You are developing a tablet formulation of a new drug $($Drug X$).$ The dose of Drug $x$ is $50mg$ and the total tablet weight is $300mg.$ You have made $12kg$ of an appropriate powder mix of Drug $x$ and other ingredients using a low shear mixer.

a$)$ You are assessing the particle size distribution of your mix and have used three techniques:

light microscopy,

sieve analysis and

laser diffraction analysis

Explain the basic principles of these three techniques, including their advantages, an indication of the particle size range each technique can analyse, and any assumptions that are made when interpreting the experimental data.

Explain why you used these three techniques in conjunction.

You wish to assess the quality of the mixing process. Your assistant has taken $10$ powder samples, each of $600mg,$ in a $3-$dimensional testing pattern throughout the powder bed. The values of the content of Drug $x($in $mg)$ in the powder samples are as follows:

$100\mathrm{.}1,94\mathrm{.}5,89\mathrm{.}2,105\mathrm{.}2,100\mathrm{.}9,99\mathrm{.}1,94\mathrm{.}6,102\mathrm{.}8,104\mathrm{.}7,92\mathrm{.}4$

b$)$ Comment on the testing protocol and the analytical data, assuming that your mixing specification is an RSD $2\mathrm{.}0\%$ around the correct content mean.

You perform tapped density analysis and angle of repose analysis on your mixed powder. The results of these tests are as follows: weight $35\mathrm{.}4g,$ initial volume $71mL,$ final volume $54mL,$ angle $45.$

c$)$ Explain why you performed these tests, how you performed these tests and what the results tell you about your powder.