Problem 50 - Chemistry The Central Science In Si Units

Question

The electron microscope has been widely used to obtain highly magnified images of biological and other types of materials. When an electron is accelerated through a particular potential field, it attains a speed of \(9.47 \times 10^{6} \mathrm{~m} / \mathrm{s}\) What is the characteristic wavelength of this electron? Is the wavelength comparable to the size of atoms?

Step-by-Step Solution

Answer

The wavelength is approximately \(7.68 \times 10^{-11} \text{ m}\), comparable to atomic sizes.

Step 1: Identify the relationship

To find the characteristic wavelength of an electron, we use the de Broglie wavelength formula: \( \lambda = \frac{h}{mv} \), where \( \lambda \) is the wavelength, \( h \) is Planck's constant \(6.626 \times 10^{-34} \text{ m}^2 \text{ kg / s}\), \( m \) is the mass of an electron \(9.109 \times 10^{-31} \text{ kg}\), and \( v \) is the velocity of the electron.

Step 2: Plug in the known values

Substitute the given values into the de Broglie wavelength formula: \[ \lambda = \frac{6.626 \times 10^{-34} \text{ m}^2 \text{ kg/s} }{9.109 \times 10^{-31} \text{ kg} \times 9.47 \times 10^{6} \text{ m/s} } \]

Step 3: Calculate the wavelength

Perform the calculation to find the wavelength:- Multiply the mass of the electron by its velocity: \( 9.109 \times 10^{-31} \text{ kg} \times 9.47 \times 10^{6} \text{ m/s} = 8.622 \times 10^{-24} \text{ kg m/s} \).- Divide Planck's constant by this product to find the wavelength: \( \lambda = \frac{6.626 \times 10^{-34} \text{ m}^2 \text{ kg/s} }{8.622 \times 10^{-24} \text{ kg m/s} } \approx 7.68 \times 10^{-11} \text{ m} \).

Step 4: Compare with atomic size

Typically, atomic sizes are on the order of \(10^{-10} \text{ m}\), or 1 Ångström. The calculated wavelength, \(7.68 \times 10^{-11} \text{ m}\), is comparable to but slightly smaller than atomic sizes, indicating that the wavelength is indeed within the scale of atoms.

Key Concepts

Electron Microscopy

Electron microscopy is a fascinating technology that uses a beam of electrons to create images of tiny objects, which are often invisible to conventional light microscopes. This is because electrons have much shorter wavelengths than visible light. This shorter wavelength allows the magnification of objects up to thousands of times larger than what can be achieved with light microscopes.

There are several types of electron microscopes, including:

• Transmission Electron Microscope (TEM): Which allows us to view the internal structures of thin specimens.
• Scanning Electron Microscope (SEM): Which provides detailed three-dimensional images of the surface of a sample.

Understanding this technology is crucial for fields such as biology, material science, and chemistry, as it provides insights into the structure and composition of materials at a molecular or atomic level.

Quantum Mechanics

Quantum mechanics is the branch of physics that studies the behavior of matter and energy on the atomic and subatomic scale. It is essential for explaining how particles, such as electrons, behave and interact. This theory encompasses the idea that particles can have properties of both a wave and a particle.

One of the key concepts in quantum mechanics is the wave-particle duality. This principle explains that particles, like electrons, exhibit both wave-like and particle-like properties. The de Broglie wavelength is a perfect example of this duality, showing that particles like electrons have a wavelength associated with their motion.

The consequences of this duality affect how we perceive the physical world. It has led to the development of numerous technologies, such as electron microscopy, which actively uses principles of quantum mechanics to resolve structures at the atomic scale.

Atomic Scale Calculations

Atomic scale calculations involve mathematical processes used to analyze the properties and behaviors of atoms and molecules. In many scientific fields, understanding the movement and interaction of atoms is crucial. Quantum mechanics is often employed in these calculations to provide accurate and reliable results.

In the context of the de Broglie wavelength, these calculations help us determine the wavelength associated with moving electrons, such as those used in electron microscopes. By calculating the wavelength using the de Broglie formula, \( \lambda = \frac{h}{mv} \), where \( \lambda \) is the wavelength, \( h \) is Planck's constant, \( m \) is the mass of the electron, and \( v \) is its velocity, we can compare the wavelength to the size of atoms.

Such calculations have practical applications, like improving imaging techniques and developing new materials at the atomic level. It underscores the fascinating intersection of mathematics, physics, and chemistry in advancing technological capabilities.