Silicon used in computer chips must have an impurity level below $10^{-9}$ (that is, fewer than one impurity atom for every $10^9 \mathrm{Si}$ atoms). Silicon is prepared by the reduction of quartz $\left(\mathrm{SiO}_2\right)$ with coke (a form of carbon made by the destructive distillation of coal) at about $2000^{\circ} \mathrm{C}$ :

$$\mathrm{SiO}_2(\mathrm{~s})+2 \mathrm{C}(\mathrm{s}) \longrightarrow \mathrm{Si}(l)+2 \mathrm{CO}(\mathrm{g})$$

Next, solid silicon is separated from other solid impurities by treatment with hydrogen chloride at $350^{\circ} \mathrm{C}$ to form gaseous trichlorosilane $\left(\mathrm{SiCl}_3 \mathrm{H}\right)$ :

$$\mathrm{Si}(\mathrm{s})+3 \mathrm{HCl}(\mathrm{g}) \longrightarrow \mathrm{SiCl}_3 \mathrm{H}(\mathrm{g})+\mathrm{H}_2(\mathrm{~g})$$

Finally, ultrapure $\mathrm{Si}$ can be obtained by reversing the above reaction at $1000^{\circ} \mathrm{C}$ :

$$\mathrm{SiCl}_3 \mathrm{H}(\mathrm{g})+\mathrm{H}_2(\mathrm{~g}) \longrightarrow \mathrm{Si}(\mathrm{s})+3 \mathrm{HCl}(\mathrm{g})$$

What types of crystals do $\mathrm{Si}$ and $\mathrm{SiO} 2$ form?

Methane and oxygen react in the presence of a catalyst to form formaldehyde. In a parallel reaction, methane is oxidized to carbon dioxide and water:

$$\begin{array}{c}{\mathrm{CH}_{4}+\mathrm{O}_{2} \rightarrow \mathrm{HCHO}+\mathrm{H}_{2} \mathrm{O}} \\ {\mathrm{CH}_{4}+2 \mathrm{O}_{2} \rightarrow \mathrm{CO}_{2}+2 \mathrm{H}_{2} \mathrm{O}}\end{array}$$

The feed to the reactor contains equimolar amounts of methane and oxygen. Assume a basis of 100 mol feed/s. a) Draw and label a flowchart. Use a degree-of-freedom analysis based on extents of reaction to determine how many process variable values must be specified for the remaining variable values to be calculated. b) Use Equation 4.6-7 to derive expressions for the product stream component flow rates in terms of the two extents of reaction, $\xi_{1} \text { and } \xi_{2}$. c) The fractional conversion of methane is 0.900 and the fractional yield of formaldehyde is 0.855. Calculate the molar composition of the reactor output stream and the selectivity of formaldehyde production relative to carbon dioxide production. d) A classmate of yours makes the following observation: “If you add the stoichiometric equations for the two reactions, you get the balanced equation $2 \mathrm{CH}_{4}+3 \mathrm{O}_{2} \rightarrow \mathrm{HCHO}+\mathrm{CO}_{2}+3 \mathrm{H}_{2} \mathrm{O}$ The reactor output must therefore contain one mole of $C O_{2}$ for every mole of HCHO, so the selectivity of formaldehyde to carbon dioxide must be 1.0. Doing it the way the book said to do it, I got a different selectivity. Which way is right, and why is the other way wrong?” What is your response?

A distillation column is being used to separate methanol and water at atmospheric pressure. The column temperature varies from approximately $65^{\circ} \mathrm{C}$ at the top to $100^{\circ} \mathrm{C}$ at the bottom. Liquid enters the top of the column and flows down to the bottom; vapor is generated in a reboiler at the bottom of the column, flows upward, and leaves at the top. The molar flow rate of vapor up the column may be assumed to be constant from top to bottom. The vapor velocity is kept below 5.0 ft/s to keep the vapor from entraining liquid (suspending and carrying away liquid droplets). a) Where in the column is the greatest risk of liquid entrainment? Explain your answer. b) Assuming that the liquid flowing down the column and the column internals (equipment inside the column) occupy a negligible fraction of the column cross-sectional area, estimate the minimum column diameter if the vapor flow rate is 25.0 lb-mole/min. c) Suppose the column is constructed with a diameter 10% greater than that determined in Part (b). What are the vapor velocities at the top and bottom of the column if the vapor molar flow rate in both locations is 25.0 lb-mole/min? How much can the vapor molar flow rate be increased without causing liquid entrainment? d) There is a need to increase process throughput, which would require the vapor molar flow rate to be doubled. It has been suggested that increasing the pressure in the column would allow that to be done without risking excessive liquid entrainment. Again applying a vapor velocity limit of 5 ft/s, what would the new pressure be?

Make a sketch to show each unit. Then use shading or some sort of marks to indicate the fractional part. Be sure that the thirds are clear. a. $\frac{2}{3}$ of the 6 children in the play were girls. b. $\frac{2}{3}$ of the 6 yards of cloth were needed for the play. c. $\frac{2}{3}$ of the 6 cartons of ice cream were chocolate. d. $\frac{2}{3}$ of each of the 6 cartons of ice cream were eaten. e. $\frac{2}{3}$ of the 6 moving vans were painted red. f. $\frac{2}{3}$ of each of the 6 moving vans were painted red. g. The store had 6 cartons of paper towels and used $\frac{2}{3}$ of them. h. The store had 6 cartons of paper goods and used $\frac{2}{3}$ of each of them.

A particular talent competition has five judges, each of whom awards a score between 0 and 10 to each performer. Fractional scores, such as 8.3, are allowed. A performer’s final score is determined by dropping the highest and lowest score received, then averaging the three remaining scores. Write a program that uses this method to calculate a contestant’s score. It should include the following functions: - void getJudgeData () should ask the user for a judge’s score, store it in a reference parameter variable, and validate it. This function should be called by main once for each of the five judges. - void calcScore () should calculate and display the average of the three scores that remain after dropping the highest and lowest scores the performer received. This function should be called just once by main and should be passed the five scores. The last two functions, described below, should be called by calcScore, which uses the returned information to determine which of the scores to drop. - int findLowest () should find and return the lowest of the five scores passed to it. - int findHighest () should find and return the highest of the five scores passed to it. Input Validation: Do not accept judge scores lower than 0 or higher than 10.

Oakland passengers 2016 Part $\mathrm{I}$ of this exercise is pretty much as it appeared in the previous edition of this textbook:

The scatterplot below shows the number of passengers departing from Oakland (CA) airport month by month since the start of 1997. Time is shown as years since 1990, with fractional years used to represent each month. (Thus, June 1997 is $7.5$-halfway through the 7 th year after 1990.)

Dependent variable is Passengers

$$\begin{array}{lc}\text { R-squared }=71.1 \% & \mathrm{~s}=104330 \\ \text { Variable } & \text { Coefficient } \\ \text { Constant } & 282584 \\ \text { Year-1990 } & 59704.4\end{array}$$

a) Interpret the slope and intercept of the model.

b) What does the value of $R^2$ say about the model?

c) Interpret $s_e$ in this context.

d) Would you use this model to predict the numbers of passengers in 2010 (YearsSince $1990=20$ ) Explain.

e) There's a point near the middle of this time span with a large negative residual. Can you explain this outlier?

Part II We updated the data through 2016 for this edition, and here is the scatterplot for data through 2016:

f) If you had used the model for the data through 2006 to predict the number of passengers in 2010 , how well would it have worked? Why?

g) Explain how this shows the danger of extrapolating. What are possible reasons for the sudden decline in airline travel in about 2008 ?

The one-half integral (or semi-integral) of a function $f(t)$ is defined as the convolution $I_{1 / 2}(f)=\frac{1}{\sqrt{\pi}}\left(t^{-1 / 2} * f\right)$. Using the detinition of the one-half integral, we have the following definition: The one-half derivative (or semi-derivative) of a function $f(t)$ is defined by $\frac{d^{1 / 2}}{d t^{1 / 2}} f(t)=\frac{d}{d t} I_{1 / 2}(f)$. As with whole derivatives, fractional derivatives may or may not exist. Find the following one-half derivatives and compare them with first derivatives. (a) $\frac{d^{1 / 2}}{d t^{1 / 2}}(1) \quad$ (b) $\frac{d^{1 / 2}}{d t^{1 / 2}}(t) \quad$ (c) $\frac{d^{1 / 2}}{d t^{1 / 2}}\left(a t^{2}+b t+c\right)$

A vapor mixture of n-butane (B) and n-hexane (H) contains 50.0 mole% butane at $120^{\circ} \mathrm{C}$ and 1.0 atm. A stream of this mixture flowing at a rate of 150.0 L/s is cooled and compressed, causing some but not all of the vapor to condense. (Treat this process as a single-unit operation.) Liquid and vapor product streams emerge from the process in equilibrium at $T\left(^{\circ} \mathrm{C}\right)$ and 110 mm Hg. The vapor product contains 60.0 mole% butane. a) Draw and label a flowchart. Perform a degree-of-freedom analysis to show that you have enough information to determine the required final temperature …T†, the composition of the liquid product (component mole fractions), and the molar flow rates of the liquid and vapor products from the given information and Antoine expressions for the vapor pressures $p_{\mathrm{B}}^{*}(T)$ and $p_{\mathrm{H}}^{*}(T)$. Just identify the equations—for example, mole balance on butane or Raoult’s law for hexane—but don’t write them yet. b) Write in order the equations you would use to determine the quantities listed in Part (a) and also the fractional condensation of hexane (mol H condensed/mol H fed). In each equation, circle the variable for which you would solve. Do no algebra or calculations. c) Complete the calculations either manually or with an equation-solving program. d) State three assumptions you made that could lead to errors in the calculated quantities.

Now that you have an efficient way to raise a complex number to an integer power, how can you raise a complex number to a fractional power? Start with a simple case, $z^{2}=i$. The goal is to determine the values of z that make the equation a true statement. These values are the two square roots of i, or $i^{1/2}$. a. Write i in polar form. b. Use De Moivre's Theorem to show that $i^{n}$ can be written as $\cos \left(\frac{n \pi}{2}\right)+i \sin \left(\frac{n \pi}{2}\right)$. c. To solve for z, each side of the equation is raised to the $\frac{1}{2}$ power. Use part (b) to write $z=i^{1 / 2}$ in polar form. d. Write $z=i^{1 / 2}$ in rectangular form. Algebraically confirm that $z^{2}=i$. e. The original equation has $z^{2}$, so there are two possible values for z. You determined one solution in part (c). Determine the other solution, then graph both solutions.

The "grad" is a unit of angle measure that is sometimes used in France. One grad is the measure of an angle that cuts off an arc that is $\frac{1}{400} \mathrm{th}$ of a circle's circumference. An angle that rotates a circle's circumference is 400 grads. (Recall that one degree is an angle with vertex at the origin that cuts off an are that is $\frac{1}{360} \mathrm{th}$ of a circle's circumference and every circle's circumference is 360 degrees.) a. Using measures of circumference and arc length, explain how to make a protractor that measures an angle's openness in grads. b. If the protractor has a radius of 4.3 inches, determine the arc length in inches of: i. grad ii. 10 grads iii. 200 grads iv. 400 grads. c. An angle measuring 50 grads subtends what percent (or fractional part) of a circle's circumference? Explain your answer. d. How many degrees are equivalent to 50 grads? 100 grads? 10.2 grads? e. Define a function that converts a number of grads to a number of degrees. Explain how you determined this function.

Continuing with the situation in the previous problem, the number of molecules in the right and left halves of a room containing $N$ molecules can be written as $N_{\mathrm{R}}=(N / 2)(1+\Delta), N_{\mathrm{L}}=(N / 2)(1-\Delta)$. For $N$ large, we expect the number of molecules in the two halves to be nearly equal so $\Delta \ll 1$. (a) Show that the probability of a fractional discrepancy of $2 \Delta$ more molecules on the right than on the left is given by

$$P(\Delta)=C e^{-\Delta^2 N}$$

where $C$ is a constant that does not depend on $\Delta$. [Hint: You will need to use Stirling's approximation, $\ln N ! \approx N \ln N-N$, and the relation $\ln (1+x) \approx x$, valid for $x \ll 1$. Begin by writing $P=N ! /\left(N_{R} ! N_{L} !\right)$ and take the In of both sides.] (b) For a room containing $10^{25}$ molecules, what is the ratio $P(\Delta=0.001) /$ $P(\Delta=0) ?$ (c) What values of $\Delta$ are likely to actually occur in a room?

Suppose that the Federal Reserve (the Fed) buys $100 million of government debt obligations from private owners. This creates$100 million of new money and sets off a chain reaction because of the "fractional reserve" banking system. When the $100 million is deposited into private bank accounts, the banks keep only 15$(.85)^2(100)$million additional dollars. This process may be repeated indefinitely. Compute the total amount of new money that can be created theoretically by this process, beyond the original$100 million. (In practice, only about an additional $300 million is created, usually within a few weeks of the action of the Fed.)

To improve the chance that the neutrons produced in fission will themselves induce further fissions, it is important to slow them down. A moderator is a substance with which neutrons can collide and lose energy. It must be chosen so that the neutrons lose as much energy as possible (but are not actually captured). In practice the best process for reducing the neutrons' energy is an elastic collision. Consider a head-on elastic collision between a neutron (mass $\mathrm{m}$ ) and a nucleus (mass $M$ ). (a) Use conservation of energy and momentum (nonrelativistic) to derive an expression for the neutron's fractional loss of energy, $\Delta K / K$, as a function of the mass ratio $\mu=m / M$. (b) Show that the loss is maximum when the masses are equal $(\mu=1)$. (c) This result suggests that hydrogen - in water, for example - would be the best moderator. Unluckily, ordinary hydrogen, ${ }^1 \mathrm{H}$, has the disadvantage that it can capture neutrons readily to form deuterium. Two practical alternatives with masses at least fairly close to the neutron mass are deuterium (in "heavy" water) and carbon (in the form of graphite). What is $\Delta K / K$ for these two moderators?

A long-distance carrier charges the following rates for telephone calls:

Write a program that asks for the starting time and the number of minutes of the call, and displays the charges. The program should ask for the time to be entered as a floating point number in the form HH.MM. For example, 07:00 hours will be entered as 07.00, and 16:28 hours will be entered as 16.28.

Input Validation: The program should not accept times that are greater than 23:59. Also, no number whose last two digits are greater than S 9 should be accepted. Hint: Assuming nunt is a floating-point variable, the following expression will give you its fractional part:

num - static_cast<int>(num)