9780073529585, Chapter 1, Problem 6

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Chapter 1, Problem 6 Problem Calculate the angle between any pair of bonds in the tetrahedral structure. Step-by-step solution Step of 5 Write the expression for the dot product between two vectors: Here, and represent vectors and respectively and Θ denotes the angle between the vectors and Rearrange the expression of the dot product between two vectors to find Write the expression for the vector between two points having coordinates and Write the expression for the magnitude of vector having coordinates (3) Consider two vectors and Then, write the expression for the dot product between vectors a and (4) Step of 5 Draw the regular tetrahedron with the help of the vertices of the cube. Take the coordinates of the origin as (0,0,0) The diagram is shown below: (0,1,1) (1,1,1) (0,1,0) (1,1,0) (0.5,0.5,0.5) (0,0,1) (1,0,1) (0,0,0) (1,0,0) x Here, the coordinates of the other vertices of the cube are shown in the diagram. Take the center of the cube which is equidistant from all the vertices. Thus, the coordinates of the center of the cube become (0.5,0.5,0.5) as shown in the diagram. Also, the vectors and are from the center to the point having coordinates (0,1,1) and (1,1,0) respectively. Consider vector along the axis as positive and vice versa. Now, to calculate the angle between any pair of bonds in the tetrahedral structure, take two vertices and draw the vectors to them from the center. Consider the two points as (0,1,1) and Then, the vector is determined using equation (2) as, (0.5,0.5,0.5) = =(-0.5,0.5,0.5) Similarly, calculate the vector =(0.5,0.5,-0.5) Step of Calculate the magnitude of vector by using equation (3): = - 0.866 Similarly, calculate the magnitude of vector = 0.866 Also, calculate the dot product of vector and v2 by using equation (4). (0.5,0.5,-0.5) (0.5x-0.5) Step 4 of 5 Now, calculate the angle between any pair of bonds in the tetrahedral structure: Step 5 Substitute -0.25 for 0.866 for and 0.866 for in equation (1) and solve, -0.25 0.866x0.866 109.5° Hence, the angle between the pair of bonds in the tetrahedral structure is 109.5°