise involved a conversion from grams to kilograms, as the standard unit for formulas like \( \lambda = \frac{h}{mv} \). Here’s how you do it:

• Gram to Kilogram Conversion: Since 1 kg = 1000 grams, you simply divide the mass in grams by 1000 to convert it into kilograms.
• Our Example: The golf ball's mass was 46 grams, which converts to \( 0.046 \text{ kg} \).

Why is this important? Using consistent units ensures accurate calculations and results. Always double-check conversions, as they form the foundation for subsequent computations.

Wavelength Calculation

Calculating the wavelength involves substituting known values into De Broglie's formula. In our example, the goal was to find the wavelength of a golf ball moving at 30 m/s. Using \( \lambda = \frac{h}{mv} \), we:

• Substitute Consistent Units: Insert Planck's constant \( h = 6.626 \times 10^{-34} \text{ m}^2 \text{ kg} / \text{s} \), mass \( m = 0.046 \text{ kg} \), and velocity \( v = 30 \text{ m/s} \) into the formula.
• Perform the Calculation: Compute \( \lambda = \frac{6.626 \times 10^{-34}}{0.046 \times 30} \).
• Result Interpretation: Convert the resulting meters to nanometers for better comprehension in the context of wavelengths. Here, 1 meter = \( 10^9 \) nanometers.

Upon calculation, the wavelength is approximately \( 4.81 \times 10^{-25} \text{ nm} \). Although this is minuscule, it provides insight into particle wave properties.

Speed Calculation

Sometimes, rather than calculating wavelength, you may need to determine the speed at which a particle must travel to achieve a specific wavelength. To do this, rearrange De Broglie's formula as follows: \( v = \frac{h}{m\lambda} \).

• Known Values: Use Planck's constant \( h = 6.626 \times 10^{-34} \text{ m}^2 \text{ kg} / \text{s} \), mass \( m = 0.046 \text{ kg} \), and a given wavelength \( \lambda = 5.6 \times 10^{-3} \text{ nm} \).
• Convert Wavelength: Translate this wavelength from nanometers to meters, as this conversion is crucial \( 1 \text{ nm} = 10^{-9} \text{ m} \).
• Calculate Speed: Substitute into the formula \( v = \frac{6.626 \times 10^{-34}}{0.046 \times 5.6 \times 10^{-12}} \), resulting in a computed speed of approximately \( 2.58 \times 10^{-22} \text{ m/s} \).

This calculation shows just how extremely slow the ball must move to achieve a wavelength in the nanometer range. It serves as an excellent example of the disparity between quantum and classical mechanics.