Problem 46 - Chemistry The Central Science

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 \(8.95 \times 10^{6} \mathrm{~m} / \mathrm{s}\). What is the characteristic wavelength of this electron? Is the wavelength comparable to the size of atoms?

Step-by-Step Solution

Answer

The characteristic wavelength of an electron with a speed of \(8.95 \times 10^6 \; m/s\) is \(8.13 \times 10^{-11}\; m\), which is comparable to the size of atoms (\(10^{-10}\; m\)). This ability to have a wavelength similar to atomic sizes allows electron microscopes to produce highly magnified and detailed images of atoms and other small biological materials.

Step 1: The mass of an electron (m) can be determined from a standard reference table. Its value is \(9.11 \times 10^{-31} \; kg\). #Step 2: Calculate the momentum of the electron#

Now, let's calculate the momentum (p) using the formula: \[ p = m \cdot v \] Substitute the values for mass (m) and speed (v): \[ p = (9.11 \times 10^{-31} \; kg) \cdot (8.95 \times 10^{6} \; m/s) \] \[ p = 8.15 \times 10^{-24} \; kg \cdot m/s \] #Step 3: Calculate the de Broglie wavelength#

Step 2: We can now use the de Broglie wavelength formula: \[ \lambda = \frac{h}{p} \] where \(h\) is the Planck's constant, which is equal to \(6.63 \times 10^{-34} \; Js\). Substituting the values of \(h\) and \(p\), we get: \[ \lambda = \frac{6.63 \times 10^{-34}\; Js}{8.15 \times 10^{-24} \; kg \cdot m/s} \] \[ \lambda = 8.13 \times 10^{-11}\; m \] #Step 4: Compare the wavelength to the size of atoms#

The size of atoms is on the order of \(10^{-10}\; m\) (1 Ångström). Comparing the calculated wavelength (\(8.13 \times 10^{-11}\; m\)) to the size of atoms, we find that the wavelength of the electron is approximately one order of magnitude smaller than the size of an atom and thus can be considered comparable to the size of atoms. This explains why electron microscopes can produce highly magnified and detailed images of atoms and other small biological materials.

Key Concepts

de Broglie wavelength

The de Broglie wavelength is a fundamental concept in quantum mechanics, which describes the wave-like nature of particles. According to de Broglie's hypothesis, every moving particle or object has an associated wavelength. The wavelength can be calculated using the formula:

• \( \lambda = \frac{h}{p} \)

Here, \( \lambda \) stands for the de Broglie wavelength, \( h \) is Planck's constant, and \( p \) is the momentum of the particle.
The idea is that not only light, which was known to exhibit wave-particle duality, but also matter, like electrons, can show similar properties.
As particles get smaller and move at higher speeds, their wave nature becomes more significant. In practical applications, like electron microscopes, the de Broglie wavelength helps us understand the scale at which we can see materials. Since the wavelength is inversely proportional to momentum, faster and more massive particles have shorter wavelengths.

momentum

Momentum is a principle in physics that describes the quantity of motion an object has. It is a vector quantity, meaning it has both magnitude and direction. The momentum \( p \) of an object is calculated using the formula:

• \( p = m \cdot v \)

where \( m \) is the mass and \( v \) is the velocity of the object.
In the context of the electron microscope problem, the electron's momentum is calculated by multiplying its known mass and velocity.
Momentum plays a crucial role in determining an object’s de Broglie wavelength. An increase in either mass or velocity increases the momentum, consequently reducing the wavelength. This relationship is important because it shows how microscopic particles can act like waves, allowing technologies like electron microscopes to probe details as minute as the atomic scale.

atomic size

Atomic size refers to the size of an atom, typically measured in terms of its radius or diameter. Atomic sizes vary across the periodic table, but they are generally on the scale of nanometers or smaller, often perceived in terms of Angstroms (

• 1 Ångström = \(10^{-10}\) meters

).
Comparing the wavelength of moving particles, like electrons, to atomic sizes helps us understand why electron microscopes are powerful.
The ability of electron microscopes to see detail at atomic scales hinges on the fact that the de Broglie wavelength of high-speed electrons is quite small, comparable to the size of atoms.
These microscopes use electrons instead of light because electrons have much shorter wavelengths than visible light, enabling them to resolve smaller structures.

Planck's constant

Planck's constant is a fundamental constant in quantum mechanics, denoted by \( h \). It describes the quantized nature of energy and plays a vital role in the field of quantum mechanics.

• Its value is approximately \( 6.63 \times 10^{-34} \; Js \)

This constant is critical in calculations involving the de Broglie wavelength because it serves as a proportionality factor between the momentum of a particle and its corresponding wavelength.
Plank’s constant bridges the gap between classical and quantum physics by describing how particles and waves interact on a quantum level.
Understanding and using Planck's constant allows for the conceptualization and calculation of various phenomena, especially in technologies such as electron microscopes, which rely on quantum principles to reveal images of minuscule atomic structures.