Limiting radius ratio for tetrahedral

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The discussion focuses on proving the limiting radius ratio for tetrahedral structures, specifically for coordination number 4, which is established between 0.225 and 0.414. The user successfully demonstrated the lower limit of 0.225 but seeks assistance in proving the upper limit of 0.414. It is noted that each coordination number has a minimum radius ratio, and larger ratios can be inferred from diagrams illustrating these relationships. The maximum radius ratio for a given coordination number is determined by the minimum radius ratio of the next higher coordination number. The conversation emphasizes the need for clarity on these mathematical relationships to complete the proof.
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Homework Statement
Prove that the tetrahedral structure (like ZnS) having a coordination number of 4 have a limiting radius ratio in the range 0.225-0.414
Relevant Equations
Pythagoras Theorem, d^2 = a^2 + a^2
I am able to prove that it is 0.225 but how do I prove that it is also 0.414?
I need to find the max. and min. packing fraction values, which I got as a function of (r1/r2)
Please help
 
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tanaygupta2000 said:
Homework Statement:: Prove that the tetrahedral structure (like ZnS) having a coordination number of 4 have a limiting radius ratio in the range 0.225-0.414
Relevant Equations:: Pythagoras Theorem, d^2 = a^2 + a^2

I am able to prove that it is 0.225 but how do I prove that it is also 0.414?
I need to find the max. and min. packing fraction values, which I got as a function of (r1/r2)
Please help
I’m definitely no expert but maybe this will help...

Each coordination number has a minimum value of the radius ratio (RR). Larger values of the RR are possible - you can get the general idea from this diagram. https://upload.wikimedia.org/wikipedia/commons/thumb/8/8b/Criticalradiusratio.png/800px-Criticalradiusratio.png

For a given coordination number, the maximum value of the RR is taken as the minimum value of the RR of the next coordination number. (I’m no expert so please don’t ask me why!)

What is the next possible coordination number after 4?
What is the minimum value of its RR?
 
Thread 'Chain falling out of a horizontal tube onto a table'

Thread 'Chain falling out of a horizontal tube onto a table'

My attempt: Initial total M.E = PE of hanging part + PE of part of chain in the tube. I've considered the table as to be at zero of PE. PE of hanging part = ##\frac{1}{2} \frac{m}{l}gh^{2}##. PE of part in the tube = ##\frac{m}{l}(l - h)gh##. Final ME = ##\frac{1}{2}\frac{m}{l}gh^{2}## + ##\frac{1}{2}\frac{m}{l}hv^{2}##. Since Initial ME = Final ME. Therefore, ##\frac{1}{2}\frac{m}{l}hv^{2}## = ##\frac{m}{l}(l-h)gh##. Solving this gives: ## v = \sqrt{2g(l-h)}##. But the answer in the book...
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