The Chemistry Maths Book, Second Edition

310 Chapter 10Functions in 3 dimensions EXAMPLE 10.13A circular helix lies on the curved surface of a (right) circular cylinder, ...

10.6 Other coordinate systems 311 EXAMPLE 10.14The hydrogen molecule In the method of ‘linear combinations of atomic orbitals’ ( ...

312 Chapter 10Functions in 3 dimensions Then, separating variables, 0 Exercises 40, 41 10.7 Exercises Section 10.2 Find the cart ...

10.7 Exercises 313 23.Calculate the average distance from the nucleus of an electron in the 2 sorbital (see Exercise 21). 24.Cal ...

11 First-order differential equations 11.1 Concepts A differential equation is an equation that contains derivatives. For exampl ...

11.2 Solution of a differential equation 315 In this chapter we consider only first-order differential equations; second-order e ...

316 Chapter 11First-order differential equations EXAMPLE 11.1The equation provides a model for a first-order rate process such a ...

11.2 Solution of a differential equation 317 EXAMPLE 11.2 (i) Show thaty 1 = 1 e −x 1 + 11 is a solution of the first-order diff ...

318 Chapter 11First-order differential equations All ordinary differential equations can be solved by the numerical methods disc ...

11.3 Separable equations 319 and, by the method of substitution for the left side (Chapter 6), this is identical to (11.11). The ...

320 Chapter 11First-order differential equations (iii). Separation of the variables and integration gives and the general soluti ...

11.3 Separable equations 321 The differential equation (11.16) is made separable by means of the substitutionu 1 = 1 y 2 x. Then ...

322 Chapter 11First-order differential equations The general solution of the equation, after substitutingu 1 = 1 y 2 x, is there ...

11.4 Separable equations in chemical kinetics 323 and theH 2 2 O 2 equation is 01 = 1 −2H 2 1 − 1 O 2 1 + 1 2H 2 O Rate of react ...

324 Chapter 11First-order differential equations and for the H 2 2 O 2 reaction, The mechanism of a chemical reaction is in gene ...

11.4 Separable equations in chemical kinetics 325 If the initial concentration of Ais[A] 0 , at timet 1 = 10 , thenc 1 = 1 ln[A] ...

326 Chapter 11First-order differential equations Separation of the variables and integration gives If x 0 is the initial concent ...

11.4 Separable equations in chemical kinetics 327 This is the same as in case (2)above except for the factor 2, which distinguis ...

328 Chapter 11First-order differential equations and the integrated rate equation is (11.46) A plot ofln[B] 2 [A]against tis a s ...

11.5 First-order linear equations 329 Then, multiplying (11.47) byF(x), (11.52) and, by virtue of (11.51), the left side of this ...