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):
where \(n\) is the number of particles of diameter \(d\), \(p\) is the index related to the size of an individual particle (1 for length, 2 for surface, 3 for volume), and \(f\) is the frequency-distribution index (0 for a number distribution, 1 for length, 2 for surface, 3 for volume/weight). Substituting \(p=1\) and \(f=2\) into the general equation gives each of the four labelled expressions a specific test to pass:
Option A: To be correct, an expression must reduce, after substituting \(p=1, f=2\), to exactly \(\bar{d}=\dfrac{\sum n d^{3}}{\sum n d^{2}}\). If option A shows a different combination of exponents or a different outer root, it does not correspond to this specific choice of indices and can be ruled out.
Option B: The same check applies, this expression must also collapse to the surface-volume mean form above; if its exponents or overall power do not match the \(p=1, f=2\) substitution, it represents a different pair of indices (for example a number or length mean rather than a surface-volume mean) and is not the answer.
Option C: Carrying out the substitution correctly, \(\bar{d}=\left(\dfrac{\sum n d^{1+2}}{\sum n d^{2}}\right)^{1/1}=\dfrac{\sum n d^{3}}{\sum n d^{2}}\). This is the surface-volume (Sauter) mean diameter, and it is this exact expression, with the cube of \(d\) weighted by number in the numerator and the square of \(d\) weighted by number in the denominator, that matches the required indices.
Option D: Again, unless this expression algebraically reduces to \(\sum n d^{3}/\sum n d^{2}\), it corresponds to a different \(p\)/\(f\) pair and does not satisfy the given condition.
Only the expression that simplifies exactly to \(\sum n d^{3}/\sum n d^{2}\) after substituting \(p=1\) and \(f=2\) satisfies the question, and that is the formula shown as option C.
