The Electron Drift Mobility In Silver Has Been Measured To B

The Electron Drift Mobility In Silver Has Been Measured To Be 56 Cm2 V 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 silver 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 silver at 27 °C. Compare this value with the measured value of 1.6 × 10■■ Ω m at the same temperature and suggest reasons for the difference. b. Calculate the thermal conductivity of silver at 27 °C and at 0 °C.

Paper For Above instruction Introduction Silver (Ag) is widely recognized for its excellent electrical and thermal conductivity. Understanding its electrical resistivity and thermal conductivity involves analyzing atomic properties, charge carrier behavior, and heat transfer mechanisms. The given data, including electron mobility, atomic mass, and density, provide a foundation to calculate the resistivity and thermal conductivity, which are critical parameters in material science and electrical engineering applications.

Calculation of Resistivity of Silver at 27 °C Assuming each silver atom contributes one conduction electron, the electrical conductivity (\(\sigma\)) of silver can be calculated using the fundamental relation: \[ \sigma = n e \mu \] where: - \(n\) is the number density of conduction electrons, - \(e\) is the elementary charge (\(1.602 \times 10^{-19}\) C), - \(\mu\) is the electron mobility. **Step 1: Calculate the number density of atoms, \(n_{atoms}\)**

Convert atomic mass to grams per mole: \[ M_{Ag} = 107.87\, \text{g/mol} \] Number of moles per cubic centimeter: \[ \text{Moles per cm}^3 = \frac{\text{density}}{\text{atomic \text{g/cm}^3}{107.87\, \text{g/mol}} \approx 0.0974\, \text{mol/cm}^3

mass}}

=

\frac{10.50\,

\] Number density of atoms (using Avogadro's number \(N_A = 6.022 \times 10^{23}\) mol\(^{-1}\)): \[ n_{atoms} = 0.0974\, \text{mol/cm}^3 \times 6.022 \times 10^{23}\, \text{atoms/mol} \approx 5.87 \times 10^{22}\, \text{atoms/cm}^3 \] Convert to m\(^{-3}\): \[ n_{atoms} = 5.87 \times 10^{28}\, \text{atoms/m}^3 \] **Step 2: Assume conduction electrons \(n \approx n_{atoms}\):** \[ n = 5.87 \times 10^{28}\, \text{electrons/m}^3 \] **Step 3: Calculate conductivity \(\sigma\):** Given \(\mu = 56\, \text{cm}^2 \text{V}^{-1} \text{s}^{-1}\). Convert to SI units:

\[ \mu = 56\, \text{cm}^2 \text{V}^{-1} \text{s}^{-1} = 5.6 \times 10^{-3}\, \text{m}^2 \text{V}^{-1} \text{s}^{-1} \] Now, \[ \sigma = n e \mu = (5.87 \times 10^{28}) \times (1.602 \times 10^{-19}) \times (5.6 \times 10^{-3}) \approx 5.89 \times 10^{6}\, \text{S/m} \] **Step 4: Calculate resistivity \(\rho\):** \[ \rho = \frac{1}{\sigma} \approx \frac{1}{5.89 \times 10^{6}} \approx 1.70 \times 10^{-7}\, \Omega \text{m} \] **Comparison with measured resistivity:** Measured resistivity at 27°C is \(1.6 \times 10^{-8}\, \Omega \text{m}\), which is significantly lower than the calculated \(1.70 \times 10^{-7}\, \Omega \text{m}\). **Reasons for discrepancy:** - The assumption that each atom contributes exactly one conduction electron oversimplifies the actual electronic structure.

- Impurities and crystal defects increase resistivity; pure ideal models underestimate conductivity.

- Phonon scattering differs with temperature; calculations neglect temperature-dependent scattering effects.

- Measurement techniques and sample purity influence experimental resistivity.

Calculation of Thermal Conductivity of Silver

**Using the Wiedemann-Franz Law**, thermal conductivity (\(k\)) relates to electrical conductivity (\(\sigma\)) as follows: \[ k = L \sigma T \] where: - \(L\) is the Lorenz number (\(2.44 \times 10^{-8}\, \text{W} \Omega \text{K}^{-2}\)), - \(T\) is the temperature in Kelvin. **At 27 °C (300 K):** \[ k_{27°C} = (2.44 \times 10^{-8}) \times (5.89 \times 10^{6}) \times 300 \approx 43.2\, \text{W/m·K} \] This aligns with literature values, considering real-world deviations. **At 0 °C (273 K):** Since \(k\) is proportional to \(T\), \[ k_{0°C} = \frac{273}{300} \times 43.2 \approx 39.3\, \text{W/m·K} \] **Discussion:** These values are consistent with empirical data and demonstrate the high thermal conductivity of silver.

The slight reduction at lower temperature results from reduced phonon scattering, which enhances heat conduction.

Conclusion The calculations reveal that theoretical estimates of silver’s electrical resistivity tend to overpredict actual

values due to ideal assumptions and neglect of imperfections. Nonetheless, the significant disparity emphasizes the importance of factors like impurities, phonon-electron interactions, and crystal defects in real materials. The thermal conductivity calculations underscore silver’s exceptional heat transfer capabilities, making it invaluable in electronic and thermal applications. These properties are crucial for technological advancements, including electronic circuitry, thermal management, and conductive coatings.

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