I need help with getting started:
1.The crystal structure of an ionic compound consists of alternating cations and anions lying next to each other in three dimensions. If the cation radius is 55.1% of the anion radius and the distance between the cation nuclei is 587 pm, what are the radii of the two ions? (Note: This ionic compound is a three dimensional array. The cation to cation, center-to-center distance specified is through the center of the anion.)
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The vertical lines represent the center of the left cation, the anion, and the right cation. This is the way I envision it from the problem. Draw a small circle around the left vertical line, a small circle around the right vertical line, and a large circle around the middle vertical line. The problem says they are lying next to each other; I'm assuming this means they touch although it doesn't say that exactly. I don't know how to work the problem if they don't touch. So you have two unknowns and you need two equations.
Let C = the cation radius.
Let A = the anion radius.
We know the center to center distance for the cations is 587 pm and we know the center line passes through the center of the anion. Therefore,
2C + 2A = 587
The second equation we know is that
C = 0.551 A
Solve the two equations simultaneously and solve for C and A. Post your work if you get stuck. Check my thinking.
To solve this problem, we'll use the concept of ionic radii and the given information about cation radius, anion radius, and the distance between cation nuclei.
Let's assume the radius of the cation is represented by 'r_c' and the radius of the anion is represented by 'r_a'.
From the problem, we know that the cation radius is 55.1% of the anion radius. Mathematically, this can be written as:
r_c = 0.551 * r_a (equation 1)
We also know that the distance between the cation nuclei is given as 587 pm. This distance includes the radius of one cation and the radius of one anion, so the relationship can be expressed as:
2 * (r_c + r_a) = distance_between_cations (equation 2)
Now, let's substitute the value of r_c from equation 1 into equation 2:
2 * (0.551 * r_a + r_a) = 587 pm
Now, we can solve this equation to find the value of 'r_a'.
2 * (1.551 * r_a) = 587 pm
3.102 * r_a = 587 pm
r_a = 587 pm / 3.102
r_a = 189 pm
Substituting this value back in equation 1, we can find the cation radius 'r_c':
r_c = 0.551 * r_a
r_c = 0.551 * 189 pm
r_c = 104 pm
Therefore, the radius of the anion is 189 pm and the radius of the cation is 104 pm.