Problem 25 - Chemistry and Chemical Reactivity

Question

Calculate the wavelength, in nanometers, associated with a \(1.0 \times 10^{2}-\mathrm{g}\) golf ball moving at \(30 . \mathrm{m} \cdot \mathrm{s}^{-1}\) (about \(67 \mathrm{mph}\) ). How fast 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 approximately \(2.21 \times 10^{-25}\) nm, and to achieve \(5.6 \times 10^{-3}\) nm, the golf ball must travel at \(1.18 \times 10^{-21}\) m/s.

Step 1: Understand the Problem

We need to calculate the wavelength of a 100 g golf ball moving at 30 m/s and find the velocity that will give it a specific wavelength of \(5.6 \times 10^{-3}\) nm.

Step 2: Identify the Formula

Use the de Broglie wavelength formula given by \(\lambda = \frac{h}{mv}\) where \(\lambda\) is the wavelength, \(h\) is Planck's constant \( (6.63 \times 10^{-34} \text{ m}^2 \text{ kg} \text{ s}^{-1}) \), \(m\) is mass, and \(v\) is velocity.

Step 3: Convert Mass

Convert the golf ball's mass from grams to kilograms. Since 1 g = 0.001 kg, we have \(100 \, \text{g} = 0.1 \, \text{kg}\).

Step 4: Calculate Initial Wavelength

Substitute the given mass \(m = 0.1 \, \text{kg}\), velocity \(v = 30 \, \text{m/s}\), and Planck's constant into the de Broglie formula to find the wavelength: \[\lambda = \frac{6.63 \times 10^{-34}}{0.1 \times 30}.\] Calculate \(\lambda\), which comes out to be \(2.21 \times 10^{-34} \, \text{m}\). Convert this to nanometers (1 m = \(10^9\) nm): \[\lambda \approx 2.21 \times 10^{-25} \, \text{nm}.\]

Step 5: Set Up Equation for Desired Wavelength

To find the velocity for a wavelength of \(5.6 \times 10^{-3}\) nm, use the de Broglie formula rearranged:\[v = \frac{h}{m\lambda}.\] Convert the wavelength from nm to meters: \(5.6 \times 10^{-3} \, \text{nm} = 5.6 \times 10^{-12} \, \text{m}\).

Step 6: Calculate Desired Velocity

Substitute \(m = 0.1 \, \text{kg}\), \(\lambda = 5.6 \times 10^{-12} \, \text{m}\), and \(h\) into the rearranged formula:\[v = \frac{6.63 \times 10^{-34}}{0.1 \times 5.6 \times 10^{-12}}.\]Calculate \(v\), which results in \(1.18 \times 10^{-21} \, \text{m/s}\).

Key Concepts

Planck's Constant

Planck's constant is a fundamental quantity symbolized by the letter "h". It is used extensively in quantum mechanics and provides a proportional relationship between the energy of a photon and the frequency of its associated electromagnetic wave.

Planck's constant has a fixed value of approximately 6.63 x 10^-34 J·s (joule-seconds). This tiny number reflects the small scale at which quantum mechanical effects occur, influencing how particles such as electrons or, in this case, a golf ball, can exhibit wave-like characteristics.

In the context of the de Broglie wavelength, Planck's constant is crucial because it is one half of the relationship that defines the particle's wave behavior:

• \( \lambda = \frac{h}{mv} \) where \( \lambda \) is wavelength.

Understanding Planck's constant is fundamental for determining how ordinary objects, like a golf ball, can have their wave nature described theoretically, despite it being immeasurably small when in motion.

Mass Conversion

Converting mass units correctly is essential for precisely applying formulas in physics problems. When dealing with problems involving de Broglie wavelengths, mass must be expressed in kilograms (kg) for consistency with other standard physics units like meters and seconds.

In our exercise, the golf ball's mass is given in grams (g), a common unit for everyday measurements. However, for scientific calculations, we convert this mass to kilograms, as in:

• 1 g = 0.001 kg
• Therefore, 100 g = 0.1 kg

This simple conversion ensures that when you substitute the mass into the formula, it's compatible with other unit measurements like Planck's constant, which already uses kg, m, and s, maintaining unit consistency throughout the calculation. Staying consistent with units prevents errors and ensures accurate calculations.

Velocity Calculation

Calculating velocity requires rearranging the de Broglie wavelength formula to solve for velocity. Velocity, in this context, explains the speed at which a particle or object, like a golf ball, must travel to exhibit a particular wavelength.

To find this velocity, the de Broglie equation can be rearranged as follows:

• v = \frac{h}{m\lambda}

Here, *v* represents velocity, *h* is Planck's constant, *m* is the mass of the object, and \( \lambda \) is the desired wavelength.

In our specific case, if we want to determine the velocity required for the golf ball to have a wavelength of \(5.6 \times 10^{-3}\) nm, first convert that wavelength into meters:

• \(5.6 \times 10^{-3}\) nm = \(5.6 \times 10^{-12}\) m

You then substitute these values back into the rearranged equation to get the required velocity. This step is crucial because it shows how seemingly immaterial wavelengths for macroscopic objects require microscopic velocities when traditional macroscopic speeds are applied.