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Chapter 11: Problem 42

In the Bohr model of the hydrogen atom, the electron moves in a circular orbit of radius \(0.529 \times 10^{-10} \mathrm{m}\) around the proton. Assuming the orbital angular momentum of the electron is equal to \(h / 2 \pi,\) calculate (a) the orbital speed of the electron, (b) the kinetic energy of the electron, and (c) the angular frequency of the electron's motion.

Short Answer

Expert verified
The orbital speed of the electron is \(v = \frac {h} {2\pi mr}\), the kinetic energy of the electron is \(K.E. = \frac {1} {2} m (\frac {h} {2\pi mr})^2\), and the angular frequency of the electron's motion is \(蠅 = \frac {h} {2\pi mr^2}\). These results are obtained by applying the concepts of the Bohr model of the hydrogen atom along with the principles of angular momentum, kinetic energy, and angular frequency.

Step by step solution

01

Calculating the orbital speed

To find the orbital speed (v), we can use the relation for the angular momentum \(L = mvr = \frac {h} {2\pi}\) where m is the electron mass, r is the orbit radius and v is the speed. Solve for v: \(v = \frac {h} {2\pi mr}\)
02

Calculating the kinetic energy

The kinetic energy (K.E.) of the electron can be found using the formula \(K.E. = \frac {1} {2} mv^2\). Substitute the value of v from the previous step into this formula to get the kinetic energy of the electron: \(K.E. = \frac {1} {2} m (\frac {h} {2\pi mr})^2\)
03

Calculating the angular frequency

The angular frequency (蠅) can be calculated using the relationship \(v = 蠅r\). Substitute the value of v from the first step into this relationship to find the angular frequency of the electron: \(蠅 = \frac {v} {r} = \frac {h} {2\pi mr^2}\)

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Orbital Angular Momentum
In the Bohr model of the hydrogen atom, the concept of orbital angular momentum is central to understanding electron orbits. Angular momentum is basically a measure of the quantity of rotation of a body, taking into account its mass and shape. For an electron orbiting a proton, in classical terms, the angular momentum can be expressed as the product of the electron's mass (\(m\)), its velocity (\(v\)), and the radius of its orbit (\(r\)).

The Bohr model simplifies this by stating that the orbital angular momentum of an electron is quantized, or comes in discrete values, specifically in multiples of \(\frac{h}{2\pi}\), where \(h\) is Planck鈥檚 constant. For the hydrogen atom, this simplification helps in determining key properties like the speed and energy level of the electron. By setting the angular momentum \(L = \frac{h}{2\pi}\), it becomes possible to derive the formula for the speed of the electron by rearranging the equation to solve for \(v\).
Kinetic Energy
Kinetic energy is the energy that an object possesses due to its motion. When considering an electron in motion around a nucleus, this kinetic energy can be calculated using the classic formula:
  • K.E. = \(\frac{1}{2}mv^2\)
In the Bohr model, after determining the speed of the electron from the angular momentum principle, this velocity can be inserted into the kinetic energy formula. The key point is that the velocity, which comes from setting the orbital angular momentum to \(\frac{h}{2\pi}\), directly influences the kinetic energy. So, by substitution, the expression for kinetic energy in terms of known constants and variables becomes:\[ K.E. = \frac{1}{2}m\left(\frac{h}{2\pi m r}\right)^2 \]This equation reflects how tightly bound the electron is to the nucleus, influenced by the electron's speed and mass.
Angular Frequency
Angular frequency, in the context of an electron orbiting a nucleus as described in the Bohr model, refers to how frequently the electron completes an orbit in radians per second. It provides a measure of the rotation rate and can be related to the orbital speed and radius. Mathematically, angular frequency, \(\omega\), is expressed as the ratio of the electron's orbital speed to its orbit radius:\(\omega = \frac{v}{r}\)By substituting the calculated velocity from the Bohr model's angular momentum equation, we get:\[\omega = \frac{h}{2\pi mr^2}\]This expression shows that the angular frequency depends on Planck鈥檚 constant \(h\), the electron mass \(m\), and the radius of orbit \(r\). Understanding angular frequency helps illustrate how often an electron in a specific orbit completes its circular path around the nucleus.

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Most popular questions from this chapter

A space station is constructed in the shape of a hollow ring of mass \(5.00 \times 10^{4} \mathrm{kg} .\) Members of the crew walk on a deck formed by the inner surface of the outer cylindrical wall of the ring, with radius 100 m. At rest when constructed, the ring is set rotating about its axis so that the people inside experience an effective free-fall acceleration equal to \(g .\) (Figure \(\mathrm{P} 11.27\) shows the ring together with some other parts that make a negligible contribution to the total moment of inertia.) The rotation is achieved by firing two small rockets attached tangentially to opposite points on the outside of the ring. (a) What angular momentum does the space station acquire? (b) How long must the rockets be fired if each exerts a thrust of \(125 \mathrm{N} ?\) (c) Prove that the total torque on the ring, multiplied by the time interval found in part (b), is equal to the change in angular momentum, found in part (a). This equality represents the angular impulse-angular momentum theorem.

A 100 -kg uniform horizontal disk of radius \(5.50 \mathrm{m}\) turns without friction at 2.50 rev/s on a vertical axis through its center, as in Figure \(\mathrm{P} 11.46 .\) A feedback mechanism senses the angular speed of the disk, and a drive motor at \(A\) maintains the angular speed constant while a 1.20 kg block on top of the disk slides outward in a radial slot. The \(1.20-\mathrm{kg}\) block starts at the center of the disk at time \(t=0\) and moves outward with constant speed \(1.25 \mathrm{cm} / \mathrm{s}\) relative to the disk until it reaches the edge at \(t=440\) s. The sliding block feels no friction. Its motion is constrained to have constant radial speed by a brake at \(B\), producing tension in a light string tied to the block. (a) Find the torque that the drive motor must provide as a function of time, while the block is sliding. (b) Find the value of this torque at \(t=440 \mathrm{s}\), just before the sliding block finishes its motion. (c) Find the power that the drive motor must deliver as a function of time. (d) Find the value of the power when the sliding block is just reaching the end of the slot. (e) Find the string tension as a function of time. (f) Find the work done by the drive motor during the 440 -s motion. (g) Find the work done by the string brake on the sliding block. (h) Find the total work on the system consisting of the disk and the sliding block.

A puck of mass \(m\) is attached to a cord passing through a small hole in a frictionless, horizontal surface (Fig. P11.49). The puck is initially orbiting with speed \(v_{i}\) in a circle of \(\mathrm{ra}-\) dius \(r_{i}\). The cord is then slowly pulled from below, decreasing the radius of the circle to \(r\). (a) What is the speed of the puck when the radius is \(r ?\) (b) Find the tension in the cord as a function of \(r .\) (c) How much work \(W\) is done in moving \(m\) from \(r_{i}\) to \(r ?\) (Note: The tension depends on \(r\) ) (d) Obtain numerical values for \(v, T,\) and \(W\) when \(r=0.100 \mathrm{m}\) \(m=50.0 \mathrm{g}, r_{i}=0.300 \mathrm{m},\) and \(v_{i}=1.50 \mathrm{m} / \mathrm{s}.\)

A particle is located at the vector position \(\mathbf{r}=(\hat{\mathbf{i}}+3 \hat{\mathbf{j}}) \mathbf{m}\) and the force acting on it is \(\mathbf{F}=(3 \hat{\mathbf{i}}+2 \hat{\mathbf{j}})\) N. What is the torque about (a) the origin and (b) the point having coordinates (0,6) m?

A wad of sticky clay with mass \(m\) and velocity \(\mathbf{v}_{i}\) is fired at a solid cylinder of mass \(M\) and radius \(R\) (Figure \(\mathrm{P} 11.37\) ). The cylinder is initially at rest and is mounted on a fixed horizontal axle that runs through its center of mass. The line of motion of the projectile is perpendicular to the axle and at a distance \(d
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