Problem 88 - Chemistry The Central Science

Question

An electron is accelerated through an electric potential to a kinetic energy of \(18.6 \mathrm{keV}\). What is its characteristic wavelength? [Hint: Recall that the kinetic energy of a moving object is \(E=\frac{1}{2} m v^{2},\) where \(m\) is the mass of the object and \(\nu\) is the speed of the object.]

Step-by-Step Solution

Answer

The characteristic wavelength of the electron is approximately \(2.428 \times 10^{-11}\) meters.

Step 1: Convert kinetic energy to SI units

The given kinetic energy is in keV, so first, we have to convert it into SI units (Joules). To do this, we use the conversion factor 1 eV = 1.6 x 10^{-19} J. \(18.6 \,\text{keV} = 18.6 \times 10^3 \,\text{eV}\) So, E = 18.6 x 10^3 eV x (1.6 x 10^{-19} J/eV) = 2.976 x 10^{-15} J

Step 2: Use the given relation E = (1/2)mv^2

Now, we have the electron's kinetic energy in Joules. Next, we'll use the given relation E = (1/2)mv^2 and rearrange it to solve for the velocity (v). The mass of an electron (m) is 9.11 x 10^{-31} kg. \(v = \sqrt{\frac{2E}{m}} = \sqrt{\frac{2 \times 2.976 \times 10^{-15}\,\text{J}}{9.11 \times 10^{-31}\,\text{kg}}}\) Calculating the value for v, we get: \(v = 2.586 \times 10^7\,\text{m/s}\)

Step 3: Apply the de Broglie wavelength formula

Now that we have the velocity of the electron, we can use the de Broglie wavelength formula to find its characteristic wavelength. The de Broglie wavelength formula is given as: \(\lambda = \frac{h}{m v}\) where h is the Planck's constant (6.626 x 10^{-34} Js), m is the mass of the electron (9.11 x 10^{-31} kg), and v is the velocity we calculated in step 2. Plugging in the values, we get: \(\lambda = \frac{6.626 \times 10^{-34}\,\text{Js}}{(9.11 \times 10^{-31}\,\text{kg}) \times (2.586 \times 10^7\,\text{m/s})}\) Calculating the value for λ, we get: \(\lambda = 2.428 \times 10^{-11}\,\text{m}\) Hence, the characteristic wavelength of the electron is approximately \(2.428 \times 10^{-11}\) meters.

Key Concepts

Kinetic Energy

Kinetic energy is an essential concept in physics that describes the energy an object has due to its motion. When it comes to electrons, or other particles, kinetic energy is important because it influences their behavior in different environments. In general, the kinetic energy ( \(E\)) for any moving object can be calculated using the formula:
\[ E = \frac{1}{2} m v^2 \]where \(m\) is the mass, and \(v\) is the velocity of the object. Understanding how to calculate this helps us determine how fast an electron is moving when its kinetic energy is known.

• Kinetic energy depends on mass and velocity.
• Electrons have a small mass, but when accelerated by electric potential, they can reach high velocities.
• To find the velocity of an electron when its kinetic energy is given, we need to rearrange the kinetic energy formula.

Being able to switch between kinetic energy and velocity calculations is key when studying particles like electrons.

Planck's Constant

Planck's constant is one of the most fundamental constants in quantum mechanics. It symbolically represents the link between the wave characteristics and particle nature of matter. The constant is denoted as \( h \) and is approximately equal to \( 6.626 \times 10^{-34} \text{ Js} \).
Planck's constant is a crucial part of the de Broglie wavelength formula, which links the momentum of a particle to its wavelength. This formula is essential for understanding phenomena at the quantum level, particularly when it comes to analyzing very small particles like electrons. De Broglie proposed that particles exhibit wave-like properties, and their wavelength \( \lambda \) is given by:
\[ \lambda = \frac{h}{mv} \] where \( m \) is mass and \( v \) is velocity.

• Planck's constant plays a pivotal role in quantum physics.
• It's used to calculate the characteristic wavelength of particles.
• This constant underscores the duality of wave-particle behavior in quantum mechanics.

Knowing how to use Planck's constant in such formulas helps students bridge classical and modern physics.

Electron Velocity

Electron velocity is crucial for understanding various properties of electrons in physics. The velocity of an electron is influenced by its kinetic energy, as demonstrated with the earlier formula:
\( v = \sqrt{\frac{2E}{m}} \)
Given the specific kinetic energy and using the mass of an electron, we can calculate how fast an electron moves. This is particularly helpful when transitioning from classical physics to quantum physics, where electron behavior doesn't always align with everyday intuition.

• The velocity of an electron is typically very high due to its small mass.
• Knowing this velocity is key in determining its de Broglie wavelength.
• This velocity provides insight into how electrons interact in different fields, impacting potential applications in technologies.

Understanding electron velocity is beneficial for visualizing and predicting the behavior of particles in different physical scenarios.