Problem 58 - College Physics

Question

\(\bullet\) A specimen of the microorganism Gastropus hyptopus measures 0.0020 \(\mathrm{cm}\) in length and can swim at a speed of 2.9 times its body length per second. The tiny animal has a mass of roughly \(8.0 \times 10^{-12} \mathrm{kg}\) . (a) Calculate the de Broglie wave- length of this organism when it is swimming at top speed. (b) Calculate the kinetic energy of the organism (in eV) when it is swimming at top speed.

Step-by-Step Solution

Answer

(a) The de Broglie wavelength is \(1.43 \times 10^{-17}\, \text{m}\). (b) The kinetic energy is \(8.41 \times 10^{-3}\, \text{eV}\).

Step 1: Calculate Velocity

First, find the speed of the microorganism by multiplying its body length by the speed factor: \[\text{Speed} = 2.9 \times 0.0020\, \text{cm/s} = 0.0058\, \text{cm/s} = 5.8 \times 10^{-5}\, \text{m/s}.\]

Step 2: Calculate de Broglie Wavelength

Use the de Broglie wavelength formula, \( \lambda = \frac{h}{mv} \), where \( h \) is the Planck's constant \( 6.626 \times 10^{-34} \text{ m}^2 \text{kg/s} \), \( m \) is the mass \( 8.0 \times 10^{-12} \text{ kg} \), and \( v \) is the speed \( 5.8 \times 10^{-5} \text{ m/s} \). Replace these into the formula:\[\lambda = \frac{6.626 \times 10^{-34}}{8.0 \times 10^{-12} \times 5.8 \times 10^{-5}} = 1.43 \times 10^{-17} \text{ m}.\]

Step 3: Calculate Kinetic Energy in Joules

Use the kinetic energy formula, \( KE = \frac{1}{2} mv^2 \), where \( m = 8.0 \times 10^{-12} \text{ kg} \) and \( v = 5.8 \times 10^{-5} \text{ m/s} \):\[ KE = \frac{1}{2} \times 8.0 \times 10^{-12} \times (5.8 \times 10^{-5})^2 = 1.3472 \times 10^{-21} \text{ J}.\]

Step 4: Convert Kinetic Energy to Electronvolts

To convert joules to electronvolts (eV), use the conversion factor \( 1\, ext{eV} = 1.602 \times 10^{-19} \text{ J} \). Thus, the kinetic energy in eV is:\[ KE = \frac{1.3472 \times 10^{-21}}{1.602 \times 10^{-19}} = 8.41 \times 10^{-3} \text{ eV}.\]

Key Concepts

Kinetic Energy in Electronvolts

Kinetic energy is the energy that an object possesses due to its motion. It is calculated using the formula \[KE = \frac{1}{2} mv^2\]where \( m \) is the mass of the object, and \( v \) is its velocity. The SI unit of kinetic energy is joules (J). However, when dealing with very small energies, such as those at the atomic or subatomic level, it is often more convenient to use electronvolts (eV).

An electronvolt is the amount of kinetic energy gained (or lost) by an electron when it is accelerated through an electric potential difference of one volt. To convert from joules to electronvolts, you use the conversion factor:

• \(1 \text{ eV} = 1.602 \times 10^{-19} \text{ J} \)

Knowing this conversion factor is essential for understanding how tiny particles behave, especially in physics and chemistry contexts. Thus, for a microorganism like Gastropus hyptopus, converting its kinetic energy into electronvolts helps us grasp the influence of its tiny mass and speed more intuitively.

de Broglie Wavelength Calculation

The de Broglie wavelength is a concept from quantum mechanics that describes the wave-particle duality of matter. According to de Broglie, every moving particle has a wavelength associated with it. This wavelength, known as the de Broglie wavelength, is calculated using the formula:\[\lambda = \frac{h}{mv}\]Here, \( \lambda \) is the wavelength, \( h \) is the Planck's constant \( (6.626 \times 10^{-34} \text{ m}^2\text{kg/s}) \), \( m \) is the mass of the particle, and \( v \) is its velocity.

In the context of particle physics, this wavelength is critical for understanding the quantum behaviors of particles like electrons. However, it can also be applied to larger objects, like microorganisms, even though their observed wavelengths are extremely small. Calculating the de Broglie wavelength helps us realize that all matter, regardless of size, exhibits wave-like properties. In the example of the microorganism Gastropus hyptopus, although its de Broglie wavelength measures at \( 1.43 \times 10^{-17} \text{ m} \), the concept is demonstrated elegantly as it brings quantum mechanics principles into the tangible world.

Planck's Constant

Planck's constant is a fundamental constant in physics that plays a crucial role in quantum mechanics. Symbolized as \( h \), it connects the energy of a photon with the frequency of its electromagnetic wave. The constant has a value of \[6.626 \times 10^{-34} \text{ m}^2\text{kg/s}\]Planck's constant is critical for calculations involving the energy of photons and other quantum phenomena.

One of its most famous applications is found in the formula for the energy of a photon:\[E = h u\]where \( E \) is the energy, and \( u \) is the frequency of the photon. Similarly, in the de Broglie wavelength formula, Planck's constant allows us to relate a particle's momentum to its wavelength. This connects macroscopic behaviors with quantum characteristics.

Understanding Planck's constant is important because it is foundational to quantum mechanics, offering insights into the size of quanta and paving the way for advancements in understanding light, atoms, and the physical universe at the smallest scales.