With a particular catalyst and at a given temperature, the oxidation of naphthalene to phthalic anhydride proceeds as follows:

$$\begin{matrix} \quad & \quad & \text{R}\\ \quad & 1 \nearrow & \text{2} & \searrow 3 & \quad & \text{4}\\ \text{A} & \quad & \rightarrow & \quad & \text{S} & \rightarrow & \text{T}\\ \end{matrix}$$

$$\begin{array}{ll}\mathrm{A}=\text { naphthalene } & k_{1}=0.21 \mathrm{s}^{-1} \\ \mathrm{R}=\text { naphthaquinone } & k_{2}=0.20 \mathrm{s}^{-1} \\ \mathrm{S}=\text { phthalic anhydride } & k_{3}=4.2 \mathrm{s}^{-1} \\ \mathrm{T}=\text { oxidation products } & k_{4}=0.004 \mathrm{s}^{-1}\end{array}$$

What reactor type gives the maximum yield of phthalic anhydride? Roughly estimate this yield and the fractional conversion of naphthalene which will give this yield. Note the word "roughly".

A pendulum clock is designed to tick off one second on each side-to-side swing of the pendulum (two ticks per complete period). $(a)$ Will a pendulum clock gain time in hot weather and lose it in cold, or the reverse? Explain your reasoning. $(b)$ A particular pendulum clock keeps correct time at $20.0^{\circ} \mathrm{C}$. The pendulum shaft is steel, and its mass can be ignored compared with that of the bob. What is the fractional change in the length of the shaft when it is cooled to $10.0^{\circ} \mathrm{C}$ ? $(c)$ How many seconds per day will the clock gain or lose at $10.0^{\circ} \mathrm{C}$ ? $(d)$ How closely must the temperature be controlled if the clock is not to gain or lose more than $1.00 \mathrm{~s}$ a day? Does the answer depend on the period of the pendulum?

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?

While on vacation, Janet traveled the full length of Interstate Highway 5 from the U.S.- Canadian border to the U.S.- Mexican border. She recorded the following travel between cities: Blaine, Washington, to Seattle: 1 hour 49 minute; Seattle to Portland: 2 hours 48 minutes; Portland to Sacramento: 9 hours 6 minutes; Sacramento to Los Angeles: 6 hours 3 minutes; Los Angeles to San Diego: 2 hours 9 minutes; San Diego to San Ysidro, California: 22 minutes. (a) Write each of these times as a fraction or as a mixed number with minutes represented as a fraction with a denominator of 60. Do not reduce the fractional part of the mixed number. (b) What is Janet's total driving time from Blaine, Washington, to San Ysidro, California? Give your answer as a mixed numb er and in terms of hours and minutes.

Harlen Industries has a simple forecasting model: Take the actual demand for the same month last year and divide that by the number of fractional weeks in that month. This gives the average weekly demand for that month. This weekly average is used as the weekly forecast for the same month this year. This technique was used to forecast eight weeks for this year, which are shown below along with the actual demand that occurred. The following eight weeks show the forecast (based on last year) and the demand that actually occurred. Compute the MAD of forecast errors.

a. Write a C++ function named fracpart () that returns the fractional part of any number passed to the function. For example, if the number 256.879 is passed to fracpart (), the number 879 should be returned. Have the fracpart () function call the whole () function you wrote in previous Exercise. The number returned can then be determined as the number passed to fracpart () less the returned value when the same argument is passed to whole(). b. Include the function written in previous Exercise in a working program. Make sure the function is called from main() and correctly returns a value to main (). Have main() use a cout statement to display the returned value. Test the function by passing various data to it. c. When you're confident the fracpart () function written for previous Exercise works correctly, save it in the same namespace and personal library selected for previous Exercise.

A. Akgerman and M. Zardkoohi (J. Chem. Eng. Data 41, 185 (1996)) examined the adsorption of phenol from aqueous solution on to fly ash at $20 ^ { \circ } \mathrm { C }$. They fitted their observations to a Freundlich isotherm of the form $c _ { \mathrm { ads } } = K c _ { \mathrm { sol } } ^ { 1 / n }$, where $c_{ads}$ is the concentration of adsorbed phenol and $c_{sol}$ is the concentration of aqueous phenol. Among the data reported are the following:

$$\begin{matrix} \ {c _ { \text { sol } } / \left( \mathrm { mg } \mathrm { g } ^ { - 1 } \right)} & \text{8.26} & \text{15.65} & \text{25.43} & \text{31.74} & \text{40.00}\\ \ {c _ { \mathrm { ads } } / \left( \mathrm { mg } \mathrm { g } ^ { - 1 } \right)} & \text{4.41} & \text{9.2} & \text{35.2} & \text{52.0} & \text{67.2}\\ \end{matrix}$$

Determine the constants K and n. What further information would be necessary in order to express the data in terms of fractional coverage, $\theta ?$

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?

The trade-in value of a car "depreciates" with time. The trade-in value of any particular make and model of car can be found in the "blue book" (which is often orange!) which is published each month. Suppose you own a car that is presently 20 months old, and whose trade-in value is $\$ 8000$. Five months ago, the trade-in value was $\$ 9000$.
a. Let $t$ be the number of months old the car is. Let $f(t)$ be the predicted trade-in value if a linear function is assumed. Let $g(t)$ and $h(t)$ be the predicted trade-in value if exponential and inverse (fractional power) variation functions are assumed, respectively. Write particular equations for $f(t), g(t)$, and $h(t)$.
b. Use each of the three functions to predict the trade-in value when the car is 60 months old. (There may be surprises!)
c. Use each of the three functions to predict the trade-in value when the car was new. (There may be other surprises!!)
d. Based on your answers to parts (b) and (c), which of the three kinds of function is most reasonable for predicting the trade-in value over a long period of time?

In the process of distillation, a mixture of two (or more) volatile liquids is first heated to convert the volatile materials to the vapor state. Then the vapor is condensed, reforming the liquid. The net result of this liquid n vapor n liquid conversion is to enrich the fraction of a more volatile component in the mixture in the condensed vapor. We can describe how this occurs using Raoult’s law. Imagine that you have a mixture of 12% (by weight) ethanol and water (as formed, for example, by fermentation of grapes).

(a) What are the mole fractions of ethanol and water in this mixture?

(b) This mixture is heated to 78.5 $^o$C (the normal boiling point of ethanol). What are the equilibrium vapor pressures of ethanol and water at this temperature, assuming Raoult’s law (ideal) behavior? (You will need to derive the equilibrium vapor pressure of water at 78.5 $^o$C from data in Appendix G.)

(c) What are the mole fractions of ethanol and water in the vapor?

(d) After this vapor is condensed to a liquid, to what extent has the mole fraction of ethanol been enriched? What is the mass fraction of ethanol in the condensed vapor?

The property $\left( k ^ { m } \right) ^ { n } = k ^ { m n }$ can help you rewrite expressions with roots and fractional exponents. For example, since $16 ^ { 3 / 2 } = \left( 16 ^ { 3 } \right) ^ { 1 / 2 }, 16 ^ { 3 / 2 }$ can be rewritten as $\sqrt { 16 ^ { 3 } }$ However, since $16 ^ { 3 / 2 } = \left( 16 ^ { 1 / 2 } \right) ^ { 3 }, 16 ^ { 3 / 2 }$ can also be rewritten as $( \sqrt { 16 } ) ^ { 3 } \text { or } 4 ^ { 3 }$ With your team, find ways to rewrite the expressions below two different ways. Be ready to justify your answers. $a. 10 ^ { 2 / 3 }$ $b. ( \sqrt [ 3 ] { 9 } ) ^ { 4 }$ $c. \sqrt [ 5 ] { x ^ { 3 } }$ $d. ( \sqrt { 2 } ) ^ { 5 }$ $e. 5 ^ { 7 / 2 }$ $f. y ^ { 3 / 3 }$

(Program) A Julian date is represented as the number of days from a known base date. The following pseudocode shows one algorithm for converting from a Gregorian date, in the form month/day/year, to a Julian date with a base date of 00/00/0000. All calculations in this algorithm use integer arithmetic, which means the fractional part of all divisions must be discarded. In this algorithm, M = month, D = day, and Y = year.

If M is less than or equal to 2 Set the variable MP = 0 and YP = Y - 1 Else Set MP = int(0.4 * M + 2.3) and YP = Y EndIf T = int(YP / 4) - int(YP / 100) + int(YP / 400) Julian date = 365 * Y + 31 * (M - 1) + D + T - MP

Using this algorithm, modify Program 12.4 to cast from a Gregorian Date object to its corresponding Julian representation as a long integer. Test your program by using the Gregorian dates 1/31/2012 and 3/16/2012, which correspond to the Julian dates 734898 and 734943.

In one form of plethysmograph (a device for measuring volume), a rubber capillary tube with an inside diameter of 1.00 mm is filled with mercury at $20 ^ { \circ } \mathrm { C }$. The resistance of the mercury is measured with the aid of electrodes sealed into the ends of the tube. If 100.00 cm of the tube is wound in a spiral around a patient’s upper arm, the blood flow during a heartbeat causes the arm to expand, stretching the tube to a length of 100.04 cm. From this observation, and assuming cylindrical symmetry, you can find the change in volume of the arm, which gives an indication of blood flow. Calculate the fractional change in resistance during the heartbeat. Take $\rho _ { \mathrm { Hg } } = 9.4 \times 10 ^ { - 7 } \Omega \cdot \mathrm { m }$. Hint: Because the cylindrical volume is constant, $V = A _ { i } L _ { i } = A _ { f } L _ { f }$ and $A _ { f } = A _ { i } \left( L _ { i } / L _ { f } \right).$

A liquid mixture containing 30.0 mole% benzene (B), 25.0% toluene (T), and the balance xylene (X) is fed to a distillation column. The bottoms product contains 98.0 mole% X and no B, and 96.0% of the X in the feed is recovered in this stream. The overhead product is fed to a second column. The overhead product from the second column contains 97.0% of the B in the feed to this column. The composition of this stream is 94.0 mole% B and the balance T. a) Draw and label a flowchart of this process and do the degree-of-freedom analysis to prove that for an assumed basis of calculation, molar flow rates and compositions of all process streams can be calculated from the given information. Write in order the equations you would solve to calculate unknown process variables. In each equation (or pair of simultaneous equations), circle the variable(s) for which you would solve. Do not do the calculations. b) Calculate (i) the percentage of the benzene in the process feed (i.e., the feed to the first column) that emerges in the overhead product from the second column and (ii) the percentage of toluene in the process feed that emerges in the bottom product from the second column.