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 binary mixture of mole fraction $z_1$ is flashed to conditions T and P. For one of the following, determine the equilibrium mole fractions $x_1$ and $y_1$ of the liquid and vapor phases formed, the molar fraction $\mathcal{V}$ of the vapor formed, and the fractional recovery $\mathcal{R}$ of species 1 in the vapor phase (defined as the ratio for species 1 of moles in the vapor to moles in the feed). Assume that Raoult’s law applies. (a) Acetone(1)/acetonitrile(2), $z_1$ = 0.75, T = 340 K, P = 115 kPa. (b) Benzene(1)/ethylbenzene(2), $z_1 = 0.50,$ $T = 100^\circ C,$ P = 0.75(atm). (c) Ethanol(1)/1-propanol(2), $z_1$ = 0.25, T = 360 K, P = 0.80(atm).  (d) 1-Chlorobutane(1)/chlorobenzene(2), $z_1$ = 0.50, $T = 125^\circ C,$ P = 1.75 bar

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?

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.

Direct measurements of the atmospheric carbon dioxide $\left(\mathrm{CO}_2\right)$ concentration have been made since about $1959$, when the measured concentration was about $316$ parts per million. In $2017$, the concentration was about $407$ parts per million. Assume that this growth can be modeled with an exponential function of the form $Q=Q_0 \times(1+r)^t$. a. By experimenting with various values of the fractional growth rate $r$, find an exponential function that fits the given data for $1959$ and $2017$. b. Use this exponential model to predict when the $\mathrm{CO}_2$ concentration will be $560$ parts per million (twice its preindustrial level). c. Research recent trends in the carbon dioxide concentration. Does your model seem to fit, overestimate, or underestimate its recent rise? Explain.

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) $\frac{d^{1 / 2}}{d t^{1 / 2}}\left(a t^{2}+b t+c\right)$

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. $a. 10 ^ { 2 / 3 }$ $b. ( \sqrt [ 3 ] { 9 } ) ^ { 4 }$ $c. \sqrt [ 5 ] { x ^ { 3 } }$ $d. ( \sqrt { 2 } ) ^ { 5 }$ $e. 5 ^ { 7 / 2 }$ $f. y ^ { 3 / 3 }$

Conditions in the ion source of a mass spectrometer can be adjusted to allow formation of multiply ionized atoms. (a) Rewrite the above equation for the case of an $n$ times ionized atom, and solve for $R$. (b) Taking the masses of ${ }^4 \mathrm{He},{ }^{12} \mathrm{C}$, and ${ }^{16} \mathrm{O}$ to be $4 \mathrm{u}, 12 \mathrm{u}$, and $16 \mathrm{u}$, respectively, write down expressions for the radii, $R$, for $\mathrm{He}^{+}, \mathrm{C}^{3+}$, and $\mathrm{O}^{4+}$. (c) Using the actual masses of the atoms, find the fractional differences in these three radii (He versus $\mathrm{C}$ and $\mathrm{C}$ versus $\mathrm{O}$ ). The tiny differences in these radii can be measured very accurately and allow precise comparisons of the three masses involved (one of which, ${ }^{12} \mathrm{C}$, defines the atomic mass scale).

A particle of mass $m_{1}$ with an initial kinetic energy ${T}_{1}$ makes an elastic collision with a particle of mass $m_{2}$ initially at rest. $m_{1}$ is deflected from its original direction with a kinetic energy $T_{1}^{\prime}$ through an angle ${\phi}_{1}$. Letting $\alpha=m_{2} / m_{1} \text { and } \gamma=\cos \phi_{1}$, show that the fractional kinetic energy lost by $m_{1}, \Delta T_{1} / T_{1}=\left(T_{1}-T_{1}^{\prime}\right) / T_{1}$, is given by $\frac{\Delta T_{1}}{T_{1}}=\frac{2}{1+\alpha}-\frac{2 \gamma}{(1+\alpha)^{2}}(\gamma+\sqrt{\alpha^{2}+\gamma^{2}-1})$

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?

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 liquid has a density $\rho$. (a) Show that the fractional change in density for a change in temperature $\Delta T$ is

$$\Delta \rho / \rho=-\beta \Delta T$$

(b) What does the negative sign signify? (c) Fresh water has a maximum density of $1.0000 \mathrm{g} / \mathrm{cm}^{3}$ at $4.0^{\circ} \mathrm{C}$. At $10.0^{\circ} \mathrm{C}$ its density is $0.9997 \mathrm{g} / \mathrm{cm}^{3}$. What is $\beta$ for water over this temperature interval? (d) At $0^{\circ} \mathrm{C},$ the density of water is $0.9999 \mathrm{g} / \mathrm{cm}^{3} .$ What is the value for $\beta$ over the temperature range $0^{\circ} \mathrm{C}$ to $4.00^{\circ} \mathrm{C}$?

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 $A _ { f } = A _ { i } \left( L _ { i } / L _ { f } \right).$

When two molecules of hydrogen combine with one molecule of oxygen to form two water molecules,

$$2 \mathrm{H}_2+\mathrm{O}_2 \longrightarrow 2 \mathrm{H}_2 \mathrm{O}$$

the energy released is $5 \mathrm{eV}$. (a) What is the mass difference between the three original molecules and the two final ones? (b) Given that the water molecule has mass about $3 \times 10^{-26} \mathrm{~kg}$, what is the fractional change in mass, $\Delta M /$ (total mass)? (Does it matter significantly whether you use the initial or final total mass?) (c) If one were to form $10 \mathrm{~g}$ of water by this process, what would be the total change in rest mass?