Electrical And Thermal Conductivity Of Silver Electron Drift

Electrical And Thermal Conductivity Of Agthe Electron Drift Mobility I

Electrical and thermal conductivity of Ag The electron drift mobility in silver has been measured to be 56 cm2 V-1 s-1 at 27 °C. The atomic mass and density of Ag are given as 107.87 amu or g mol–1 and 10.50 g cm-3, respectively. a. Assuming that each Ag atom contributes one conduction electron, calculate the resistivity of Ag at 27 °C. Compare this value with the measured value of 1.6 × 10^-8 Ω·m at the same temperature and suggest reasons for the difference.

Paper For Above instruction

Introduction

The electrical conductivity of metals is fundamentally tied to the movement of free electrons within the metal's lattice structure. Silver (Ag), renowned for its exceptional electrical conductivity, offers a compelling subject for analyzing the relationship between microscopic electron mobility and macroscopic resistivity. The goal of this paper is to utilize the given electron drift mobility and metallurgical parameters of silver to calculate its resistivity at room temperature and compare this computed value with the accepted measured resistivity. The analysis will include assumptions, calculations, and discussions on potential discrepancies.

Understanding Electron Mobility and Resistivity

Electron drift mobility (μ_e) reflects how quickly an electron can move through a metal when subjected to an electric field. It directly influences electrical conductivity (σ), which is inversely proportional to resistivity (ρ). The relationship between electrical conductivity and electron mobility is expressed as:

σ = n e μ_e

where n is the number density of conduction electrons, e is the elementary charge, and μ_e is the electron mobility.

Calculating the Electron Density in Silver

Assuming each silver atom contributes one conduction electron, the electron density (n) can be calculated using the atomic mass (A), density (ρ_d), and Avogadro's number (N_A).

Given data:

• Atomic mass, A = 107.87 g/mol
• Density, ρ_d = 10.50 g/cm³
• Avogadro's number, N_A = 6.022 × 10²³ mol^-1

First, convert the volume density of atoms to number density of electrons:

n = (ρ_d / A) × N_A

Calculations:

n = \frac{10.50 \text{ g/cm}^3}{107.87 \text{ g/mol}} \times 6.022 \times 10^{23} \text{ atoms/mol}

n \approx 0.0974 \text{ mol/cm}^3 \times 6.022 \times 10^{23} \text{ atoms/mol}

n \approx 5.87 \times 10^{22} \text{ electrons/cm}^3

Converting to SI units for consistency:

1 \text{ cm}^3 = 1 \times 10^{-6} \text{ m}^3

n \approx 5.87 \times 10^{28} \text{ electrons/m}^3

Calculating Electrical Conductivity and Resistivity

Using the relation:

σ = n e μ_e

where e = 1.602 × 10^-19 C, and μ_e = 56 cm² V^-1 s^-1.

Converting μ_e to SI units:

μ_e = 56 cm² V^-1 s^-1 = 5.6 × 10^-3 m² V^-1 s^-1

Calculating σ:

σ = (5.87 × 10²⁸ m^-3) × (1.602 × 10^-19 C) × (5.6 × 10^-3 m² V^-1 s^-1)

σ ≈ 5.87 × 10²⁸ × 1.602 × 10^-19 × 5.6 × 10^-3

σ ≈ (5.87 × 1.602 × 5.6) × 10^{28 - 19 - 3}

σ ≈ (5.87 × 1.602 × 5.6) × 10⁶

σ ≈ (5.87 × 8.971) × 10⁶

σ ≈ 52.69 × 10⁶ S/m

σ ≈ 5.27 × 10⁷ S/m

Finally, resistivity (ρ) is the reciprocal:

ρ = 1 / σ ≈ 1 / 5.27 × 10⁷ ≈ 1.90 × 10^-8 Ω·m

Comparison with Measured Resistivity and Analysis

The calculated resistivity of approximately 1.90 × 10^-8 Ω·m exceeds the experimentally measured value of 1.6 × 10^-8 Ω·m. The close proximity of these figures validates the assumptions but also highlights the possible sources of discrepancy.

Factors influencing the difference include:

• Impurities and defects in real silver specimens increase electron scattering, leading to higher resistivity (Ashcroft & Mermin, 1976).
• Variations in electron mobility due to phonon interactions at different temperatures (Kittel, 2005).
• Simplifications in assuming a free electron model, ignoring band structure complexities (Hansen, 2012).
• Measurement uncertainties and sample purity affecting the actual resistivity (Weast & Astle, 1980).

These factors suggest that fundamental calculations based purely on idealized models tend to underestimate the resistivity observed in real-world samples. Additionally, lattice vibrations (phonons) significantly affect electron mobility at room temperature, decreasing it from the idealized value.

Thermal Conductivity Considerations

Silver's high electrical conductivity correlates with its thermal conductivity through the Wiedemann-Franz law, which states:

\(\frac{\kappa}{\sigma} = L T\)

where κ is thermal conductivity, L is the Lorenz number (~2.44 × 10^-8 WΩK^-2), and T is the temperature in Kelvin. This relation underpins the intrinsic link between a metal's electrical and thermal transport properties, emphasizing how electron mobility affects both, reaffirming the importance of conduction electrons in mediating thermal transfer as well as electrical conductivity.

Conclusion

The calculated resistivity, based on the provided electron mobility and metallurgical data, aligns closely with experimentally measured values, validating the free electron model assumptions. Minor differences are attributable to real-world imperfections, phonon interactions, and inherent material complexities. Understanding these foundational relationships aids in designing materials with tailored electrical and thermal properties for applications in electronics and thermal management. Future studies could incorporate more detailed band structure analyses and temperature-dependent scattering mechanisms to refine these estimates further.

References

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