Problem 104 - General Chemistry: Principles and Modern Applications

Question

The angular momentum of an electron in the Bohr hydrogen atom is mur , where \(m\) is the mass of the electron, \(u,\) its velocity, and \(r,\) the radius of the Bohr orbit. The angular momentum can have only the values nh/2 \(\pi\), where \(n\) is an integer (the number of the Bohr orbit). Show that the circum frences of the various Bohr orbits are integral multiples of the de Broglie wavelengths of the electron treated as a matter wave.

Step-by-Step Solution

Answer

The circumference for each Bohr orbit is an integral multiple of an electron's de Broglie wavelength.

Step 1: Write down the known facts

Angluar Momentum L = \(mur = \frac{nh}{2\pi}\), where m is mass of electron, u its velocity, r the radius, n is an integer denoting orbit number, and h is Planck's constant. According to de Broglie hypothesis, \(\lambda = \frac{h}{mu}\), where \(\lambda\) is the wavelength.

Step 2: Substituting for velocity

From the angular momentum equation, the velocity can be written as \(u = \frac{nh}{2\pi m r}\). Substitute this into the wavelength formula, and we find \(\lambda = \frac{2\pi r}{n}\)

Step 3: Relate the circumference to the wavelength.

The circumference of an orbit is given by \(2\pi r\). Thus, each orbit's circumference can be written as \(n\lambda\), meaning that each orbit's circumference is an integral multiple of the electron's de Broglie wavelength.

Key Concepts

Angular Momentum

In the Bohr model of the hydrogen atom, we find that angular momentum is a crucial component. Angular momentum refers to the amount of rotation an object has, taking into account its mass, shape, and speed. In this context, an electron orbiting a nucleus carries angular momentum. The formula to determine this is given by:

• Angular Momentum, \(L = mur = \frac{nh}{2\pi}\)

Where \(m\) stands for the electron's mass, \(u\) its velocity, and \(r\) the radius of the orbit. The constant \(h\) is Planck's constant, which plays a significant role in quantum mechanics. Here, \(n\) is an integer, representing the quantum number, which shows that angular momentum is quantized - it can only take specific values, specifically multiples of \(\frac{h}{2\pi}\).
In this framework, each orbit corresponds to a particular value of angular momentum. This quantization is fundamental to understanding how electrons manage to 'stay' in their specific orbits without spiraling into the nucleus or flying away.

De Broglie Wavelength

The de Broglie wavelength is a concept that bridges classical and quantum physics, mainly associated with particles, showcasing wave-like behavior. According to the de Broglie hypothesis, matter exhibits both particle and wave characteristics. For an electron in motion, the wavelength \(\lambda\) can be determined as:

• \(\lambda = \frac{h}{mu}\)

Where \(h\) is Planck's constant, \(m\) is the mass of the electron, and \(u\) is its velocity. By determining this wavelength, we can understand how particles such as electrons exhibit wave properties. Substituting the expression for velocity derived from the angular momentum equation, \(u = \frac{nh}{2\pi m r}\), into the wavelength equation, we get \(\lambda = \frac{2\pi r}{n}\).
This formulation implies that the wavelength of the electron, as it behaves like a wave, is directly related to its orbital path in an atom, giving a new dimension to our understanding of atomic structure.

Matter Waves

The idea of matter waves is a revolutionary concept introduced by Louis de Broglie. It describes the wave nature of all particles, positing that particles like electrons have wave characteristics. De Broglie's theory suggests that every moving particle or object has an associated wavelength - a key idea in quantum mechanics.
An electron in a Bohr orbit can be treated as a wave encircling the nucleus. The circumference of the orbit equals an integral number of wavelengths, resulting from:

• Orbit circumference = \(2\pi r = n\lambda\)

This relationship ensures that the wave aligns coherently with itself after one complete loop around the nucleus, leading to stable electron paths. This condition means that the electron's orbit acts like a standing wave, requiring the length of each orbit to be an integer multiple of its wavelength. This integration of wave behavior into particles marks a paradigm shift in understanding atomic and subatomic structures.
Thus, the concept of matter waves not only helps explain atomic stability in Bohr’s model but also deepens our grasp of the dual nature of particles as both waves and particles.