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Find the number of inches in 1.000 m.

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 speed of light in miles per second.

Find the speed of light in miles per hour.

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.

The distance by road from Memphis, Tennessee to Nashville, Tennessee is 206 mi. Express this distance in meters and in kilometers.

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.

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 100 yd, assuming a constant speed.

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.

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 Problem**

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.

The volume of a sphere is given by

where 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 check your first answer for the volume.

The volume of a right circular cylinder is given by

V = πr^{2}h,

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 volumes that the cylinder might have with the given information and check your first answer for the volume.

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.

Some elementary chemistry textbooks give the value of R, the ideal gas constant, as 0.0821l atm K^{−1}mol^{−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 = 10.000 l, T = 298.15 K.

The van der Waals equation of state gives better accuracy than the ideal gas equation of state. It is

where a and b are parameters that have different values for different gases and where V_{m} = V/n, the molar volume. For carbon dioxide, a = 0.3640 Pa m^{6} mol^{âˆ’2}, b = 4.267 Ã— 10^{âˆ’5} m^{3} mol^{âˆ’1}. Calculate the pressure of carbon dioxide in pascals, assuming that n = 0.13678 mol, V = 10.00l, and T = 298.15 K. Convert your answer to atmospheres and torr.

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^{−1}g^{−1}. Find the rise in temperature if 100.0 J of heat is transferred to 1.000 kg of water.

The volume of a cone is given by

where 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.

The volume of a sphere is equal to 4/3π r^{3} 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 cubic miles and in cubic meters. Using a value of π with at least six digits give the correct number of significant digits in your answer.

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 4πr^{2}, where r is the radius of the sphere.

**Previous Problem**

^{3} 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 cubic miles and in cubic meters. Using a value of π with at least six digits give the correct number of significant digits in your answer.

The hectare is a unit of land area defined to equal exactly 10,000 m^{2}, 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.

Enter a formula into cell D2 that will compute the mean of the numbers in cells A2, B2, and C2.

Use Excel or Mathematica to construct a graph representing the function

y(x) = x^{3} − 2x^{2} + 3x + 4.

Generate the negative logarithms in the short table of common logarithms.

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.

Without using a calculator or a table of logarithms, find the following:

(a) ln (100.000),

(b) ln (0.0010000),

(c) log_{10} (e)

For a positive value of b find an expression in terms of b for the change in x required for the function e^{bx} to double in size.

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 s^{âˆ’1} find the time required for the concentration to drop to 21.0% of its initial value.

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 repeat the calculation. Set your calculator to use angles measured in radians. Find the value of sin (0.3000). Find the value of sin (0.3000 + 2π). See if there is any round-off error in the last digit.

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.

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%.

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.

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}.

Convert the following numbers to scientific notation:

(a) 0.00000234.

(b) 32.150.

Round the following numbers to three significant digits:

(a) 123456789.

(b) 46.45.

Find the pressure P of a gas obeying the ideal gas equation

PV = nRT,

if the volume V is 0.200 m^{3}, 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 by V the smallest value of the quotient will correspond to the largest value of V.

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).

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).

Sketch graphs of the arcsine function, the arccosine function, and the arctangent function. Include only the principal values.

Make a graph of tanh(x) and coth(x) on the same graph for values of x ranging from 0.1 to 3.0.

Determine the number of significant digits in sin (95.5^{◦}).

Sketch rough graphs of the following functions. Verify your graphs using Excel or Mathematica.

(a) e^{−x/5} sin (x).

(b) sin^{2} (x) =[sin (x)]^{2}.

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 points. Draw a smooth curve nearly through the points. Use Excel to construct another graph and notice how much work the spreadsheet saves you.

**Temperature ( ^{◦}C) .......................Vapor pressure (torr)**25.00 ..........................................................55.9

30.00...........................................................70.0

35.00 ..........................................................97.0

40.00 ........................................................117.5

45.00 ........................................................154.1

50.00 .......................................................190.7

55.00 .......................................................241.9

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 problem**

**Temperature ( ^{◦}C) .......................Vapor pressure (torr)**25.00 ..........................................................55.9

30.00...........................................................70.0

35.00 ..........................................................97.0

40.00 ........................................................117.5

45.00 ........................................................154.1

50.00 .......................................................190.7

55.00 .......................................................241.9

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 s^{âˆ’1} find the time required for the concentration to drop to 21.0% of its initial value.

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.

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).

Sketch rough graphs of the following functions. Verify your graphs using Excel or Mathematica:

(a) x^{2}e^{−x/2}.

(b) 1/x^{2}.

(c) (1 − x)^{e−x}.

(d) xe^{−x2} .

Tell where each of the following functions is discontinuous. Specify the type of discontinuity:

(a) tan (x).

(b) csc (x).

(c) |x|.

Tell where each of the following functions is discontinuous. Specify the type of discontinuity:

(a) cot (x).

(b) sec (x).

(c) ln (x − 1).

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 by

y = a cosh (x/a),

where a = T /gρ and where ρ is the mass per unit length, g is the acceleration due to gravity, and T is the tension force on the chain. The variable x is equal to zero at the center of the chain. Construct a graph of this function such that the distance between the two points of support is 10.0 m, the mass per unit length is 0.500 kg m^{−1}, and the tension force is 50.0 N.

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 Problem**

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 by

y = a cosh (x/a),

where a = T /gρ and where ρ is the mass per unit length, g is the acceleration due to gravity, and T is the tension force on the chain. The variable x is equal to zero at the center of the chain. Construct a graph of this function such that the distance between the two points of support is 10.0 m, the mass per unit length is 0.500 kg m^{−1}, and the tension force is 50.0 N.

Construct a graph of the two functions: 2 cosh (x) and e^{x} for values of x from 0 to 3. At what minimum value of x do the two functions differ by less than 1%?

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.

Verify the trigonometric identity

cos (2x) = 1 − 2 sin^{2} (x)

for x = 0.50000 rad. Use as many digits as your calculator will display and check for round-off error.

Write the following expression in a simpler form:

Manipulate the van der Waals equation into an equation giving P as a function of T and V_{m}.

(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.

Find the spherical polar coordinates of the point whose Cartesian coordinates are (2.00, 3.00, 4.00).

Find the Cartesian coordinates of the point whose cylindrical polar coordinates are ρ = 25.00, ϕ = 60.0^{◦}, z = 17.50

Find the cylindrical polar coordinates of the point whose Cartesian coordinates are (−2.000,−2.000, 3.000).

Find the cylindrical polar coordinates of the point whose spherical polar coordinates are r = 3.00, θ = 30.00^{◦} , ϕ = 45.00^{◦}.

Simplify the expression

(4 + 6i )(3 + 2i ) + 4i.

Express the following complex numbers in the form r e^{iϕ}:

(a) 4.00 + 4.00i.

(b) −1.00.

Express the following complex numbers in the form x + iy:

(a) z = e^{3πi/2}.

(b) z = 3e^{πi/2}.

Find the complex conjugates of

(a) A = (x + iy)^{2} − 4e^{ixy}.

(b) B = (3 + 7i )^{3} − (7i )^{2}.

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.

If z = (3.00 + 2.00i)^{2}, find R(z),I (z),r, and ϕ.

Find the square roots of x = 4.00 + 4.00i. Sketch an Argand diagram and locate the roots on it.

Find the four fourth roots of −1.

Estimate the number of house painters in Chicago.

Manipulate the van der Waals equation into a cubic equation in V_{m}. That is, make a polynomial with terms proportional to powers of V_{m} up to V^{3}_{m} on one side of the equation.

Find the value of the expression

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 the tree?

The equation x^{2 }+ y^{2 }+ z^{2} = c^{2}, 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?

Express the equation y = b, where b is a constant, in plane polar coordinates.

Express the equation y = mx + b, where m and b are constants, in plane polar coordinates.

Find the values of the plane polar coordinates that correspond to x = 3.00, y = 4.00.

A surface is represented in cylindrical polar coordinates by the equation z = ρ^{2}. Describe the shape of the surface.

The solutions to the SchrÃ¶dinger 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 Ïˆ_{nlm}.

(a)** **The Ïˆ_{210 }function is given by

where a_{0} = 0.529 Ã— 10^{âˆ’10} m is called the Bohr radius. Write this function in terms of Cartesian coordinates.

(b)** **The Ïˆ_{211} function is given by

Write an expression for the magnitude of this complex function.

(c) The Ïˆ_{211} function is sometimes called Ïˆ_{2p1}. Write expressions for the real and imaginary parts of the function, which are proportional to the related functions. which are called Ïˆ_{2px} and Ïˆ_{2py}.

Find the complex conjugate of the quantity e^{2.00i} + 3e^{iπ}.

Find the sum of 4e^{3i} and 5e^{2i}.

Find the difference 3.00e^{πi} − 2.00e^{2i}.

Find the three cube roots of 3.000 − 2.000i .

Find the four fourth roots of 3.000i.

Find the real and imaginary parts of (3.00 + i)^{3} + (6.00 + 5.00i)^{2}. Find z^{∗}.

if we

find R(z),I (z), r, and Ï•.

Obtain the famous formulas

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.

A gas has a molar volume of 20 l. Estimate the average distance between nearest-neighbor molecules.

Estimate the number of blades of grass in a lawn with an area of 1000 m^{2}.

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 the volume of water on the earth and estimate how many comets would have collided with the earth to supply this much water.

Find A − B if A = 2.00i + 3.00j and B = 1.00i + 3.00j − 1.00k.

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 = (0,2) and B = (2,0).

Find |A|if A = 3.00i + 4.00j − 5.00k.

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 · B if A = (1.00,1.00,1.00) and B = (2.00,2.00,2.00).

Find A × B if A = (0.00,1.00,2.00) and B = (2.00,1.00,0.00).

Find A × B if A = (1,1,1) and B = (2,2,2).

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