Trigonometry Problem: Finding Distances in Sodium Chloride Crystal Structure

In summary, the author attempted to find the distance from the lower left corner of the front face to the upper right corner of the rear face of a cube using the Pythagorean theorem and one of the trig functions. The distance was found to be 0.48670627692685451948121242196315 nm.
  • #1
1irishman
243
0
1. Homework Statement
A drawing shows sodium and chloride ions positioned at the corners of a cube that is part of the crystal structure of sodium chloride. The edge of the cube is 0.281nm in length. Find the distance between the sodium ion located at one corner of the cube and the chloride ion located on the diagonal at the opposite corner.


2. Homework Equations
I'm thinking pythagorean theorem and one or more of the trig functions perhaps?


3. The Attempt at a Solution
Well...I figured the diameter if the cube is drawn inside a circle is 2(0.281)=0.562nm.
Not sure what to do this...it's difficult for me to conceptualize.

(The answer in the book is 0.487nm)
 
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  • #2
One way to approach this is as a distance problem in 3-D space. Put on corner of the cube at the origin - (0, 0, 0) - and the opposite corner at (.281, .281, .281).

The distance d from a point (x1, y1, z1) to another point (x2, y2, z2) is
[tex]d~=~\sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2}[/tex]

BTW, I get a distance of 0.48670627692685451948121242196315 nm.
 
  • #3
I kind of understand, but what would this look like as a picture on paper? Imagine no numbers...
 
  • #4
Unfortunately, I am unable to draw you a picture. The distance formula in my previous post is the 3-D counterpart to the length of the hypotenuse in the Theorem of Pythagoras.

I'll try to describe what this formula does in words, and leave the drawing to you. In your NaCl cube, let's say that you want to find the distance from the lower left corner (O) of the front face, to the upper right corner (P) of the rear face. To find this distance, consider the triangle whose vertices are O, P, and the point at the lower right corner of the front face (call this point Q). Triangle OPQ is a right triangle, with the right angle formed by sides OQ and OP. By the theorem of Pythagoras, |OP| = sqrt(|OQ|2 + |QP|2). |OQ| is given in your problem, and |OQ| is the length of the diagonal across one of the square faces.

Hopefully I have described this well enough so that you can draw the picture.
 
  • #5
Thank you for this very detailed help. I appreciate the effort you put into helping me construct a picture if you will. It does make sense to me after I drew it based on your instructions. Thanks again.
 

FAQ: Trigonometry Problem: Finding Distances in Sodium Chloride Crystal Structure

1. How is trigonometry used to find distances in a sodium chloride crystal structure?

Trigonometry is used to find distances in a sodium chloride crystal structure by using the principles of right triangle trigonometry. This involves using the known lengths of sides and angles in a right triangle to calculate the unknown distances within the crystal structure.

2. What are the most important concepts in trigonometry for solving this problem?

The most important concepts in trigonometry for solving this problem include the Pythagorean theorem, basic trigonometric ratios (sine, cosine, and tangent), and the unit circle. These concepts are essential for understanding the relationships between angles, sides, and distances within the crystal structure.

3. How do you determine the angle between two points in a sodium chloride crystal structure?

The angle between two points in a sodium chloride crystal structure can be determined by using the inverse trigonometric functions (arcsine, arccosine, and arctangent). This involves finding the ratio of the opposite and adjacent sides of the triangle formed by the two points and using the inverse function to solve for the angle.

4. What is the significance of finding distances in a sodium chloride crystal structure?

Finding distances in a sodium chloride crystal structure is significant because it allows scientists to understand the arrangement of atoms and molecules within the crystal. This information can provide insights into the physical, chemical, and electrical properties of the crystal, which can be useful for various applications in materials science and engineering.

5. Can trigonometry be used to find distances in other crystal structures besides sodium chloride?

Yes, trigonometry can be used to find distances in other crystal structures besides sodium chloride. The principles and concepts of trigonometry can be applied to any crystal structure that can be represented as a three-dimensional lattice, such as diamond, quartz, and graphite. However, the specific calculations and formulas may vary depending on the structure's characteristics and symmetries.

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