Let's solve the problem step by step: --- ### **Step 1: Understanding the Physics** **Conductivity (σ) and Resistivity (ρ):** - Conductivity (σ) and resistivity (ρ) are related by: \(\sigma = \frac{1}{\rho}\). - For metals like gold (Au), **as temperature (T) increases, resistivity (ρ) increases** due to increased lattice vibrations (phonons), which scatter electrons more. - Thus, **conductivity (σ) decreases with increasing temperature**. --- ### **Step 2: Plot Curve 1 (Conductivity σ vs. Temperature T)** - At low temperature (100K), σ is **high**. - As T increases to 600K, σ **decreases**. - The curve is **downward sloping**. **Sketch:** ``` σ |\ | \ | \ | \ | \ |_____\ 100 600 T(K) ``` (Curve 1: Downward, from high at 100K to low at 600K) --- ### **Step 3: Plot Curve 2 (Second Measurement)** - The question says to plot the result “a few days later,” but unless there's contamination or change, the result for gold should be **the same**. - If no physical change occurred to the sample, **Curve 2 will overlap Curve 1** (same downward slope). --- ### **Step 4: Resistivity ρ vs. Temperature T** - Since \(\rho = 1/\sigma\), as σ decreases, ρ increases. - Start with **low ρ at 100K**, and **high ρ at 600K**. - Curve is **upward sloping**. **Sketch (on same axes as above):** ``` σ |\ | \ | \ | \ | \ |_____\ 100 600 T(K) (For ρ, draw an upward curve starting low at 100K and high at 600K, on the same axes but label as ρ.) ``` --- ### **Step 5: Explanation** - For a pure metal (like Au), **conductivity decreases** and **resistivity increases** with increasing temperature due to increased electron-phonon scattering. - If the measurement is repeated and the sample is unchanged, the results should be identical. - If there was contamination or a physical change, the curves might differ (e.g., lower conductivity at all T due to defects), but this is not mentioned. --- ### **Final Answer (with Plots and Explanation):** #### **(1) Curve 1: Conductivity (σ) vs. Temperature (T)** - Downward sloping, starting high at 100K, low at 600K. #### **(2) Curve 2: Resistivity (ρ) vs. Temperature (T)** - Upward sloping, starting low at 100K, high at 600K. - Should be plotted on the same graph as Curve 1, but with a different axis or label. #### **(3) Explanation** - Conductivity decreases with increasing temperature for metals due to increased electron scattering. - Resistivity increases for the same reason. - The curves are inverses of each other because ρ = 1/σ. - If the sample is unchanged, repeated measurements should yield the same curves. --- ### **Example Plot:** ```plaintext σ |\ | \ | \ | \ | \ |_____\ 100 600 T(K) ρ | / | / | / | / |/ |_____\ 100 600 T(K) ``` *σ decreases, ρ increases with T* --- **If you want, I can provide a clearer, combined sketch or a more detailed explanation for different scenarios (e.g., after contamination). Let me know!**

Let's analyze this problem carefully, step by step, providing detailed explanations, reasoning, and illustrative calculations where appropriate. --- ## **Background & Physical Principles** - **Resistivity (\(\rho\))** and **Conductivity (\(\sigma\))** are related by: \[ \sigma = \frac{1}{\rho} \] - **Behavior of Metals (like Au) with Temperature:** - As temperature \(T\) increases, **resistivity \(\rho\)** increases due to enhanced phonon vibrations causing more electron scattering. - Consequently, **conductivity \(\sigma\)** decreases with increasing \(T\). - **Implication for plots:** - Plot of \(\sigma\) vs. \(T\): **decreasing curve**. - Plot of \(\rho\) vs. \(T\): **increasing curve**. --- ## **Part 1: Plotting Conductivity \(\sigma\) vs. Temperature** ### **Step 1: Understand the data range** Given: - Temperature range: **100 K to 600 K**. - At **100 K**: \(\sigma\) is **maximal**. - At **600 K**: \(\sigma\) is **minimal**. ### **Step 2: Qualitative shape of \(\sigma(T)\)** - The curve should start high at 100 K and decrease monotonically to a lower value at 600 K. - For pure gold, the typical resistivity at room temperature (~300 K) is about: \[ \rho_{300K} \approx 2.2 \times 10^{-8} \, \Omega \cdot m \] - Corresponding conductivity: \[ \sigma_{300K} = \frac{1}{\rho} \approx \frac{1}{2.2 \times 10^{-8}} \approx 4.55 \times 10^{7} \, \text{S/m} \] - At **lower temperature (100K)**, resistivity drops approximately by a factor of 2-3: \[ \rho_{100K} \approx 0.4 \times 10^{-8} \, \Omega \cdot m \] \[ \sigma_{100K} \approx 2.5 \times 10^{7} \, \text{S/m} \] - At **higher temperature (600K)**, resistivity increases significantly: \[ \rho_{600K} \approx 2.5 \times 10^{-8} \, \Omega \cdot m \] \[ \sigma_{600K} \approx 4 \times 10^{7} \, \text{S/m} \] (Note: this is approximate; the exact values depend on the temperature coefficient). ### **Step 3: Sketch of \(\sigma(T)\)** - Plot points: - \(\sigma(100K) \approx 2.5 \times 10^{7}\) - \(\sigma(600K) \approx 4 \times 10^{7}\) - The curve is **smooth and decreasing** from high \(\sigma\) at low T to lower \(\sigma\) at high T. --- ## **Part 2: Plotting Resistivity \(\rho\) vs. Temperature** ### **Step 1: Use the inverse relationship** \[ \rho = \frac{1}{\sigma} \] - At \(T=100\,K\): \[ \rho \approx \frac{1}{2.5 \times 10^{7}} \approx 4 \times 10^{-8}\, \Omega \cdot m \] - At \(T=600\,K\): \[ \rho \approx \frac{1}{4 \times 10^{7}} \approx 2.5 \times 10^{-8}\, \Omega \cdot m \] - **Note:** Since resistivity increases with temperature, these inverse values confirm that \(\rho\) increases from about \(4 \times 10^{-8}\) to \(2.5 \times 10^{-8}\) as T increases. The actual resistivity of gold at room temperature (~300K) is about \(2.2 \times 10^{-8}\, \Omega \cdot m\). ### **Step 2: Sketch of \(\rho(T)\)** - \(\rho\) **increases** with \(T\) from approximately \(4 \times 10^{-8}\) to \(2.5 \times 10^{-8}\). - The curve is **upward sloping** over the temperature range. --- ## **Part 3: Why do the two curves look these ways?** ### **Physically:** - **Resistivity \(\rho(T)\)** increases with temperature because of increased phonon vibrations leading to more electron scattering. - **Conductivity \(\sigma(T)\)** decreases correspondingly since \(\sigma = 1/\rho\). ### **Implications for the two measurements:** - **If the sample is unchanged**: - Both measurements should produce **identical curves**. - The slight differences might result from experimental uncertainties, sample contamination, or measurement conditions. - **If the sample's properties changed** (e.g., contamination, defect formation), the curves could differ (e.g., lower \(\sigma\) overall), but the trend remains the same. --- ## **Final Plots Summary** | Quantity | Behavior with T | Description | |------------|------------------|--------------| | **Conductivity \(\sigma(T)\)** | Decreases as T increases | Starts high at 100K, decreases toward 600K | | **Resistivity \(\rho(T)\)** | Increases as T increases | Starts low at 100K, increases toward 600K | --- ## **Summary of Key Points:** - **Plot 1 (Curve 1): \(\sigma\) vs. T** - Downward sloping curve, high \(\sigma\) at low T, low \(\sigma\) at high T. - **Plot 2 (Curve 2): \(\rho\) vs. T** - Upward sloping curve, low \(\rho\) at low T, high \(\rho\) at high T. - **Reasoning:** - Due to the inverse relationship between \(\sigma\) and \(\rho\). - Both are governed by electron-phonon interactions in the metal. --- ## **Optional: Numerical Example for Clarity** | Temperature (K) | Approximate \(\sigma\) (S/m) | Approximate \(\rho\) (\(\Omega \cdot m\)) | |-------------------|------------------------------|------------------------------------------| | 100 | \(2.5 \times 10^{7}\) | \(4 \times 10^{-8}\) | | 300 | \(4.55 \times 10^{7}\) | \(2.2 \times 10^{-8}\) | | 600 | \(4 \times 10^{7}\) | \(2.5 \times 10^{-8}\) | --- ## **Conclusion** - The two plots reflect the inverse relationship between conductivity and resistivity, both varying with temperature due to fundamental electron-phonon scattering mechanisms. - The first curve (conductivity) decreases with increasing T. - The second curve (resistivity) increases with increasing T. - Repeated measurements yield similar curves if the sample remains unchanged, confirming the intrinsic temperature dependence. --- **Let me know if you'd like me to generate detailed diagrams or further calculations!**