Work out the algebra to solve for k in equation 4 in the man

Work out the algebra to solve for k in equation (4) in the manual. Your equation for k should be in terms of d, t, C_K and C_0. Check to see if you have done the algebra correctly, noting that setting C_K = C_0 should yield k = 1. You will use this equation to calculate k from measurements in step 4.

Solution

Given you only want the equation to be resolved, I will take you through the step wise approach to do the same.

The equation we have here is:

1 / Ck = [1 / (koA/t)] + [1 / ( oA / d-t)]

We will arrange the right hand side to get:

1 / Ck = [t / (koA)] + [d - t / oA] [We have simply transferred the denominator of denominator to numerator]

As we need to single out K, I will transfer the first term on the right hand side to the left and bring 1 / Ck to the right and change the signs.

Then I will take LCM of the denominators on the right and add the fractions as follows:

t / (koA) = 1/Ck - [(d - t) / oA]

or t / (koA) = (oA - dC_k + tC_k )/ oAC_k

Now we can bring all the variables except k from the left hand side to the right to get:

1/ k = (oA - dC_k + tC_k )/ t*C_k

Take reciprocal of both the sides to obtain:

k =  t*C_k /(oA - dC_k + tC_k )

Now, we know that Co = oA/ d, putting this in above equation, we get:

k = t(oA/ d) / oA - oA + (toA/d) = 1

Therefore, the equation as mentioned gives k = 1 fro Ck = Co