Question

In the Bohr model of the hydrogen atom, the electron revolves in a circular orbit around the nucleus. If the radius is \(5.3 \times 10^{-11} \mathrm{~m}\) and the electron makes \(6.6 \times 10^{15} \mathrm{rev} / \mathrm{s}\), find (a) the speed of the electron, \((b)\) the acceleration of the electron, and \((c)\) the force acting on the electron. (This force is the result of the attraction between the positively charged nucleus and the negatively charged electron.)

Short answer

The speed of the electron is obtained from the formula for speed in a circular path. The acceleration is calculated using the formula for centripetal acceleration. The force acting on the electron is found by substituting the mass of the electron and the calculated acceleration into the formula for centripetal force.

Step-by-step solution

Calculate Speed

Since the electron is moving in a circular path, it is covering a certain distance per unit time, which signifies its speed. In this context, the speed \( v \) is given by the formula: \( v = 2 \pi r f \), where \( r \) is the radius, and \( f \) is the frequency, which is the number of revolutions per unit time. With \( r = 5.3 \times 10^{-11} \, m \) and \( f = 6.6 \times 10^{15} \, rev/s \), substituting these values into the formula gives the speed of the electron.

Calculate Acceleration

Acceleration for circular motion which is also called centripetal acceleration, \( a \), is given by the formula: \( a = v^2 / r \), where \( v \) is the speed from the previous step and \( r \) is the radius. Substituting the speed and the radius value into the formula provides the acceleration of the electron.

Calculate Force

The force \( F \) acting on the electron is calculated using the formula for centripetal force: \( F = m \cdot a \), where \( m \) is the mass of the electron (\( 9.1 \times 10^{-31} \, kg \)), and \( a \) is the acceleration obtained from the previous step. Substituting these values into the formula gives the force acting on the electron.

Key concepts

Electron Circular Motion

The Bohr model of the hydrogen atom paints a picture where the electron orbits the nucleus similar to how planets orbit the Sun. However, unlike planets, the electron's orbit is fixed and quantized in discrete circles known as energy levels. In our context, when an electron is said to be in 'circular motion', it means that it moves in a path that forms a circle around the nucleus.

Understanding the electron's circular motion involves grasping that despite the motion being continuous, the electron maintains a constant distance from the nucleus, known as the radius of its orbit. This circular motion is a result of the electrostatic force of attraction between the oppositely charged electron and nucleus, a concept consistent with Coulomb's law. Moreover, due to this motion, the electron also possesses angular momentum, which, according to Bohr's postulates, is quantized.

• The radius of the electron's orbit in the hydrogen atom is given to be approximately 0.053 nanometers.
• The frequency, or the number of revolutions per second, is exceptionally high, illustrating the electron's swift motion around the nucleus.

Centripetal Acceleration

The concept of centripetal acceleration is vital when discussing circular motion. It refers to the acceleration that is directed towards the center of the circle in which an object is moving. This type of acceleration is necessary for an object to maintain its circular path and not move off in a straight line, as would be suggested by Newton's first law of inertia.

To put it simply, centripetal acceleration keeps the electron tethered to its circular orbit around the nucleus. This inward acceleration is continuous as the electron travels along its circular path and remains at a constant magnitude since the electron's speed is constant in its orbit.

• Mathematical Expression: The centripetal acceleration a is calculated by the equation \( a = \frac{v^2}{r} \), where v is the electron's orbital speed, and r is the radius of the orbit.
• To get the sense of this acceleration's magnitude, one can use the given values of the electron's speed and orbit radius from the Bohr model to evaluate it numerically.

Centripetal Force

Building on the concept of centripetal acceleration, there is a corresponding force that results in this acceleration, known as the 'centripetal force'. Centripetal force is not a new kind of force but rather the net force acting towards the center of the circle, caused by other forces. In the case of an electron orbiting the nucleus, this force is the electrostatic attraction between the positively charged nucleus and the negatively charged electron.

This force ensures that the electron stays in its circular orbit and does not fly away due to inertia. It is characterized by the following features:

• Direction: Centripetal force always points towards the center of the circular path.
• Calculation: It is given by the equation \( F = m \cdot a \), where m is the electron's mass and a is the centripetal acceleration.
• In the Bohr model, using the formula with the known mass of the electron and computed acceleration, the exact force acting on the electron in its orbit can be obtained. This force magnitude allows scientists to understand the strength of the electron's bond to the nucleus.