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mathematics
mathematics physical chemistry

Mathematics for Physical Chemistry 4th Edition Robert G. Mortimer - Solutions

Carry out the line integral of du = dx + x dy from (0,0) to (x1,y1):(a) On the rectangular path from (0,0) to (0,y1) and then to (x1,y1).(b) On the rectangular path from (0,0) to (x1,0) and then to (x1,y1).
(a) Show that the following differential is exact:dz = (yexy)dx + (xeey)dy.(b) Calculate the line integral ʃc dz on the line segment from (0,0) to (2,2). On this line segment, y = x and x = y.(c) Calculate the line integral ʃc dz on the path going from (0,0) to (0,2) and then to (2,2) (a
Show that the differential in the preceding example is exact.
Find an expression for the Laplacian of the functionf = r2 sin (θ) cos (ϕ).
Complete the formula = ), as as +? Т,п дV P,n
If u = RT ln (aT Vn) find du in terms of dT , dV, and dn, where R and a are constants.
Test each of the following differentials for exactness.(a) du = xy dx + xy dy,(b) du = y eaxy dx + x eaxy dy.
A certain nonideal gas is described by the equation of statewhere T is the temperature on the Kelvin scale, Vm is the molar volume, P is the pressure, and R is the gas constant. For this gas, the second virial coefficient B2 is given as a function of T byFind (ˆ‚ P/ˆ‚Vm)T and
Write the expression for the Laplacian of the function e−r2.
Write the formula for the divergence of a vector function F expressed in terms of cylindrical polar coordinates. Note that ez is the same as k.
(a) Find the h factors for cylindrical polar coordinates.(b) Find the expression for the gradient of a function of cylindrical polar coordinates, f = f (Ï,Ï•,z).(c) Find the gradient of the function |f = e-(p²+z²)/a² sin (6).
Show that ∇ × ∇f = 0 if f is a differentiable scalar function of x,y, and z.
Find an expression for the divergence of the functionF = i sin2 (x) + j sin2 (y) + k sin2 (z).
Find an expression for the gradient of the functionf (x,y,z) = cos (xy) sin (z).
Neglecting the attractions of all other celestial bodies, the gravitational potential energy of the earth and the sun is given bywhere G is the universal gravitational constant, equal to 6.673 Ã— 10ˆ’11 m3 sˆ’2 kgˆ’1, ms is the mass of the sun, me is the
Find the maximum in the function of the previous problem subject to the constraint x + y = 2. Do this by substitution and by Lagrange’s method of undetermined multipliers.Previous ProblemFind the location of the maximum in the functionf = f (x,y) = x2 − 6x + 8y + y2considering the region 0 <
Find the location of the maximum in the functionf = f (x,y) = x2 − 6x + 8y + y2considering the region 0 < x < 2 and 0 < y < 2.What is the value of the function at the maximum?
Find the minimum in the function of the previous problem subject to the constraint x + y = 2. Do this by substitution and by the method of undetermined multipliers.Previous ProblemFind the location of the minimum in the functionf = f (x,y) = x2 − x − y + y2considering all real values of x and
Find the minimum of the previous example without using the method of Lagrange.
Find the location of the minimum in the functionf = f (x,y) = x2 − x − y + y2considering all real values of x and y. What is the value of the function at the minimum?
(a) Find the local minimum of the functionf(x,y) = x2 + y2 + 2x.(b) Without using the method of Lagrange, find the constrained minimum subject to the constraint,x = −y.(c) Find the constrained minimum using the method of Lagrange.
Find (∂2 f /∂x2)y, (∂2 f /∂x∂y), (∂2 f /∂y∂x), and (∂2 f /∂y2), for each of the following functions,where a, b, and c are constants.a. f = e(ax2+by2),b. f = ln (bx2 + cy2).
Evaluate D at the point (0, 0) for the function of the previous example and establish that the point is a local maximum.
Show that the differentialis inexact, and that y/x is an integrating factor. [ x In (x) x2 dy (1+x)dx + y
Test each of the following differentials for exactness.(a) du = y/1+x2 dx − tan−1 (x)dy,(b) du = (x2 + 2x + 1)dx + (y2 + 5y + 4)dy.
Show that the following is not an exact differential du = (2y)dx + (x)dy + cos(z)dz.
Test each of the following differentials for exactness.(a) du = sec2 (xy)dx + tan (xy)dy,(b) du = y sin(xy)dx + x sin(xy)dy.
Find (∂2 f /∂x2)y, (∂2 f /∂x∂y), (∂2 f /∂y∂x), and (∂2 f /∂ y2), for each of the following functions,where a, b, and c are constants(a) f = (x2 + y2)−1,(b) f = sin (xy).
Determine whether the following differential is exact.du = (2ax + by2)dx + (bxy)dy.
Find (∂2 f /∂x2)y, (∂2 f /∂x∂y), (∂2 f /∂y∂x), and (∂2 f /∂y2) , for each of the following functions,where a, b, and c are constants.(a) f = (x + y)−2,(b) f = cos(xy).
Find (∂f/∂x)y, and (∂f /∂y)x for each of the following functions, where a, b, and c are constants.(a) f = a cos2 (bxy),(b) f = a exp(−b(x2 + y2).
Find (∂f/∂x)y, and (∂f/∂y)x for each of the following functions, where a, b, and c are constants.(a) f = (x + y)/(c + x),(b) f = (ax + by)−2.
Show that the reciprocal identity is satisfied by (∂z/∂x)y and (∂x/∂z)y if z = sin and x = y sin (z) = y arcsin (z).
Find (∂f/∂x)y, and (∂f/∂y)x for each of the following functions, where a, b, and c are constants.(a) f = axy ln (y),(b) f = c sin(x2 y).
Complete the following equations.a.b.c. Apply the equation of part b if z = cos(x/u) + 4y/u and w = y/u. (е#) р — (е) мл+? () = ан ЭТ )Р,п ат эт уп Эz ди / х,у :)x.+? ,+? ди / х,ш
For a certain system, the thermodynamic energy U is given as a function of S,V,and n byU = U(S,V,n) = Kn5/3V−2/3e2S/3nR,where S is the entropy, V is the volume, n is the number of moles, K is a constant, and R is the gas constant.(a) According to thermodynamic theory, T = (∂U/∂S)V ,n. Find an
The volume of a right circular cylinder is given byV = πr2h,where r is the radius and h the height. Calculate the percentage error in the volume if the radius and the height are measured and a 1.00% error is made in each measurement in the same direction. Use the formula for the differential, and
Use Simpson€™s rule with at least four panels to evaluate the following definite integral. Use Mathematica to check your results. e&r2 ,2r² dx.
Use Simpson€™s rule to evaluate the following definite integral. Use Mathematica to check your results. -Зх3 -3x dx.
The entropy change to bring a sample from 0 K (absolute zero) to a given state is called the absolute entropy of the sample in that state.where Sm(T′) is the absolute molar entropy at temperature T′,CP,m is the molar heat capacity at constant pressure, and T is the absolute temperature. Using
When a gas expands reversibly, the work that it does on its surroundings is given by the integralwhere V1 is the initial volume, V2 is the final volume, and P is the pressure of the gas. Certain nonideal gases are described by the van der Waals equation of state,where V is the volume, n is the
Find the integral: ʃ sin [x(x + 1)] (2x + 1)dx.
Write Mathematica entries to obtain the following integrals:(a) ʃ cos3 (x)dx.(b) ʃ21 e5x2 dx.
Using Simpson€™s rule, evaluate erf (2.000):Compare your answer with the correct value from a more extended table than the table in Appendix G, er f (2.000) = 0.995322265. -1² dt. erf(2) 2.
Approximate the integralusing Simpson€™s rule. You will have to take a finite upper limit, choosing a value large enough so that the error caused by using the wrong limit is negligible. The correct answer isˆšÏ€/2 = 0.886226926 · · ·. dx,
Apply Simpson€™s rule to the integralusing two panels. Since the integrand curve is a parabola, your result should be exactly correct. 20.00 x² dx, J10.00
Determine whether the following improper integrals converge. Evaluate the convergent integrals,a. ʃ∞0 sin (x) dx.b. ʃπ/2-π/2 tan (x) dx.
Using the trapezoidal approximation, evaluate the following integral, using five panels. 2.00 cosh (x)dx. 1.00
Show that the expressions for G and H are correct. Verify your result using Mathematica if it is available.
Determine whether the following improper integrals converge. Evaluate the convergent integralsa. ʃ10 1/x in (x) dx.b. ʃ∞1 (1/x) dx.
Solve the simultaneous equations to obtain the result of the previous example.
Determine whether each of the following improper integrals converges, and if so, determine its value:a. ʃ∞0 1/x3 dx.b. ʃ0-∞ ex dx.
Evaluate the integralwithout using a table. You will have to apply partial integration twice. ел x² sin (x)dx,
Find the following area by computing the values of a definite integral. The area bounded by the parabola y = 4 − x2 and the x axis. You will have to find the limits of integration.
Evaluate the integralwithout using a table of integrals. esin (0) cos (0)d0, *IT/2 0,
Determine whether each of the following improper integrals converges, and if so, determine its value:a.b. So ) dx. х () dx 1+x
Find the area bounded by the curve representing y = x3, the positive x axis, and the line x = 3.
Evaluate the definite integral: ʃπ/20 x sin (x2) dx.
(a) By drawing rough graphs, satisfy yourself that ψ1 is even about the center of the box. That is, ψ1(x) = ψ1(a − x). Satisfy yourself that ψ2 is odd about the center of box.(b) Draw a rough graph of the product ψ1ψ2 and satisfy yourself that the integral of this product from x = 0 to x =
Evaluate the definite integral: ʃπ/20 sin (x) cos2 (x) dx.
Drawa rough graph of f (x) = eˆ’x2. Satisfy yourself that this is an even function. Identify the area in the graph that is equal to the definite integraland satisfy yourself that this integral is equal to twice the integral e-x² dx I1 -3 e=x² dx = 211. I2
Draw a rough graph of f(x) = x eˆ’x2and satisfy yourself that this is an odd function. Identify the area in this graph that is equal to the following integral and satisfy yourself that the integral vanishes: xe x*dx = 0. -2
Find the approximate value of the integralby making a graph of the integrand function and measuring an area. dx,
Evaluate the definite integrals, using a table of indefinite integralsa. ʃ2.0001.000 in(3x)/x dx.b. ʃ5.0000.000 4x dx.
Evaluate the definite integral e* dx. х 0.
Find the function whose derivative is −(10.00)e−5.00x and whose value at x = 0.00 is 10.00.
Find the indefinite integral without using a table:(a) ʃ x ln (x)dx.(b) ʃ x sin2 (x)dx.
Find the maximum height for the particle in the preceding example.
Solve the following equations by hand, using the Newton–Raphson method. Verify your results using Excel or Mathematica:(a) e−x − 0.5x = 0.(b) sin (x)/x − 0.75 = 0.
Carry out Newton’s method to find the smallest positive root of the equation5.000x − ex = 0.Do the calculation by hand and verify your result by use of Excel.
The van der Waals equation of state isWhen the temperature of a given gas is equal to its critical temperature, the gas has a state at which the pressure as a function of V at constant T and n exhibits an inflection point at which dP/dV = 0 and d2P/dV2 = 0. This inflection point corresponds to the
Find the relative maxima and minima of the function f(x) = x3 + 3x2 − 2x for all real values of x.
Draw a rough graph of the function y = tan (x)/x in the interval −π/2 < x < π/2. Use l’Hôpital’s rule to evaluate the function at x = 0.
The probability that a molecule in a gas will have a speed v is proportional to the functionwhere m is the mass of the molecule, kB is Boltzmann€™s constant, and T is the temperature on the Kelvin scale. The most probable speed is the speed for which this function is at a maximum. Find
The sum of two nonnegative numbers is 100. Find their values if their product plus twice the square of the first is to be a maximum.
The thermodynamic energy of a collection of N harmonic oscillators (approximate representations of molecular vibrations) is given by(a) Drawa rough sketch of the thermodynamic energy as a function of T.(b) The heat capacity of this system is given by C = dU/dT.Show that the heat capacity is given
According to the Planck theory of black-body radiation, the radiant spectral emittance is given by the formulawhere λ is the wavelength of the radiation, h is Planck’s constant, kB is Boltzmann’s constant, c is the speed of light, and T is the temperature on the Kelvin scale. Treat T as a
If a hydrogen atom is in a 2s state, the probability of finding the electron at a distance r from the nucleus is proportional to 4Πr2Ψ22swhere Ψ represents the orbital (wave function):where a0 is a constant known as the Bohr radius, equal to 0.529 × 10-10m.(a) Locate the maxima and minima
Find the maximum and minimum values of the functiony = x3 + 4x2 − 10xin the interval −5 < x < 5.
Find the following limits:a. limx→∞ (e−x2/e−x).b. limx→’0 [x2/(1 − cos (2x))]c. limx→Π [sin (x)/ sin (3x/2)].
Find the following limits:a. limx→0+[ln (1+x)/sin (x)].b. limx†’0 + [sin (x) ln (x)].
Find the following limits:a. limx→∞[ln (x)/x2].b.c. lim,--3 [(x – 27)/(x² – 9)]. |
A cylindrical tank in a chemical factory is to contain 2.000 m3 of a corrosive liquid. Because of the cost of the material, it is desirable to minimize the area of the tank. Find the optimum radius and height and find the resulting area.
(a) A rancher wants to enclose a rectangular part of a large pasture so that 1.000 km2 is enclosed with the minimum amount of fence. Find the dimensions of the rectangle that he should choose. The area isA = xybut A is fixed at 1.000 km2, so that y = A/x.(b) The rancher now decides that the fenced
The mean molar Gibbs energy of a mixture of two enantiomorphs (optical isomers of the same substance) is given at a constant temperature T byGm = Gm(x) = G◦m + RTx ln (x) + RT(1 − x) ln (1 − x),where x is the mole fraction of one of them. G◦m is a constant, R is the ideal gas constant,
Draw a rough graph of the functiony = y(x) = cos (|x|).Is the function differentiable at x = 0? Draw a rough graph of the derivative of the function.
Draw a rough graph of the functiony = y(x) = sin (|x|).Is the function differentiable at x = 0? Draw a rough graph of the derivative of the function.
A collection of N harmonic oscillators at thermal equilibrium at absolute temperature Tis shown by statistical mechanics to have the thermodynamic energywhere kB is Boltzmann’s constant, h is Planck’s constant, T is the absolute temperature, and ν is the vibrational frequency. Find the limit
Draw a rough graph of the functiony = y(x) = e−|x|.Is the function differentiable at x = 0? Draw a rough graph of the derivative of the function.
Find the limit In (x) lim X00
Investigate the limit
Find the value of the limit:If a limiting expression appears to approach 0 × ∝, it can be put into a form that appears to approach 0/0 or ∝/∝ by using the reciprocal of one factor. In addition the rule must sometimes be applied more than once in order to find the value of the limit. tan (x)
Find the following derivatives:(a) d(yz)/dx, where y = ax2, z = sin (bx).(b) dP/dV, where(c) dη/dλ, where nRT an? P = (V – nb) V2 *
Decide which of the following limits exists and find the values of those that do exist:(a) limx→π/2[x tan (x)].(b) limx→0[ln (x)].
Find the following derivatives and evaluate them at the points indicated:(a) (dy/dx)x=1, if y =(ax3 + bx2 + cx + 1)−1/2, where a,b, and c are constants.(b) (d2y/dx2)x=0, if y = ae−bx, where a and b are constants.
Find the inflection points for the function y = sin (x).
Find the following derivatives and evaluate them at the points indicated:(a) (dy/dx)x=0 if y = sin (bx), where b is a constant.(b) (d f /dt)t=0 if f = Ae−kt, where A and k are constants.
For the interval −10 ≤ x ≤ 10, find the maximum and minimum values of y = −1.000x3 + 3.000x2 − 3.000x + 8.000.
Find the second and third derivatives of the following functions. Treat all symbols except for the specified independent variable as constants:(a) νrms = νrms(T) =√3RT/M.(b) P = P(V) = nRT/(V − nb) − an2/V2.
Find the curvature of the function y = cos (x) at x = 0 and at x = π/2.

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