Problem 23 - Chemistry and Chemical Reactivity

Question

An electron moves with a velocity of \(2.5 \times 10^{8} \mathrm{cm} \cdot \mathrm{s}^{-1}\) What is its wavelength?

Step-by-Step Solution

Answer

The electron's wavelength is approximately \(2.91 \times 10^{-10} \text{ m}\).

Step 1: Understanding the problem

We need to calculate the wavelength of an electron given its velocity using the de Broglie wavelength formula. The velocity of the electron is \(v = 2.5 \times 10^{8} \text{ cm/s}\).

Step 2: Recall the de Broglie wavelength formula

The de Broglie wavelength \(\lambda\) of a particle is given by the 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 of the electron (\(9.109 \times 10^{-31} \text{ kg}\)), and \(v\) is the velocity of the electron.

Step 3: Convert velocity units

The velocity of the electron is given in cm/s. Convert this to m/s for consistency with the units of Planck's constant and electron mass: \[ 2.5 \times 10^{8} \text{ cm/s} = 2.5 \times 10^{6} \text{ m/s} \]

Step 4: Calculate the de Broglie wavelength

Substitute the values into the de Broglie formula: \[ \lambda = \frac{6.626 \times 10^{-34}}{(9.109 \times 10^{-31})(2.5 \times 10^{6})} \] Calculate the result to find \(\lambda\): \[ \lambda \approx 2.91 \times 10^{-10} \text{ m} \]

Step 5: Conclusion

The de Broglie wavelength of the electron is approximately \(2.91 \times 10^{-10} \text{ m}\).

Key Concepts

Planck's constant

Planck's constant is a fundamental quantity in quantum mechanics that helps bridge the equations of some classical physics, like energy, with wave-like properties of particles. Its value is extremely small, precisely \(6.626 \times 10^{-34} \text{ m}^2\text{kg/s}\).

The Importance of Planck's Constant

- Planck's constant determines the scale at which quantum effects become noticeable.- It is key in the de Broglie wavelength formula to associate a wave property with particles.This tiny constant allows particles such as electrons to display wave-like behaviors, driving concepts like interference and diffraction. Remember, this is why even atoms and subatomic particles like electrons can act like 'particles' and 'waves' in different conditions.

Electron mass

The electron mass represents another crucial factor in the de Broglie wavelength calculation and is a tiny 9.109 × 10^{-31} kilograms. Despite its small mass, it plays a vital role in how electrons behave at quantum scales.

Why Electron Mass Matters

- Contributes to calculating the momentum of electrons when paired with velocity. - Low mass allows electrons to exhibit significant wave-like properties. Every time you calculate an electron's de Broglie wavelength, its mass contributes to the particles’ incredible flexibility in behaving as waves or particles—central to understanding quantum mechanics.

Unit conversion

Unit conversion is often necessary when performing physics calculations, especially in quantum mechanics. Here, converting units ensures consistency across calculations to prevent errors. For example, velocity was provided in cm/s, but needs to be aligned with SI units (m/s).

Why Conversion is Crucial

- SI units like meters ensure compatibility with constants, like Planck's constant, which are defined in these units. - Accurate conversions ensure correct calculations of physical quantities. Always convert quantities to compatible units before substituting into equations. Missing this step could lead to incorrect results, which in science can significantly change the outcome and interpretation of an experiment.

Quantum mechanics

Quantum mechanics is a branch of physics that deals with physical phenomena at a nanoscale, where classical mechanics fails to provide accurate predictions. Fundamental to quantum mechanics is the concept that particles like electrons can behave both as waves and as particles.

Core Principles of Quantum Mechanics

- **Duality**: Particles exhibit both wave and particle characteristics. - **Uncertainty Principle**: Exact position and momentum cannot both be precisely determined at the same time. Understanding quantum mechanics is essential for explaining phenomena like electron orbitals in chemistry and semiconductors in physics. The de Broglie wavelength equation is one example of applying quantum mechanics to explain the behavior of particles that classical mechanics cannot account for.