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Ch 42: Molecules and Condensed Matter
Young & Freedman Calc - University Physics 15th Edition
Young & Freedman Calc15th EditionUniversity PhysicsISBN: 9780135159552Non è quello che usi tu?Cambia libro di testo
Tutti i libri di testoYoung & Freedman Calc 15th EditionCh 42: Molecules and Condensed MatterProblema 21
Capitolo 41, Problema 21

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.

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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 \( C_V = \text{(some value)} \cdot R \).

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Concetti chiave

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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.
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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.
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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.
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