Let's solve the problem step by step: ### **Step 1: Understanding the Ternary Diagram** - Each corner (A, B, C) represents 100% of that component. - Each side represents a binary combination (no third component). - Each point inside the triangle gives a specific composition \((X_A, X_B, X_C)\) where \(X_A + X_B + X_C = 1\). ### **Step 2: Identify the Coordinates for Points 1, 11, 6, 8** We need to determine \(X_B/X_C\) for points 1, 11, 6, 8. #### **How to Read the Ternary Diagram** - **Horizontal lines**: Represent constant \(X_A\). - **Lines parallel to AB**: Represent constant \(X_C\). - **Lines parallel to AC**: Represent constant \(X_B\). Each small triangle typically represents an increment of 0.1 in composition. --- Let's estimate the coordinates for the required points: #### **Point 1** (near A) - Closest to A (100% A, 0% B, 0% C) - \(X_A \approx 1, X_B \approx 0, X_C \approx 0\) - \(X_B/X_C = 0/0\) (undefined, but both are zero) #### **Point 11** (on AB, close to A) - A bit further from A along AB, but still near A. - Estimate: \(X_A \approx 0.8, X_B \approx 0.2, X_C = 0\) - \(X_B/X_C = 0.2/0 = \infty\) (since \(X_C = 0\), undefined, but conceptually infinite) #### **Point 6** (center-ish) - Near the center, so all components are roughly equal. - Estimate: \(X_A \approx 0.33, X_B \approx 0.33, X_C \approx 0.33\) - \(X_B/X_C = 0.33/0.33 = 1\) #### **Point 8** (closer to B and C) - Along BC, roughly two-thirds B, one-third C. - Estimate: \(X_A = 0, X_B \approx 0.67, X_C \approx 0.33\) - \(X_B/X_C = 0.67/0.33 = 2\) --- ### **Step 3: Summarize the Ratios** | Point | \(X_B\) | \(X_C\) | \(X_B/X_C\) | |-------|--------|--------|------------| | 1 | 0 | 0 | undefined | | 11 | 0.2 | 0 | ∞ (infinite)| | 6 | 0.33 | 0.33 | 1 | | 8 | 0.67 | 0.33 | 2 | --- ### **Step 4: Conclusion** **Along the line from point 1 → 11 → 6 → 8:** - At point 1 and 11 (close to A and AB), \(X_C\) is zero, so \(X_B/X_C\) starts as undefined/infinite. - As we proceed towards the center (point 6), the ratio becomes 1. - Approaching point 8 (closer to B), the ratio increases to 2. **Conclusion:** **As you move along the line from A towards BC, the ratio \(X_B/X_C\) increases from undefined/infinite (when \(X_C = 0\)) to 1 (at the center), and further increases to 2 (towards B). This shows that the relative proportion of B to C increases along this line.** --- **If you want the exact values, you would need to count the grid lines, but the above gives a good estimation and the required trend. Let me know if you want me to count the lines for precise values!**

Let's proceed with detailed calculations based on the ternary phase diagram provided. Our goal is to determine the compositions of points 1, 11, 6, and 8, compute the ratio \( X_B / X_C \) for each, and analyze the trend. --- ### **Step 1: Understanding the Diagram** - The ternary diagram has three axes: A (bottom left), B (bottom right), C (top). - The composition at any point is given by the triangle's coordinate system. - The grid lines help estimate component fractions. Typically, each small subdivision represents 0.1 (10%) increments. --- ### **Step 2: Coordinates of the Points** --- #### **Point 1** - Located near the corner A (bottom left). - From the diagram, point 1 appears very close to A, so: \[ X_A \approx 1.00 \\ X_B \approx 0.00 \\ X_C \approx 0.00 \] - **Ratios:** \[ X_B / X_C = 0/0 \Rightarrow \text{undefined (or 0/very small)}. \] --- #### **Point 11** - Located along the line between A and B, closer to A. - Counting grid lines: Suppose each small subdivision from A toward B is 0.1 in \(X_B\): - The point appears roughly 1/5th of the way from A towards B (since it's about 1/5th along the line): \[ X_A \approx 0.8 \\ X_B \approx 0.2 \\ X_C \approx 0.00 \] - **Ratios:** \[ X_B / X_C = 0.2 / 0 \Rightarrow \text{tends to infinity} \] --- #### **Point 6** - Located near the center of the triangle, roughly equally spaced among the three corners. - Estimated as: \[ X_A \approx 0.33 \\ X_B \approx 0.33 \\ X_C \approx 0.34 \] - **Ratios:** \[ X_B / X_C \approx 0.33 / 0.34 \approx 0.97 \approx 1 \] --- #### **Point 8** - Located between B and C, closer to B. - Counting subdivisions: Suppose it's about two-thirds along from B to C: \[ X_A \approx 0.00 \\ X_B \approx 0.67 \\ X_C \approx 0.33 \] - **Ratios:** \[ X_B / X_C = 0.67 / 0.33 \approx 2.03 \] --- ### **Step 3: Summary of Compositions and Ratios** | Point | \(X_A\) | \(X_B\) | \(X_C\) | \(X_B / X_C\) | Remarks | |---------|---------|---------|---------|--------------|--------------------------------------------| | 1 | 1.00 | 0.00 | 0.00 | 0/0 (undefined) | Near pure A, ratio undefined/infinite | | 11 | 0.80 | 0.20 | 0.00 | ∞ | Near A, along AB, \(X_C=0\) | | 6 | 0.33 | 0.33 | 0.34 | ~1 | Center point, roughly equal components | | 8 | 0.00 | 0.67 | 0.33 | ~2 | Closer to B-C side, B dominant | --- ### **Step 4: Analysis of the \(X_B / X_C\) Ratio** - **At point 1:** ratio is undefined (since \(X_C=0\)), indicating B/C ratio tends to infinity as we approach the B side. - **At point 11:** ratio is also very high, tending to infinity. - **At point 6:** ratio is approximately 1, indicating equal amounts of B and C. - **At point 8:** ratio is about 2, indicating B is twice as much as C. **Trend:** As we move from point 1 (near A) toward point 8 (near B-C side), the ratio \(X_B / X_C\) increases from undefined/infinite to about 2. This shows an increasing proportion of B relative to C along the line. --- ### **Final Conclusions:** - **Along the line from point 1 to point 8**, the ratio \(X_B / X_C\) **increases monotonically** from very high (approaching infinity) to about 2. - **At the center point (6)**, the components are nearly equal, with a ratio close to 1. - **This indicates** that moving along this line, the B component becomes increasingly dominant relative to C. --- ### **Summary:** | Point | \(X_B / X_C\) | Interpretation | |---------|--------------|------------------------------------------------------------| | 1 | Undefined/infinite | Near pure A; B and C negligible | | 11 | Infinite | Close to A, with a small B component, negligible C | | 6 | ~1 | Components evenly distributed | | 8 | ~2 | B component dominates over C | **This analysis reveals the compositional trend and ratio behavior along the specified points on the ternary phase diagram.**