Calculate the density of states g ( E ) g(E) for the free-electron model of a metal if E = 7.0 E = 7.0 eV and V = 1.0 V = 1.0 cm 3 . Express your answer in units of states per electron volt.

Hi, everyone. In this practice problem, we are being asked to consider electrons as completely free particles within the conductor and determine the number of states per unit energy range or G E where the energy E of states at the surface is 6.5 E V and volume V is 1.5 centimeter cube. And we are being asked to state G E in terms of states per electron fault or E V. And the options given are a 101.27 times 10 to the power of 40 states per E V B, one point oh one times 10 to the power of 24 states per E V C six th 632.95 times 10 to the power of states per E V. And lastly, D 632.95 times 10 to the power of 22 states per E V. So the number of states per unit energy range D N divided by D E is called the energy of states or G E. So G E, if you want to recall can be calculated by multiplying in parenthesis to M to the power of three half multiply that by V multiply that by E to the power of one half and dividing all of this we to pi squared multiplied by H bar cube. In this uh case M is going to be the mass of electrons which will come out to be 9. times 10 to the power of negative 31 kg, which is just a constant Essentially V is going to be the volume, which is given in the problem statement to be 1.5 centimeter cube. We want to definitely convert this into meter cubes. So we want to multiply this with one m cube divided by 10 to the power of six centimeter cube. And that will give us the volume to then be equal to 1.5 times 10 to the power of negative six m cube. And then E is going to be the energy of states. And um each bar is going to be equals to H divided by two pi or the plans constant divided by two pi which will come up to a value of 1.54 times 10 to the power of negative 34 seconds. So now we can actually um substitute all of our values into our G E equation. So let's do that G E will then be equal to in parentheses, two multiplied by mass of electrons which is 9.11 times 10 to the power of negative 31 kg to the power of three healths multiplied that by the volume which is going to be 1.5 times to the power of negative six m cube multiply that again with the E or the energy of states which is given in the problem statement to be 6.5 E V. So we wanna input 6.5 E V multiply that and convert that into Joles, which is 1.6 oh two times 10 to the power of negative Joles divide that with one E V for every 1.6 oh two times 10 to the power of negative 19 Joles and put that to the power of health. And then from there, we want to divide everything, want to divide everything with two pi squared and then multiply that with H bar cube which will be one point oh 54 times 10 to the power of negative seconds. So these are all of the variables that we need in order to calculate this G E. So calculating all of this, we will then be able to get the G E value to then be equal to 632.95 times 10 to the power of 40 states per Joel. The the answer or the options given are in states per E V. So we wanna uh convert this into E V by multiplying where 1.6 oh two times 10 to the power of negative 19, just divide that by one E V. And calculating that or converting, we will then get our G E values to then be equal to one point oh one times 10 to the power of 24 states B E V. So that will be the final answer to this problem statement with the G E value of one point oh one times 10 to the power of 24 states per E V which will correspond to option B in our answer choices. So option B will be the answer to the number of states per you need energy range or G E and that will be it for this video. If you guys still have any sort of confusion, please feel free to check out our adolescent videos on similar topics available on our website. But other than that, that will be it for this one. Thank you.

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Problem 21

Textbook Question

Calculate the density of states $g (E)$ for the free-electron model of a metal if $E equals 7.0$ eV and $V equals 1.0$ cm³. Express your answer in units of states per electron volt.

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Step 1: Recall the formula for the density of states in the free-electron model: $ g(E) = \frac{V}{2\pi^2} \left( \frac{2m}{\hbar^2} \right)^{3/2} \sqrt{E} $, where $ V $ is the volume, $ m $ is the electron mass, $ \hbar $ is the reduced Planck's constant, and $ E $ is the energy.

Step 2: Convert the given volume $ V = 1.0 \, \text{cm}^3 $ into cubic meters (SI units). Use the conversion factor $ 1 \, \text{cm}^3 = 10^{-6} \, \text{m}^3 $.

Step 3: Substitute the known constants into the formula: $ m = 9.11 \times 10^{-31} \, \text{kg} $, $ \hbar = 1.055 \times 10^{-34} \, \text{J·s} $, and $ E = 7.0 \, \text{eV} $. Convert $ E $ from electron volts to joules using $ 1 \, \text{eV} = 1.602 \times 10^{-19} \, \text{J} $.

Step 4: Calculate the term $ \left( \frac{2m}{\hbar^2} \right)^{3/2} \sqrt{E} $. This involves squaring $ \hbar $, dividing $ 2m $ by $ \hbar^2 $, taking the cube root, and multiplying by $ \sqrt{E} $.

Step 5: Multiply the result from Step 4 by $ \frac{V}{2\pi^2} $ to find $ g(E) $. Ensure the final units are in states per electron volt by verifying the dimensional consistency of the formula.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Density of States

The density of states (g(E)) is a crucial concept in solid-state physics that quantifies the number of electronic states available at a given energy level per unit volume. It helps in understanding how many electrons can occupy specific energy levels in a material, which is essential for calculating various properties of metals and semiconductors.

Free-Electron Model

The free-electron model is a simplified representation of electrons in a metal, treating them as non-interacting particles in a uniform potential. This model assumes that electrons can move freely throughout the metal, allowing for the derivation of important properties such as conductivity and the density of states, which are foundational for understanding metallic behavior.

Energy and Volume Relationship

In the context of the density of states, the relationship between energy (E) and volume (V) is significant. The density of states is often expressed in terms of energy levels per unit volume, and knowing the volume of the system allows for the calculation of how many states are available at a specific energy, which is critical for determining electronic properties in materials.

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Calculate the density of states g(E)g(E) for the free-electron model of a metal if E=7.0E = 7.0 eV and V=1.0V = 1.0 cm3. Express your answer in units of states per electron volt.

Key Concepts

Density of States

Free-Electron Model

Energy and Volume Relationship

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Calculate the density of states $g (E)$ for the free-electron model of a metal if $E equals 7.0$ eV and $V equals 1.0$ cm³. Express your answer in units of states per electron volt.