化學數學期末考
(2017/1/9)
1. Please solve the following ordinary differential equations. (25%)
(i) y y’ + 25 x = 0
(ii) y' + 2y = 2x
(iii) y” + 9 y = 0, y(0) = 0.2, y’(0) = -1.5
(iv) yiv – 10 y'' + 9 y = 0
(v) y’ + 2 y = 4 cos 2x, y(1/4 p) = 3
(i) y y’ + 25 x = 0/ (ii) y' + 2y = 2x
(iii) y” + 9 y = 0, y(0) = 0.2, y’(0) = -1.5
/ (iv) yiv – 10 y'' + 9 y = 0
(v) y’ + 2 y = 4 cos 2x, y(1/4 π) = 3
2. Please derive the integrated rate law for the reaction and plot the concentrations of all the species as a function of time. (Initially [A] = 1.0 M, [B] = [C] = 0.0 M, k1 = 1 s-1, k2 = 2 s-1, k-1 = 0.5 s-1) (20%)
(i) A ® B (k1) ® C (k2)
(ii) A Û B (k1, k-1)
Please also show how to solve the problems using Mathematica. (10%)
(i) A ® B (k1) ® C (k2)Mathematica
=1; =1;=2; y[t] /. sol
(ii) A Û B (k1, k-1)
Mathematica
=1; =1;=0.5;
sol=DSolve[{eqn,y[0]==1},y,t]
3. Please prove:
(a) if B is similar to A, then B has the same eigenvalues as A
(b) An orthogonal transformation preserves the value of the inner product of vectors a · b. (20%)
(a)(b)
4. A = B =
Please find (i) 2A-B (ii) AT × B (iii) A-1 (iv) det (ATB) (20%)
(b) AT × B
(c) A-1
(d) det (ATB)
5. Transform 17x12 - 30 x1x2 + 17 x22 = 128 to principal axes. What kind of conic section is it? What are the directions of the principal axes in the original coordinate system? (15%)
6. The energy difference between the two lowest rotational level of 12C32S is 3.246 × 10−23 J. Please calculate the bond length of the CS molecule. (in Å, 32S = 31.972 amu, 1 amu = 1.66054 × 10−27 kg,
h = 6.62608 × 10−34 J s,) (15%)
7. The vibrational frequency of H2 is 4401 cm-1. Please calculate its vibrational zero-point energy (in kcal/mol), and please estimate the vibrational frequency of HD molecule (in cm-1)。(H = 1.0078 amu, D = 2.0141 amu, c = 2.998 × 1010 cm/s ) (15%)