Colloidal Particles and their size

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Discussion Overview

The discussion revolves around the process of subdividing a cube to determine how many subdivisions are necessary to reduce its size to that of colloidal particles, specifically 100 nm. The focus includes mathematical reasoning and calculations related to exponential decay in size through successive subdivisions.

Discussion Character

  • Mathematical reasoning

Main Points Raised

  • One participant proposes a method to calculate the number of subdivisions needed by expressing the new length after 'n' subdivisions as (1/2)^n * l, where l is the original length.
  • The same participant sets the new length equal to 100 nm (10^-7 m) and derives the equation 10^-7 = (1/2)^n, leading to a logarithmic calculation to find n.
  • Another participant agrees with the calculation and provides context about the powers of 2 in relation to orders of magnitude, suggesting that the derived value of n is reasonable.
  • A later reply expresses gratitude for the clarification, indicating that the discussion has been helpful.

Areas of Agreement / Disagreement

There appears to be agreement on the mathematical approach and the derived value of n, although no explicit consensus on the correctness of the final answer is stated.

Contextual Notes

The discussion does not address potential assumptions about the properties of the subdivisions or the implications of using logarithmic calculations in this context.

Prashasti
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Suppose we have a cube, of length 1 metre. It is cut in all the three directions so that 8 cubes, each having 0.5 m as its length. Then, these cubes are again subdivided in the same manner to get cubes with length 0.25m and so on.

HOW MANY OF THESE SUCCESSIVE SUBDIVISIONS ARE REQUIRED BEFORE THE SIZE OF THE CUBES IS REDUCED TO THE SIZE OF COLLOIDAL PARTICLES, which is 100 nm??
 
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Come on, we all know that all indians are brilliant mathematicians, so show us at least your intent of solution.
 
Ok, then, what I did was,

Let original length be 'l',
Now one subdivision reduces the length to half of its original value, and 2, to one fourth. So, 'n' subdivisions will lead to reduction of the length to ( 1/2)^n.
Let new length be 'a'.
So, a = (1/2)^n *l.
in the given case,
a = 10^-7 *l
which means,
10^-7 = (1/2)^n
taking log,
n log 2 = 7 log 10
Which gives n = 23.253.
So, am I right??
 
This seems to be correct. It is helpful to have some orders of magnitude in mind. The powers of 2 we all know from informatics: ##2^{10}=1024\approx 10^3## and ##2^3=8\approx 10## so ##10^7=10^3*10^3*10\approx 10^{23}##.
 
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Got it. Thanks.
 
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