Consider the scale up of a chromatography column for purification of protein. A column of $40$cm length is used with a superficial velocity of $40$cm $/$ hr$.$ The peak concentration of the target protein is at a time of $100$min$.$ the standard deviation of the distribution is $14$ min. a$.$ If you collect your chromatography elution at t $1=0,$ How long must you wait to collect $90\%$ yield of the protein? b$.$ If the same column is used, but velocity is increased to $60$cm $/$ hr and internal diffusion controlled, what will be the value of the new $\backslash$sigma $?$ c$.$ If the same column is lengthened to $60$cm while increase the velocity is at $60$ cm $/$ hr$,$ how will $\backslash$sigma change compare to the original condition $(40$cm $/$ h with a $40-$ cm column$)?$