Question

In Bohr's model of the hydrogen atom, an electron revolves around a proton in a circular orbit of radius \(5.29 \times 10^{-11} \mathrm{~m}\) with a speed of \(2.18 \times 10^{6} \mathrm{~m} / \mathrm{s}\). (a) What is the acceleration of the electron in this model of the hydrogen atom? (b) What is the magnitude and direction of the net force that acts on the electron?

Short answer

The acceleration of the electron in Bohr's model of the hydrogen atom is given by \(a = v^{2}/r\), and can be calculated by substituting the provided values for velocity and radius. The net force acting on the electron is given by \(F = m \times a\), where \(m\) is the mass of the electron and the acceleration was calculated in the first step. Both the acceleration and the net force are directed towards the center of the atom/path

Step-by-step solution

Calculate the centripetal acceleration

The formula for calculating the centripetal acceleration of an object moving in a circular path is given by \(a = v^{2}/r\), where \(v\) is the speed of the electron and \(r\) is the radius of the circular path. Substituting the provided values, \(v = 2.18 \times 10^{6} m/s\) and \(r = 5.29 \times 10^{-11} m\), into the formula we get \(a = (2.18 \times 10^{6})^{2} / (5.29 \times 10^{-11})\)

Compute the net force on the electron

The net force on an object moving in a circular path, also known as the centripetal force, is given by \(F = m \times a\), where \(m\) is the mass of the object (electron in this case) and \(a\) is the calculated acceleration. The mass of an electron is a known constant, \(9.11 \times 10^{-31} kg\). Inserting the values we have \(F = 9.11 \times 10^{-31} \times a\)

Determine the direction of the net force and the acceleration

Since the electron is moving in a circular path, the net force, as well as its acceleration, is centripetal, meaning they are directed towards the center of the path. This is a property of circular motion

Key concepts

Centripetal Acceleration

Centripetal acceleration is fundamental to understanding how an object moves along a curved trajectory, such as circular motion. In any circular path, a constant inward acceleration keeps the object moving along the desired path rather than in a straight line. This acceleration always points toward the center of the circular path and ensures the object does not fly off due to its inertia.

To determine the centripetal acceleration, we use the formula:

• \( a = \frac{v^2}{r} \)

Here, \(v\) is the velocity (speed and direction) of the object, and \(r\) is the radius of the circle. For an electron in Bohr's model of the hydrogen atom:

• Given: \( v = 2.18 \times 10^{6} \text{ m/s} \)
• Given: \( r = 5.29 \times 10^{-11} \text{ m} \)

We calculate the acceleration \(a\) using the formula mentioned above. This centripetal acceleration helps keep the electron in its circular orbit around the proton, a crucial aspect of maintaining atomic structure in the Bohr model.

Understanding centripetal acceleration helps establish why electrons move the way they do in atomic orbits and allows for deeper insights into atomic structure and behavior.

Circular Motion

Circular motion might seem like an advanced concept, but it's simply about objects moving in a path that forms a circle or part of a circle. This concept helps describe a variety of motions, from planets around the sun to electrons around a nucleus. In our example, the electron circles the proton, forming part of what early 20th-century scientists visualized as routes orbiting a sun-like scenario with the nucleus as the sun.

Key traits of circular motion include:

• Constant speed along the circular path
• Changing direction at every point in the motion
• An inward centripetal force and acceleration keeping the object on its path

For circular motion, like that described in Bohr's hydrogen atom model, the centripetal force is crucial in ensuring that the electron remains in orbit. Without it, the electron would deviate and follow a straight path due to inertia. Therefore, the concept of circular motion ties closely with centripetal elements—acceleration and force—showing the interconnectedness of these phenomena in governing atomic motions.

Centripetal Force

Centripetal force is what pulls an object toward the center of a circular path, preventing it from flying off straight due to inertia. In Bohr's hydrogen model, this force ensures that the electron keeps orbiting the proton.

This force can be calculated with the equation:

• \( F = m \times a \)

Here, \( m \) is the mass of the electron, and \( a \) is the centripetal acceleration previously calculated.

• Mass of an electron: \( 9.11 \times 10^{-31} \text{ kg} \)

By inputting these values, we identify the quantum scale forces maintaining electrons in stable orbits. The centripetal force is also always directed inward towards the center of the circle, such that in our atom model example, it draws the electron even more tightly around the proton.

This understanding illuminates how forces act at a distance on minute particles. The continuous inward force ensures the tight, cohesive structure of the hydrogen atom, key to countless chemical reactions and phenomena in our universe.