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 dr/dt = a(p – x)(q – x), (5) where a is a positive constant. = 0, determine the limiting value of x(t) as t → o without solving the differential (a) If æ(0) equation. Then solve the initial value problem and find x(t) for any t.

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 dr/dt = a(p – x)(q – x), (5) where a is a positive constant. = 0, determine the limiting value of x(t) as t → o without solving the differential (a) If æ(0) equation. Then solve the initial value problem and find x(t) for any t.

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Differential Equations

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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
dr/dt = a(p – x)(4 – x),
(5)
where a is a positive constant.
(a) If x(0) = 0, determine the limiting value of x(t) as t → oo without solving the differential
equation. Then solve the initial value problem and find x(t) for any t.

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