Ch 42: Molecules and Condensed Matter

Young & Freedman Calc - University Physics 15th Edition

Young & Freedman Calc15th EditionUniversity PhysicsISBN: 9780135159552Not the one you use?Change textbook

Young & Freedman Calc 15th Edition

Ch 42: Molecules and Condensed Matter

Problem 21

Chapter 41, Problem 21

Silver has a Fermi energy of $5.48$ eV. Calculate the electron contribution to the molar heat capacity at constant volume of silver, $C_{V}$ , at $300$ K. Express your result as a multiple of $R$ .

Verified step by step guidance

Step 1: Understand the problem. The molar heat capacity at constant volume, CV, for electrons in a metal can be calculated using the Fermi energy and temperature. The formula for the electron contribution to CV is derived from the Sommerfeld expansion and is given by: $ C_V = \frac{\pi^2}{2} \cdot \frac{k_B^2 T}{E_F} \cdot R $, where $ k_B $ is the Boltzmann constant, $ T $ is the temperature, $ E_F $ is the Fermi energy, and $ R $ is the gas constant.

Step 2: Convert the Fermi energy $ E_F $ from electron volts (eV) to joules (J). Use the conversion factor $ 1 \text{ eV} = 1.602 \times 10^{-19} \text{ J} $. Multiply $ E_F = 5.48 \text{ eV} $ by this factor to express it in joules.

Step 3: Substitute the values for $ k_B $, $ T $, and $ E_F $ into the formula. The Boltzmann constant $ k_B $ is $ 1.38 \times 10^{-23} \text{ J/K} $, the temperature $ T $ is $ 300 \text{ K} $, and $ E_F $ is the converted value from Step 2.

Step 4: Simplify the expression to calculate $ C_V $ as a multiple of $ R $. The gas constant $ R $ is $ 8.314 \text{ J/(mol·K)} $. Divide the calculated value of $ C_V $ by $ R $ to express the result as a multiple of $ R $.

Step 5: Verify the units and ensure the result is dimensionless when expressed as a multiple of $ R $. The final expression should be $ C_V = \text{(some value)} \cdot R $.

Verified video answer for a similar problem:

This video solution was recommended by our tutors as helpful for the problem above.

Video duration:

Was this helpful?

Key Concepts

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

Fermi Energy

Fermi energy is the highest energy level occupied by electrons in a solid at absolute zero temperature. It is a crucial concept in solid-state physics, as it helps determine the distribution of electrons in metals and semiconductors. The Fermi energy influences various properties, including electrical conductivity and heat capacity, as it defines the energy states available for electrons at higher temperatures.

Molar Heat Capacity at Constant Volume (CV)

The molar heat capacity at constant volume (CV) is the amount of heat required to raise the temperature of one mole of a substance by one degree Celsius while keeping the volume constant. For metals, CV can be influenced by the contributions from both lattice vibrations and free electrons. In the case of metals like silver, the electron contribution is significant due to the presence of conduction electrons that can absorb energy.

Debye Model

The Debye model is a theoretical approach used to describe the heat capacity of solids at low temperatures. It accounts for the quantized vibrational modes of the lattice and provides a framework for understanding how heat capacity varies with temperature. In metals, the electron contribution to heat capacity can be analyzed using this model, particularly at temperatures where the electron gas behaves classically, allowing for calculations of CV in relation to the Fermi energy.

Recommended video:

Guided course

07:21

Ideal Gases and the Ideal Gas Law

Related Practice

Textbook Question

Suppose a piece of very pure germanium is to be used as a light detector by observing, through the absorption of photons, the increase in conductivity resulting from generation of electron–hole pairs. If each pair requires $0.67$ eV of energy, what is the maximum wavelength that can be detected? In what portion of the spectrum does it lie?

views

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.

views

Textbook Question

At the Fermi temperature $T_{F}$ , $E_{F} = k T_{F}$ (see Exercise $42.22$ ). When $T = T_{F}$ , what is the probability that a state with energy $E equals 2 E sub F$ is occupied?

views

Textbook Question

Pure germanium has a band gap of $0.67$ eV. The Fermi energy is in the middle of the gap. For temperatures of $250$ K, $300$ K, and $350$ K, calculate the probability $f (E)$ that a state at the bottom of the conduction band is occupied.

views

Textbook Question

The maximum wavelength of light that a certain silicon photocell can detect is $1.11$ mm. What is the energy gap (in electron volts) between the valence and conduction bands for this photocell?

views

Textbook Question

Potassium bromide (KBr) has a density of $2.75 times 10 cubed$ kg/m³ and the same crystal structure as NaCl. The mass of a potassium atom is $6.49 \times 10^{- 26}$ kg, and the mass of a bromine atom is $1.33 \times 10^{- 25}$ kg. Calculate the average spacing between adjacent atoms in a KBr crystal.

views

Silver has a Fermi energy of 5.485.48 eV. Calculate the electron contribution to the molar heat capacity at constant volume of silver, CVC_V, at 300300 K. Express your result as a multiple of RR.

Verified video answer for a similar problem:

Key Concepts

Fermi Energy

Molar Heat Capacity at Constant Volume (CV)

Debye Model

Silver has a Fermi energy of $5.48$ eV. Calculate the electron contribution to the molar heat capacity at constant volume of silver, $C_{V}$ , at $300$ K. Express your result as a multiple of $R$ .