Carburizing Calculation

The problem is stated as follows:
*The surface of a steel gear made of 1022 steel (0.22 wt% C) is to be gas-carburized at $927$ °C. Calculate the time necessary to increase the carbon content to 0.30 wt% at 0.030 in below the surface of the gear. Assume the carbon content of the surface to be 1.20 wt% and* $D\_{927^{\circ}C}=1.28 \times 10^{-11}\frac{m^{2}}{s}$


The formula is $\frac{C\_{s}-C\_{x}}{C\_{s}-C\_{0}}=ERF(\frac{x}{2\sqrt{Dt}})$


The solution manual is setting $C\_{x}=0.30$, but should it not be $C\_{x}=0.30+0.22=0.52$, since the initial carbon content in the steel is $C\_{0}=0.22$?




No.


In the equation $C\_x$ simply means the concentration at depth x. It does not mean the *change* of concentration. There is no need to add its initial concentration. 


Besides, concentration cannot be linearly added. (Does adding a 1M solution to a 2M solution give 3M?)


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