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For each figure determine if the encounter is successful or unsuccessful and describe what you are looking for to know if it is successful or unsuccessful; determine if it has translational or vibrational energy and describe what you see for translational and vibrational energy.

For each figure determine if the encounter is successful or unsuccessful and describe what you are looking for to know if it is successful or unsuccessful; determine if it has translational or vibrational energy and describe what you see for translational and vibrational energy.

Chemistry
Chemistry
10th Edition
ISBN:9781305957404
Author:Steven S. Zumdahl, Susan A. Zumdahl, Donald J. DeCoste
Publisher:Steven S. Zumdahl, Susan A. Zumdahl, Donald J. DeCoste
Chapter1: Chemical Foundations
Section: Chapter Questions
Problem 1RQ: Define and explain the differences between the following terms. a. law and theory b. theory and...
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For each figure determine if the encounter is successful or unsuccessful and describe what you are looking for to know if it is successful or unsuccessful; determine if it has translational or vibrational energy and describe what you see for translational and vibrational energy.

### Contour Plots with Critical Points The following figure consists of four contour plots (a, b, c, and d), each illustrating the relationship between two variables \( R_{BC} \) and \( R_{AB} \). These plots are used to visualize the behavior of functions in multivariable calculus and optimization problems with a focus on critical points. #### Plot Descriptions: 1. **Plot (a):** - **Axes:** Horizontal axis (\( x \)-axis) represents \( R_{AB} \). Vertical axis (\( y \)-axis) represents \( R_{BC} \). - **Contours and Critical Point:** The contours are curved lines representing constant values of the function. The inset arrow points to a critical point marked as \( C^* \). 2. **Plot (b):** - **Axes:** Same as in plot (a). - **Contours and Critical Point:** Similar to plot (a) but showing more intricate contour patterns. A critical point \( C^* \) is marked with an arrow, indicating a region of interest where the function has a potentially significant behavior. 3. **Plot (c):** - **Axes:** Same as in plot (a) and (b). - **Contours and Critical Point:** Contours are displayed, showing the function's topography around the critical point indicated by \( C_1 \). 4. **Plot (d):** - **Axes:** Same as in previous plots. - **Contours and Critical Point:** Contours illustrate the function's variations, marking a critical point \( C_1 \) within the pattern. ### Understanding Critical Points: Critical points can either be maxima, minima, or saddle points of a function where the gradient is zero. - **Saddle Point (S):** A point on the surface of a graph where the slopes (gradients) are zero but it is neither a maximum nor a minimum. - **Local Maximum (M):** A point where the function value is higher than that of all nearby points. - **Local Minimum (m):** A point where the function value is lower than that of all nearby points. These plots help in understanding the nature of critical points and the function behavior in multivariable systems. By analyzing the contours and the configuration around the critical points, one can infer important characteristics of the function under study.
Transcribed Image Text:### Contour Plots with Critical Points The following figure consists of four contour plots (a, b, c, and d), each illustrating the relationship between two variables \( R_{BC} \) and \( R_{AB} \). These plots are used to visualize the behavior of functions in multivariable calculus and optimization problems with a focus on critical points. #### Plot Descriptions: 1. **Plot (a):** - **Axes:** Horizontal axis (\( x \)-axis) represents \( R_{AB} \). Vertical axis (\( y \)-axis) represents \( R_{BC} \). - **Contours and Critical Point:** The contours are curved lines representing constant values of the function. The inset arrow points to a critical point marked as \( C^* \). 2. **Plot (b):** - **Axes:** Same as in plot (a). - **Contours and Critical Point:** Similar to plot (a) but showing more intricate contour patterns. A critical point \( C^* \) is marked with an arrow, indicating a region of interest where the function has a potentially significant behavior. 3. **Plot (c):** - **Axes:** Same as in plot (a) and (b). - **Contours and Critical Point:** Contours are displayed, showing the function's topography around the critical point indicated by \( C_1 \). 4. **Plot (d):** - **Axes:** Same as in previous plots. - **Contours and Critical Point:** Contours illustrate the function's variations, marking a critical point \( C_1 \) within the pattern. ### Understanding Critical Points: Critical points can either be maxima, minima, or saddle points of a function where the gradient is zero. - **Saddle Point (S):** A point on the surface of a graph where the slopes (gradients) are zero but it is neither a maximum nor a minimum. - **Local Maximum (M):** A point where the function value is higher than that of all nearby points. - **Local Minimum (m):** A point where the function value is lower than that of all nearby points. These plots help in understanding the nature of critical points and the function behavior in multivariable systems. By analyzing the contours and the configuration around the critical points, one can infer important characteristics of the function under study.
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