Now we resort to a mass balance. In very simple terms, we know how much carbon and how much solution we add to this batch process at the start. What ever disappears from the solution can only end up adsorbed on the carbon. Let's establish some notation for this:

V is the volume of solution with initial concentration C_{o}.

B is the weight of carbon with initial loading qo. This loading is zero for fresh carbon.

C_{1} is the concentration of the final solution, and q1 is the
final loading of the carbon.

We know C_{o} and q_{o}. The design problem is either to
estimate the final
concentration and final loading when we are given values for V and B or to calculate what V and B should be to achieve a given removal of material. The latter is the most common
situation, so let's do the calculation. Each large stub line on the abscissa marks 5 more
concentration units.
We mark a point for the start; in this case the feed solution is
25 mg per liter while the carbon is fresh with zero loading.
The point is on the abscissa ( X-axis) at 25 units on the scale.

The final point must lie on the equilibrium line at the desired final concentration. In
this case, we have decided that 5 mg per liter is our target.
We move up from 5 units (first big stub line)
on the abscissa until we intersect the isotherm line. We can read the loading by
moving horizontally to the ordinate (Y-axis).
This is about 15 mg/g.
Now we know everything in the equation
but V and B. Probably we do know V because we have so much solution to treat per
day or per batch. Then we merely plug into the equation to find B, the amount of
carbon needed.

The orange line is known as the operating line. Even if we were not at equilibrium, the material that leaves the solution is on the carbon, and this operating line satisfies the mass balance. If we know V and B, we know the slope of the line. We can extend from the starting point and draw a line with this slope. When it intersects the equilibrium isotherm line, we have the values to solve the design problem for the situation where we are given the starting concentration and loading and must estimate what a given amount of carbon will do.