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mathematics physical chemistry

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

Manipulate the van der Waals equation into a cubic equation in Vm. That is, make a polynomial with terms proportional to powers of Vm up to V3m on one side of the equation.
Estimate the number of house painters in Chicago.
Find the four fourth roots of −1.
Find the square roots of x = 4.00 + 4.00i. Sketch an Argand diagram and locate the roots on it.
If z = (3.00 + 2.00i)2, find R(z),I (z),r, and ϕ.
Write a complex number in the form x + iy and show that the product of the number with its complex conjugate is real and nonnegative.
Find the complex conjugates of(a) A = (x + iy)2 − 4eixy.(b) B = (3 + 7i )3 − (7i )2.
Express the following complex numbers in the form x + iy:(a) z = e3πi/2.(b) z = 3eπi/2.
Express the following complex numbers in the form r eiϕ:(a) 4.00 + 4.00i.(b) −1.00.
Simplify the expression(4 + 6i )(3 + 2i ) + 4i.
Find the cylindrical polar coordinates of the point whose spherical polar coordinates are r = 3.00, θ = 30.00◦ , ϕ = 45.00◦.
Find the cylindrical polar coordinates of the point whose Cartesian coordinates are (−2.000,−2.000, 3.000).
Find the Cartesian coordinates of the point whose cylindrical polar coordinates are ρ = 25.00, ϕ = 60.0◦, z = 17.50
Find the spherical polar coordinates of the point whose Cartesian coordinates are (2.00, 3.00, 4.00).
(a) Find x and y if ρ = 6.00 and ϕ = π/6 rad.(b) Find ρ and ϕ if x = 5.00 and y = 10.00.
Manipulate the van der Waals equation into an equation giving P as a function of T and Vm.
Write the following expression in a simpler form: (x² + 2x)? – x²(x – 2)² + 12x4 6x3 + 12x4 B =
Verify the trigonometric identitycos (2x) = 1 − 2 sin2 (x)for x = 0.50000 rad. Use as many digits as your calculator will display and check for round-off error.
Verify the trigonometric identity sin (x + y) = sin (x) cos (y) + cos (x) sin (y)for the angles x = 1.0000 rad, y = 2.00000 rad. Use as many digits as your calculator will display and check for round-off error.
Construct a graph of the two functions: 2 cosh (x) and ex for values of x from 0 to 3. At what minimum value of x do the two functions differ by less than 1%?
For the chain in the previous problem, find the force necessary so that the center of the chain is no more than 0.500 m lower than the ends of the chain.Previous ProblemIf the two ends of a completely flexible chain (one that requires no force to bend it) are suspended at the same height near the
If the two ends of a completely flexible chain (one that requires no force to bend it) are suspended at the same height near the surface of the earth, the curve representing the shape of the chain is called a catenary. It can be shown that the catenary is represented byy = a cosh (x/a),where a = T
Tell where each of the following functions is discontinuous. Specify the type of discontinuity:(a) cot (x).(b) sec (x).(c) ln (x − 1).
Tell where each of the following functions is discontinuous. Specify the type of discontinuity:(a) tan (x).(b) csc (x).(c) |x|.
Sketch rough graphs of the following functions. Verify your graphs using Excel or Mathematica:(a) x2e−x/2.(b) 1/x2.(c) (1 − x)e−x.(d) xe−x2 .
Express the following with the correct number of significant digits. Use the arguments in radians: tan (0.600) sin (0.100)(a) tan(0.600).(b) sin(0.100).(c) cosh(12.0).(d) sinh(10.0).
Find the value of the hyperbolic sine, cosine, and tangent for x = 0 and x = π/2. Compare these values with the values of the ordinary (circular) trigonometric functions for the same values of the independent variable.
A reactant in a first-order chemical reaction without back reaction has a concentration governed by the same formula as radioactive decay,where [A]0 is the concentration at time t = 0,[A]t is the concentration at time t, and k is a function of temperature called the rate constant. If k = 0.123
Using the data from the previous problem, construct a graph of the natural logarithm of the vapor pressure as a function of the reciprocal of the Kelvin temperature. Why might this graph be more useful than the graph in the previous problem?Previous problemTemperature (◦C)
The following is a set of data for the vapor pressure of ethanol taken by a physical chemistry student. Plot these points by hand on graph paper, with the temperature on the horizontal axis (the abscissa) and the vapor pressure on the vertical axis (the ordinate). Decide if there are any bad data
Sketch rough graphs of the following functions. Verify your graphs using Excel or Mathematica.(a) e−x/5 sin (x).(b) sin2 (x) =[sin (x)]2.
Determine the number of significant digits in sin (95.5◦).
Make a graph of tanh(x) and coth(x) on the same graph for values of x ranging from 0.1 to 3.0.
Sketch graphs of the arcsine function, the arccosine function, and the arctangent function. Include only the principal values.
For an angle that is nearly as large as π/2, find an approximate equality similar to Eq. (2.38) involving (π/2) − α, cos (α), and cot (α).
Calculate the following to the proper numbers of significant digits:(a) 17.13 + 14.6751 + 3.123 + 7.654 − 8.123.(b) ln (0.000123).
Find the pressure P of a gas obeying the ideal gas equationPV = nRT,if the volume V is 0.200 m3, the temperature T is 298.15 K, and the amount of gas n is 1.000 mol. Take the smallest and largest values of each variable and verify your number of significant digits. Note that since you are dividing
Round the following numbers to three significant digits:(a) 123456789.(b) 46.45.
Convert the following numbers to scientific notation:(a) 0.00000234.(b) 32.150.
Express the following in terms of SI base units. The electron volt (eV), a unit of energy, equals 1.6022 × 10−19 J:(a) 13.6 eV.(b) 24.17 mi.(c) 55 mi h−1.(d) 7.53 nm ps−1.
Take a few fractions, such as 2/3, 4/9, or 3/7, and represent them as decimal numbers, finding either all of the nonzero digits or the repeating pattern of digits.
Construct an accurate graph of sin (x) and tan (x) on the same graph for values of x from 0 to 0.4 rad and find themaximum value of x forwhich the two functions differ by less than 1%.
Using a calculator and displaying as many digits as possible, find the values of the sine and cosine of 49.500◦. Square the two values and add the results. See if there is any round-off error in your calculator.
Using a calculator, find the value of the cosine of 15.5◦ and the value of the cosine of 375.5◦. Display as many digits as your calculator is able to display. Check to see if your calculator produces any round-off error in the last digit. Choose another pair of angles that differ by 360◦ and
A reactant in a first-order chemical reaction without back reaction has a concentration governed by the same formula as radioactive decay,where [A]0 is the concentration at time t = 0, [A]t is the concentration at time t, and k is a function of temperature called the rate constant. If k = 0.123
For a positive value of b find an expression in terms of b for the change in x required for the function ebx to double in size.
Without using a calculator or a table of logarithms, find the following:(a) ln (100.000),(b) ln (0.0010000),(c) log10 (e)
Using a calculator or a spreadsheet, evaluate the quantity (1+ 1/n)n for several integral values of n ranging from 1 to 1,000,000. Notice how the value approaches the value of e as n increases and determine the value of n needed to provide four significant digits.
Generate the negative logarithms in the short table of common logarithms.
Use Excel or Mathematica to construct a graph representing the functiony(x) = x3 − 2x2 + 3x + 4.
Enter a formula into cell D2 that will compute the mean of the numbers in cells A2, B2, and C2.
The hectare is a unit of land area defined to equal exactly 10,000 m2, and the acre is a unit of land area defined so that 640 acre equals exactly one square mile. Find the number of square meters in 1.000 acre, and find the number of acres equivalent to 1.000 hectare.
Using the radius of the earth in the previous problem and the fact that the surface of the earth is about 70% covered by water, estimate the area of all of the bodies of water on the earth. The area of a sphere is equal to four times the area of a great circle, or 4πr2, where r is the radius of
The volume of a sphere is equal to 4/3π r3 where r is the radius of the sphere. Assume that the earth is spherical with a radius of 3958.89 miles. (This is the radius of a sphere with the same volume as the earth, which is flattened at the poles by about 30 miles.) Find the volume of the earth in
The volume of a cone is given bywhere h is the height of the cone and r is the radius of its base. Find the volume of a cone if its radius is given as 0.443 m and its height is given as 0.542 m. h, κπι
The specific heat capacity (specific heat) of a substance is crudely defined as the amount of heat required to raise the temperature of unit mass of the substance by 1 degree Celsius (1 ◦C). The specific heat capacity of water is 4.18 J ◦C−1g−1. Find the rise in temperature if 100.0 J of
The van der Waals equation of state gives better accuracy than the ideal gas equation of state. It iswhere a and b are parameters that have different values for different gases and where Vm = V/n, the molar volume. For carbon dioxide, a = 0.3640 Pa m6 mol–2, b = 4.267 × 10–5 m3 mol–1.
Some elementary chemistry textbooks give the value of R, the ideal gas constant, as 0.0821l atm K−1mol−1.(a) Obtain the value of R in l atm K−1 mol−1 to five significant digits.(b) Calculate the pressure in atmospheres and in (N m−2 Pa) of a sample of an ideal gas with n = 0.13678 mol, V
The value of an angle is given as 31◦. Find the measure of the angle in radians. Find the smallest and largest values that its sine and cosine might have and specify the sine and cosine to the appropriate number of digits.
The volume of a right circular cylinder is given byV = πr2h,where r is the radius and h is the height. If a right circular cylinder has a radius given as 0.134 m and a height given as 0.318 m, find its volume, specifying it with the correct number of digits. Calculate the smallest and largest
The volume of a sphere is given bywhere V is the volume and r is the radius. If a certain sphere has a radius given as 0.005250 m, find its volume, specifying it with the correct number of digits. Calculate the smallest and largest volumes that the sphere might have with the given information and
The Rankine temperature scale is defined so that the Rankine degree is the same size as the Fahrenheit degree, and absolute zero is 0 ◦R, the same as 0 K:(a) Find the Rankine temperature at 0.00 ◦C.(b) Find the Rankine temperature at 0.00 ◦F.
A light year is the distance traveled by light in one year:(a) Express this distance in meters and in kilometers. Use the average length of a year as described in the previous problem. How many significant digits can be given?(b) Express a light year in miles.Previous ProblemFind the average length
Find the average length of a century in seconds and in minutes. Use the rule that a year ending in 00 is not a leap year unless the year is divisible by 400, in which case it is a leap year. Therefore, in four centuries there will by 97 leap years. Find the number of minutes in a microcentury.
In the USA, footraces were once measured in yards and at one time, a time of 10.00 s for this distance was thought to be unattainable. The best runners now run 100 m in 10 s or less. Express 100 m in yards, assuming three significant digits. If a runner runs 100.0 m in 10.00 s, find his time for
A US gallon is defined as 231.00 cubic in.(a) Find the number of liters in one gallon.(b) The volume of 1.0000 mol of an ideal gas at 25.00 ◦C (298.15 K) and 1.0000 atm is 24.466 l. Express this volume in gallons and in cubic feet.
The distance by road from Memphis, Tennessee to Nashville, Tennessee is 206 mi. Express this distance in meters and in kilometers.
A furlong is exactly one-eighth of a mile and a fortnight is exactly 2 weeks. Find the speed of light in furlongs per fortnight, using the correct number of significant digits.
Find the speed of light in miles per hour.
Find the speed of light in miles per second.
Find the number of meters in 1.000 mile and the number of miles in 1.000 km, using the definition of the inch.
Find the number of inches in 1.000 m.

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