Modeling Tools for Environmental Engineers and Scientists

modeling, in diagnosis and troubleshooting, and in verifying and interpretingthe results obtained.EXERCISE PROBLEMS3.1. The relationship between pressure, P, volume, V, and temperature, T, formany real gases can be described by the van der Waals equation: P aV2(V– b) RTIdentify the problems with solving the above equation using elementaryalgebraic methods. Prepare an Excel ® worksheet to solve this equationfor V for known values of P, T, a, and b.3.2. The rate expressions for the reversible reaction A S B result in the followingcoupled differential equations: d [Ad ]t= –k [A] k [B] and 1 2 d [B]d = k 1 [A] – k 2 [B]tRecognizing that the total concentration of A and B = C T , the rate equationfor A can be found as follows: d [A] = –k 1 [A] k 2 [C T – A]dtShow that the solution to the above ODE isA = A 0 exp[–k 1 k 2 )t] k 2 C Tk1 k 2{1 – exp[–(k 1 k 2 )t]}where A 0 is the initial concentration of A.Assuming k 1 = 0.2, k 2 = 0.1, C T = 0.01, and A 0 = 0.009, find the timerequired for A to decrease by 50%.3.3. Consecutive reactions A → B → C such as can be found in environmentalsystems (e.g., NH 3 → NO – 2 → NO – 3). Starting from the individualrate equations:and d [A] = –k 1 [A];dt d [B] = k 1 [A] k 2 [B]dt d[ C] = k 3 [B]dt© 2002 by CRC Press LLC