Which of the following is a correct expression of average particle size with value of p=1 i.e index related to the size of an individual particle and frequency index (f=2):
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To find the average particle size, use the formula:
\[
d_{1f} = \frac{\sum n d^{f+1}}{\sum n d^f}
\]
where \( p \) is the size index and \( f \) is the frequency index.
The problem asks for the correct expression of average particle size with an index value of \( p = 1 \). The index \( p \) is related to the size of an individual particle, and the frequency index is given as \( f = 2 \). This suggests that we need to use the appropriate formula for particle size distribution. The general formula for the size index is:
\[
d_{pq} = \frac{\sum n d^{p+q}}{\sum n d^{p+q-1}}
\]
For \( p = 1 \) and \( q = n \), we substitute these values:
\[
d_{s1} = \frac{\sum n d^2}{\sum n d^3}
\] This matches option (B), making it the correct answer.
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Approach Solution -2
This question asks for the correct expression of the average particle size when the size related index is p=1 and the frequency related index is f=2. In particle size analysis, an average diameter is built from a weighted ratio of particle counts and diameters, where p sets how the individual particle's size enters the sum and f sets how the counting is weighted, so substituting these two index values into the general relation gives the specific formula being asked for.
Option B: The formula shown under this label does not correspond to substituting p=1 and f=2 into the general average diameter relation; the exponents on the particle diameter d in its numerator and denominator do not match what those index values require.
Option C: Working from the general relation for average particle diameter, \[ \bar{d} = \frac{\Sigma n d^{p+f}}{\Sigma n d^{f}} \] and substituting \(p=1\) and \(f=2\) gives \[ \bar{d} = \frac{\Sigma n d^{3}}{\Sigma n d^{2}} \] This is the surface-volume mean diameter, a standard expression used to calculate the specific surface area of a powder from its size distribution, and it is the formula that appears under this label.
Option D: The exponents in this expression correspond to a different pair of index values than p=1 and f=2, so it does not represent the quantity the question is asking for.
Option A: This expression also uses a different combination of exponents on d, matching neither p=1 nor f=2 as substituted into the general relation.
Substituting the given index values into the general average particle diameter relation produces \(\Sigma n d^{3} / \Sigma n d^{2}\), the surface-volume mean diameter, which is the formula given under option C.
