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

Find the first and second derivatives of the following functions:(a) y = 3 sin2 (2x) = 3 sin (2x)2.(b) y = a0 + a1x + a2x2 + a3x3 + a4x4 + a5x5, where a0, a1, and so on are constants.(c) y = a cos (e−bx), where a and b are constants.
Find the second and third derivatives of the following functions. Treat all symbols except for the specified independent variable as constants:(a) y = y(x) = axn.(b) y = y(x) = aebx .
Find the first and second derivatives of the following functions:(a) y = (1/x)(1/(1 + x)).(b) f = f(v) = ce−mv2/(2kT) where m, c, k, and T are constants.
Carry out Newton’s method by hand to find the smallest positive root of the equation1.000x2 − 5.000x + 1.000 = 0.
Find the first and second derivatives of the following functions:(a) y = y(x) = 3x3 ln (x).(b) y = y(x) = 1/(c − x2), where c is a constant.(c) y = y(x) = ce−a cos (bx), where a, b, and c are constants.
Find the following derivatives. All letters stand for constants except for the dependent and independent variables indicated:a. dy/dx, where y = (ax2 + bx + c)−3/2.b. d ln (P)/dT, where P = ke−Q/T.c. dy/dx, where y = a cos (bx3).
Find the first and second derivatives of the following functions:(a) P = P(Vm) = RT(1/Vm + B/V2m +C/V3m) where R,B, and C are constants.(b) G = G(x) = G◦ + RTx ln (x) + RT(1 − x) ln (1 − x), where G◦, R, and T are constants.(c) y = y(x) = a ln (x1/3), where a is a constant.
Assume that y = 3.00x2 − 4.00x + 10.00. If x = 4.000 and Δx = 0.500, find the value of Δy using Eq. (6.10). Find the correct value of Δy.
Use the definition of the derivative to derive the formulawhere y and z are both functions of x. d(yz) dy y- +z dr dz dr
Decide where the following functions are differentiable:(a) y = 1/1−x·(b) y = x + 2√x.(c) y = tan (x).
The sine and cosine functions are represented by the two seriesDifferentiate each series to show thatand x3 sin (x) = x - 3! x7 5! 7! x2 + 2! x4 46 cos (x) = 1- 4! 6!
Using graph paper plot the curve representing y = sin (x) for values of x lying between 0 and π/2 radians. Using a ruler, draw the tangent line at x = π/4. By drawing a right triangle on your graph and measuring its sides, find the slope of the tangent line.
Use Mathematica to solve the simultaneous equations2x + 3y = 13, x − 4y = −10.
Determine whether the set of equations has a nontrivial solution, and find the solution if it exists5x + 12y = 0,15x + 36y = 0.
Solve the set of equations3x + 2y = 40,2x − y = 10.
Solve the simultaneous equations by the method of substitution:x2 − 2xy − x = 0,x + y = 0.
Use the NSolve statement in Mathematica to find the numerical values of the roots of the equationx3 + 5.000x − 42.00 = 0.Use the FindRoot statement to find the real root of the same equation.
Verify the real solutions in the preceding example by substituting them into the equation.
In the study of the rate of the chemical reaction of substances A and B:aA + bB †’ products,the quotient occurs.where [A]0 and [B]0 are the initial concentrations of A and B, a and b are the stoichiometric coefficients of these reactants, and x is a variable specifying the extent to which
Write Mathematica expressions for the following:(a) The complex conjugate of (10)e2.657i.(b) ln (100!) − (100 ln (100) − 100).(c) The complex conjugate of (1 + 2i)2.5.
Use Excel to find the real root of the equationx3 + 5.000x − 42.00 = 0.
Use the method of trial and error to find the two positive roots of the equationex − 3.000x = 0to five significant digits. Begin by making a graph of the function to find the approximate locations of the roots.
Stirling's approximation for ln (N!) isDetermine the validity of this approximation and of the less accurate versionln (N!) ≈ N ln (N) − Nfor several values of N up to N = 100. Use a calculator, Excel, or Mathematica. In (N!) 1 In (27 N) + N In (N) – N. =
Solve the simultaneous equations by hand, using the method of substitution:x2 + x + 3y = 15,3x + 4y = 18.Use Mathematica to check your result. Since the first equation is a quadratic equation, there will be two solution sets.
In the theory of black-body radiation, the following equationx = 5(1 ˆ’ eˆ’x)needs to be solved to find the wavelength of maximum spectral radiant emittance. The variable x iswhere Î»max is the wavelength of maximum spectral radiant emittance, h is
Find the real roots of the equationx2 − 2.00 − cos (x) = 0.
Find two positive roots of the equationln (x) − 0.200x = 0.
Find the root of the equationx − 2.00 sin (x) = 0.
Find the real root of the equationx2 − e−x = 0.
Solve the cubic equation by trial and error, factoring, or by using Mathematica or Excel:x3 + x2 − 4x − 4 = 0.
Find the smallest positive root of the equation.sinh(x) − x2 − x = 0.
An approximate equation for the ionization of a weak acid, including consideration of the hydrogen ions from water is[H+]/c◦ =√Kac/c◦ + Kw,where c is the gross acid concentration. This equation is based on the assumption that the concentration of unionized acid is approximately equal to
The van der Waals equation of state iswhere a and b are temperature-independent parameters that have different values for each gas. For carbon dioxide, a = 0.3640 Pa m6 molˆ’2 and b = 4.267 Ã— 10ˆ’5 m3 molˆ’1.(a) Write this equation as a cubic equation in
Write an Excel worksheet that will convert a list of distance measurements in meters to miles, feet, and inches. If the length in meters is typed into a cell in column A, let the corresponding length in miles appear on the same line in column B, the length in feet in column C, and the length in
The following data were taken for the thermal decomposition of N2O3at a constant temperature:Using Excel, make three graphs: one with ln ([N2O3]) as a function of t, one with 1/[N2O3] as a function of t, and one with 1/[N2O3]2 as a function of t. Determine which graph is most nearly linear. If the
Using a graphical method, find the two positive roots of the following equation.ex − 3.000x = 0.
When expressed in terms of €œreduced variables,€ the van der Waals equation of state isUsing Excel, construct a graph containing three curves of Pr as a function of Vr: for the range 0.4 < Vr < 2: one for Tr = 0.6, one for Tr = 1, and one for Tr = 1.4. (*+*)(v --)- 3 P, +
Make a properly labeled graph of the function y(x) = ln (x) + cos (x) for values of x from 0 to 2π.Use Excel.
Find the real roots of the following equations by graphing:(a) x3 − x2 + x − 1 = 0.(b) e−x − 0.5x = 0.(c) sin (x)/x − 0.75 = 0.
The acid ionization constant of chloroacetic acid is equal to 1.40 × 10−3 at 25 ◦C. Assume that activity coefficients are equal to unity and find the hydrogenion concentration at the following stoichiometric molarities:(a) 0.100 mol l−1.(b) 0.0100 mol l−1.
The pH is defined for our present purposes aspH = - log10 ([H+]/c°).Find the pH of a solution formed from 0.075 mol of NH3 and enough water to make 1.00 l of solution. The ionization that occurs isNH3 + H2O ⇋ NH+4 + OH-.The equilibrium expression in terms of molar concentrations iswhere x(H2O)
Rewrite the factored quadratic equation (x − x1)(x − x2) = 0 in the form x2 − (x1 + x2)x + x12x = 0. Apply the quadratic formula to this version and show that the roots are x = x1 and x = x2.
Solve the following equations by factoring:(a) x3 + x2 − x − 1 = 0.(b) x4 − 1 = 0.
Solve the quadratic equations:(a) x2 − 3x + 2 = 0.(b) x2 − 1 = 0.(c) x2 + 2x + 2 = 0.
Solve the set of equations using Mathematica or by hand with the method of substitution:x2 − 2xy + y2 = 0,2x + 3y = 5.
Determine which, if any, of the following sets o equations is inconsistent or linearly dependent. Draw a graph for each set of equations, showing both equations. Find the solution for any set that has a unique solution.(a) x + 3y = 4,2x + 6y = 8.(b) 2x + 4y = 24,x + 2y = 8.(c) 3x1 + 4x2 = 10,4x1
The Dieterici equation of state isPea/Vm RT (Vm − b) = RT,where P is the pressure, T is the temperature, Vm is the molar volume, and R is the ideal gas constant. The constant parameters a and b have different values for different gases. For carbon dioxide, a = 0.468 Pa m6 mol−2, b = 4.63 ×
Using a graphical procedure, find the most positive real root of the quartic equation:x4 − 4.500x3 − 3.800x2 − 17.100x + 20.000 = 0.
A boy is swinging a weight around his head on a rope. Assume that the weight has a mass of 0.650 kg, that the rope plus the effective length of the boy’s arm has a length of 1.45 m, and that the weight makes a complete circuit in 1.34 s. Find the magnitude of the angular momentum, excluding the
The magnitude of the earth’s magnetic field ranges from 0.25 to 0.65 G (gauss). Assume that the average magnitude is equal to 0.45 G,which is equivalent to 0.000045 T. Find the magnitude of the force on the electron in the previous example due to the earth’s magnetic field, assuming that the
Show that the vector C is perpendicular to B.
From the definition, show thatA × B = −(B × A)
(a) Find the Cartesian components of the position vector whose spherical polar coordinates are r = 2.00, θ = 90◦, ϕ = 0◦. Call this vector A.(b) Find the scalar product of the vector A from part a and the vector B whose Cartesian components are (1.00,2.00,3.00).(c) Find the angle between
Find the magnitude of the vector A = (− 3.00,4.00,− 5.00).
If A = 2.00i − 3.00j and B = −1.00i + 4.00j(a) Find |A| and |B.(b) Find the components and the magnitude of 2.00A − B.(c) Find A · B.(d) Find the angle between A and B.
If A = (3.00)i−(4.00)j and B = (1.00)i+ (2.00)j.(a) Draw a vector diagram of the two vectors.(b) Find A · B and (2A) · (3B).
Let |A| = 4.00,|B| = 2.00, and let the angle between them equal 45.0◦. Find A · B.
Find A − B if A = (2.50,1.50) and B = (1.00, − 7.50).
Find approximately the smallest positive root of the equationtan (x) − x = 0.
Substitute the value of the molar volume obtained in the previous example and the given temperature into the Dieterici equation of state to calculate the pressure. Compare the calculated pressure with 10.00 atm, to check the validity of the linearization approximation used in the example.
Verify the prediction of the ideal gas equation of state given in the previous example.
Solve for the hydrogen-ion concentration in solutions of acetic acid with stoichiometric molarities equal to 0.00100 mol l−1. Use the method of successive approximations.
Carry out the algebraic manipulations to obtain the cubic equation in Eq. (5.9).
For hydrocyanic acid (HCN), Ka = 4.9 × 10−10 at 25 ◦C. Find [H+] if 0.1000 mol of hydrocyanic acid is dissolved in enough water to make 1.000 l. Assume that activity coefficients are equal to unity and neglect hydrogen ions from water.
Show by substitution that the quadratic formula provides the roots to a quadratic equation.
According to the Bohr theory of the hydrogen atom, the electron in the atom moves around the nucleus in one of various circular orbits with radius r = a0n2, where a0 is a distance equal to 5.29 × 10−11 m, called the Bohr radius, and n is a positive integer. The mass of the electron is 9.109 ×
The potential energy of a magnetic dipole in a magnetic field is given by the scalar productV = −μ · B,where B is the magnetic induction (magnetic field) and μ is the magnetic dipole. Make a graph of V/|μ||B| as a function of the angle between μ and B for values of the angle from 0◦ to
An object of mass 12.000 kg is moving in the x direction. It has a gravitational force acting on it equal to −mgk, where m is the mass of the object and g is the acceleration due to gravity, equal to 9.80 m s−1. There is a frictional force equal to (0.240 N)i. What is the magnitude and
An object has a force on it given by F = (4.75 N)i + (7.00 N)j + (3.50 N)k.(a) Find the magnitude of the force.(b) Find the projection of the force in the x-y plane. That is, find the vector in the x-y plane whose head is reached from the head of the force vector by moving in a direction
A spherical object falling in a fluid has three forces acting on it: (1) The gravitational force, whose magnitude is Fg = mg, where m is the mass of the object and g is the acceleration due to gravity, equal to 9.80 m s−2; (2) The buoyant force, whose magnitude is Fb = mfg, where mf is the mass
Find the angle between A and B if A = 3.00i + 2.00j + 1.00k and B = 1.00i + 2.00j + 3.00k.
Find the angle between A and B if A = 1.00i + 2.00j + 1.00k and B = 1.00i − 1.00k.
Find A × B if A = (1,1,1) and B = (2,2,2).
Find A × B if A = (0.00,1.00,2.00) and B = (2.00,1.00,0.00).
Find A · B if A = (1.00,1.00,1.00) and B = (2.00,2.00,2.00).
Find A · B if A = (1.00)i + (2.00)j + (3.00k and B = (1.00)i + (3.00)j − (2.00)k.
Find |A|if A = 3.00i + 4.00j − 5.00k.
Find A · B if A = (0,2) and B = (2,0).
An object of mass m = 10.0 kg near the surface of the earth has a horizontal force of 98.0 N acting on it in the eastward direction in addition to the gravitational force. Find the vector sum of the two forces (the resultant force).
Find A − B if A = 2.00i + 3.00j and B = 1.00i + 3.00j − 1.00k.
Since in its early history the earth was too hot for liquid water to exist, it has been the orized that all of the water on the earth came from collisions of comets with the earth. Assume an average diameter for the head of a comet and assume that it is completely composed of water ice. Estimate
Estimate the number of blades of grass in a lawn with an area of 1000 m2.
A gas has a molar volume of 20 l. Estimate the average distance between nearest-neighbor molecules.
Estimate the number of grains of sand on the beaches of the major continents of the earth. Exclude islands and inland bodies of water. You should come up with a number somewhere near Avogadro’s number.
Obtain the famous formulas eiφ ίφ 2i = I(e'®), sin (4) eid +e-io cos (ø) R(e®).
If wefind R(z),I (z), r, and ϕ. 3+2i 4 + 5i 2.
Find the real and imaginary parts of (3.00 + i)3 + (6.00 + 5.00i)2. Find z∗.
Find the four fourth roots of 3.000i.
Find the three cube roots of 3.000 − 2.000i .
Find the difference 3.00eπi − 2.00e2i.
Find the sum of 4e3i and 5e2i.
Find the complex conjugate of the quantity e2.00i + 3eiπ.
The solutions to the Schrödinger equation for the electron in a hydrogen atom have three quantum numbers associated with them, called n, l, and m, and these solutions are denoted by ψnlm. (a)?The ψ210?function is given by where a0 = 0.529 × 10??10 m is called the Bohr radius. Write this
A surface is represented in cylindrical polar coordinates by the equation z = ρ2. Describe the shape of the surface.
Find the values of the plane polar coordinates that correspond to x = 3.00, y = 4.00.
Express the equation y = mx + b, where m and b are constants, in plane polar coordinates.
Express the equation y = b, where b is a constant, in plane polar coordinates.
The equation x2 + y2 + z2 = c2, where c is a constant, represents a surface in three dimensions. Express the equation in spherical polar coordinates. What is the shape of the surface?
A Boy Scout finds a tall tree while hiking and wants to estimate its height. He walks away from the tree and finds that when he is 35 m from the tree, he must look upward at an angle of 32◦ to look at the top of the tree. His eye is 1.40 m from the ground, which is perfectly level. How tall is
Find the value of the expression 3 (2+ 4) – 6 (7+ |–17)* + (/37=|-1)* (1+22)* – (1–71 + 6³)² + /T2+|-4| 3 2

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

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