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Chapter 39: Problem 19

A hydrogen atom is in a state with energy \(-1.51 \mathrm{eV}\). In the Bohr model, what is the angular momentum of the electron in the atom, with respect to an axis at the nucleus?

Short Answer

Expert verified
The angular momentum of the electron in the hydrogen atom is geometrically \(L = n\hbar\). After calculating the value of \(n\), we will multiply it by Planck's constant divided by \(2\pi\) to obtain the sought after angular momentum.

Step by step solution

01

Conversion of energy from eV to Joule

The energy of the orbit given in the question is in electron-volt (eV) while the energy formula for the Bohr model requires energy in joules. So we convert \(-1.51\) eV to Joules. We know that \[1 \text{ eV} = 1.6 \times 10^{-19} \text{ Joule}\]. So, \[E = -1.51 \times 1.6 \times 10^{-19}\] Joules.
02

Calculation of Principal Quantum Number (n)

The energy for any level in a hydrogen atom (Bohr Model) can be given by \[E = -13.6/n^2 \times 1.6 \times 10^{-19}\] Joule where \(n\) is the principal quantum number. Aligning this equation with the calculated energy from step 1, we can solve for \(n\).
03

Calculate the Angular Momentum (L)

According to Bohr's model, the angular momentum is quantized and it's given by \[L = n\hbar\] where \(\hbar\) is the reduced Planck's constant equal to \(1.054 \times 10^{-34}\) Js. Use the calculated \(n\) to find the angular momentum \(L\).

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Key Concepts

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

Hydrogen Atom Energy Levels
The hydrogen atom showcases a fascinating aspect of quantum mechanics through its energy levels. Unlike the continuous range of energies that classical physics would suggest, the energy of an electron in a hydrogen atom is quantized. This means that the electron can only occupy certain levels of energy, with no in-betweens. These levels are negative, indicating that energy must be supplied to free the electron from the atom鈥攈ence, the negative value points to a bound state.

Now let's get into why we care about these energy levels. In the Bohr model of the hydrogen atom, these energy levels are represented by specific orbits around the nucleus, and each orbit corresponds to a particular energy level. When an electron jumps from one energy level to a lower one, it emits a photon, and the energy of this photon corresponds to the difference in energy between the two levels. This is the basis for the concept of emission spectra. Conversely, when absorbing energy, the electron can jump to a higher energy level. Understanding this model is crucial for everything from chemistry to quantum physics as it is the simplest atom to study.
Principal Quantum Number
The principal quantum number, denoted as n, is an integer that plays a pivotal role in the Bohr model. It determines the size and energy of the orbit in which the electron resides. Essentially, n can be any positive integer (1, 2, 3, ...), where n=1 corresponds to the ground state, or the lowest possible energy level of the atom. As n increases, so does the radius of the electron's orbit and its energy.

Why Principal Quantum Number Matters

Understanding the principal quantum number is vital because it helps us predict the behavior and properties of atoms. For instance, larger quantum numbers mean electrons are further from the nucleus, which can lead to less stability and greater reactivity in chemical reactions. Additionally, the principal quantum number allows us to calculate an electron's probable distance from the nucleus, which is essential in understanding atomic structure and the chemical bonding process.
Conversion of Energy Units
Working with energy units can sometimes be tricky, especially when moving between the different scales and constants used in physics and chemistry. For the Bohr model, energy is most often calculated in electron volts (eV), which is a unit of energy commonly used in atomic and particle physics, due to its convenience when dealing with subatomic particles.

One electron-volt is defined as the amount of energy gained (or lost) by the charge of a single electron moved across an electric potential difference of one volt. In more universal terms, its value can be converted to joules鈥攖he SI unit of energy鈥攗sing the conversion factor where 1 eV = 1.6 脳 10^-19 Joules. This conversion is essential when performing calculations related to the energy of electrons in atoms since it allows us to align our results with the standard scientific units of measure.

Importance of the Conversion

It's worth noting that this conversion isn't just academic; it allows for a common language across different areas of physics and engineering. Assimilating this process helps students avoid mistakes when calculating and comparing energies, and ensures that the significance of energy changes within atoms is comprehended at a universal scale.

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Most popular questions from this chapter

A beam of electrons is accelerated from rest through a potential difference of \(0.100 \mathrm{kV}\) and then passes through a thin slit. When viewed far from the slit, the diffracted beam shows its first diffraction minima at \(\pm 14.6^{\circ}\) from the original direction of the beam. (a) Do we need to use relativity formulas? How do you know? (b) How wide is the slit?

Through what potential difference must electrons be accelerated if they are to have (a) the same wavelength as an x ray of waveIength \(0.220 \mathrm{nm}\) and (b) the same energy as the x ray in part (a)?

Sirius \(\mathrm{B}\). The brightest star in the sky is Sirius, the Dog Star. It is actually a binary system of two stars, the smaller one (Sirius B) being a white dwarf. Spectral analysis of Sirius B indicates that its surface temperature is \(24,000 \mathrm{~K}\) and that it radiates energy at a total rate of \(1.0 \times 10^{25} \mathrm{~W}\). Assume that it behaves like an ideal blackbody. (a) What is the total radiated intensity of Sirius \(\mathrm{B}\) ? (b) What is the peak-intensity wavelength? Is this wavelength visible to humans? (c) What is the radius of Sirius B? Express your answer in kilometers and as a fraction of our sun's radius. (d) Which star radiates more total energy per second, the hot Sirius \(\mathrm{B}\) or the (relatively) cool sun with a surface temperature of \(5800 \mathrm{~K}\) ? To find out, calculate the ratio of the total power radiated by our sun to the power radiated by Sirius B.

CP CALC You have entered a contest in which the contestants drop a marble with mass \(20.0 \mathrm{~g}\) from the roof of a building onto a small target \(25.0 \mathrm{~m}\) below. From uncertainty considerations, what is the typical distance by which you'll miss the target, given that you aim with the highest possible precision? (Hint: The uncertainty \(\Delta x_{f}\) in the \(x\) -coordinate of the marble when it reaches the ground comes in part from the uncertainty \(\Delta x_{i}\) in the \(x\) -coordinate initially and in part from the initial uncertainty in \(v_{x}\). The latter gives rise to an uncertainty \(\Delta v_{x}\) in the horizontal motion of the marble as it falls. The values of \(\Delta x_{i}\) and \(\Delta v_{x}\) are related by the uncertainty principle. A small \(\Delta x_{i}\) gives rise to a large \(\Delta v_{x}\), and vice versa. Find the value of \(\Delta x_{i}\) that gives the smallest total uncertainty in \(x\) at the ground. Ignore any effects of air resistance.

\(\mathrm{CP}\) (a) A particle with mass \(m\) has kinetic energy equal to three times its rest energy.What is the de Broglie wavelength of this particle? (Hint: You must use the relativistic expressions for momentum and kinetic energy: \(E^{2}=(p c)^{2}+\left(m c^{2}\right)^{2}\) and \(\left.K=E-m c^{2} .\right)\) (b) Determine the numerical value of the kinetic energy (in MeV) and the wavelength (in meters) if the particle in part (a) is (i) an electron and (ii) a proton.
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