Problem 23 - Chemistry and Chemical Reactivity

Question

An electron moves with a velocity of \(2.5 \times 10^{8} \mathrm{cm} / \mathrm{s}\) What is its wavelength?

Step-by-Step Solution

Answer

The wavelength of the electron is approximately 0.291 nm.

Step 1: Understand the Problem

We need to find the wavelength of an electron moving with a given velocity. This problem involves using the de Broglie wavelength formula, which relates the wavelength to the momentum of a particle.

Step 2: Recall de Broglie's Equation

According to de Broglie, the wavelength \( \lambda \) of a particle is given by the equation: \[ \lambda = \frac{h}{p} \] where \( h \) is the Planck's constant \((6.626 \times 10^{-34} \, \text{J s})\) and \( p \) is the momentum of the particle.

Step 3: Calculate Momentum

The momentum \( p \) of a particle is calculated as the product of its mass \( m \) and velocity \( v \): \[ p = mv \] For an electron, its mass \( m \) is approximately \(9.109 \times 10^{-31} \, \text{kg}\), and the given velocity \( v \) is \(2.5 \times 10^{8} \, \text{cm/s} = 2.5 \times 10^{6} \, \text{m/s}\).

Step 4: Insert Values into Momentum Formula

Calculate the momentum: \[ p = (9.109 \times 10^{-31} \, \text{kg}) \times (2.5 \times 10^{6} \, \text{m/s}) = 2.27725 \times 10^{-24} \, \text{kg m/s} \]

Step 5: Calculate the Wavelength

Now, apply the de Broglie equation to find the wavelength: \[ \lambda = \frac{6.626 \times 10^{-34} \, \text{J s}}{2.27725 \times 10^{-24} \, \text{kg m/s}} = 2.91 \times 10^{-10} \, \text{m} \]

Step 6: Convert the Wavelength to More Convenient Units

Since 1 nm = \(10^{-9}\) m, this wavelength is equivalent to: \[ \lambda = 0.291 \times 10^{-9} \, \text{m} = 0.291 \, \text{nm} \]

Key Concepts

electron velocity

An electron's velocity is a key piece of information when working with de Broglie wavelengths. Velocity refers to the speed at which the electron is moving in a particular direction.
When calculating de Broglie wavelength, the velocity is crucial for determining the electron's momentum.

• In the given exercise, we have a velocity of \(2.5 \times 10^{8}\) cm/s.
• Make sure to convert this velocity into meters per second for further calculations. In this case, we convert it to \(2.5 \times 10^{6}\) m/s.

Understanding the velocity helps us understand how fast the particle is moving. Hence, it significantly impacts the resultant momentum, which is pivotal in calculating the wavelength.

Planck's constant

Planck's constant, denoted as \(h\), plays a critical role in quantum mechanics and the de Broglie wavelength formula. It is a fundamental constant of nature that helps in describing the quantized nature of energy and matter.

• The value of Planck's constant is approximately \(6.626 \times 10^{-34}\) J·s.
• This constant is essential in the correlating equation for de Broglie wavelength: \(\lambda = \frac{h}{p}\), where \(\lambda\) is the wavelength and \(p\) is the momentum.

Utilizing Planck's constant allows us to bridge the gap between particle characteristics and wave-like features as per the wave-particle duality concept.

momentum calculation

Momentum is a critical factor in determining the de Broglie wavelength. It is calculated as the product of mass and velocity, employing the equation:

• \(p = mv\), where \(m\) represents mass and \(v\) denotes velocity.

For particles such as electrons:

• The mass of an electron is approximately \(9.109 \times 10^{-31}\) kg.
• Using the given velocity \(2.5 \times 10^{6}\) m/s, we calculate the momentum: \(p = (9.109 \times 10^{-31} \, \text{kg}) \times (2.5 \times 10^{6} \, \text{m/s}) = 2.27725 \times 10^{-24} \, \text{kg m/s}\).

By calculating the momentum, we bridge the macroscopic and microscopic realms, fundamental for progressing to solve the de Broglie wavelength.