Solved: Chemical Reactions. A second order chemical reaction involves the interaction

Chapter 2, Problem 28

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Chemical Reactions. A second order chemical reaction involves the interaction (collision) of one molecule of a substance P with one molecule of a substance Q to produce one molecule of a new substance X; this is denoted by P + Q X. Suppose that p and q, where p = q, are the initial concentrations of P and Q, respectively, and let x(t) be the concentration of X at time t. Then p x(t) and q x(t) are the concentrations of P and Q at time t, and the rate at which the reaction occurs is given by the equation dx/dt = (p x)(q x), (i) where is a positive constant. (a) If x(0) = 0, determine the limiting value of x(t) as t without solving the differential equation. Then solve the initial value problem and find x(t) for any t. (b) If the substances P and Q are the same, then p = q and Eq. (i) is replaced by dx/dt = (p x) 2 . (ii) If x(0) = 0, determine the limiting value of x(t) as t without solving the differential equation. Then solve the initial value problem and determine x(t) for any t