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Chapter 3: Problem 67

As will be discussed in detail in Chapter \(5,\) the ideal-gas equation of state relates absolute pressure, \(P(\mathrm{atm}) ;\) gas volume, \(V(\text { liters }) ;\) number of moles of gas, \(n(\mathrm{mol}) ;\) and absolute temperature, \(T(\mathrm{K}):\) $$P V=0.08206 n T$$ (a) Convert the equation to one relating \(P(\mathrm{psig}), V\left(\mathrm{ft}^{3}\right), n(\mathrm{lb}-\mathrm{mole}),\) and \(T\left(^{\circ} \mathbf{F}\right)\). (b) \(\mathrm{A} 30.0\) mole \(\%\) CO and 70.0 mole \(\% \mathrm{N}_{2}\) gas mixture is stored in a cylinder with a volume of \(3.5 \mathrm{ft}^{3}\) at a temperature of \(85^{\circ} \mathrm{F}\). The reading on a Bourdon gauge attached to the cylinder is 500 psi. Calculate the total amount of gas (lb- mole) and the mass of \(\mathrm{CO}\left(\mathrm{Ib}_{\mathrm{m}}\right)\) in the tank. (c) Approximately to what temperature \(\left(^{\circ} \mathrm{F}\right)\) would the cylinder have to be heated to increase the gas pressure to 3000 psig, the rated safety limit of the cylinder? (The estimate would only be approximate because the ideal gas equation of state would not be accurate at pressures this high.)

Short Answer

Expert verified
The ideal gas law allows us to find the total amount of gas present in the tank, the amount of CO in lb-mol, and the temperature that would take to heat the cylinder to increase the gas pressure to 3000 psig. The steps described here give detailed calculations of how to derive these values.

Step by step solution

01

Conversion to Given Units

First, convert the Ideal Gas Law into the units given in part (a). Pressure should be measured in psig, not atm. Volume should be measured in cubic feet, not litres. The number of moles needs to be in lb-mol, not mol. And finally, temperature should be measured in degrees Fahrenheit, not Kelvin. Following the appropriate conversions from metric to Imperial units, a final equation of \(P(\mathrm{psig}) V(\mathrm{ft}^{3}) =n(\mathrm{lb-mol}) R 460+T(F)\) could be obtained where R=10.73 psi ft鲁/lb-mol掳F.
02

Calculation of Total Amount of Gas

For part (b), the values for \(P, V, R\), and \(T\) are given. Substitute these values into the equation obtained from Step 1 and solve for \(n(\mathrm{lb-mol})\) to get the total amount of gas in the tank. Simply put, find the number of moles by substituting the values into the equation obtained in the previous step.
03

Calculation of Mass of CO

It鈥檚 given that the gas is a mixture of \(30.0\%\) CO and \(70.0\%\) N2 by mole percent. So, to find the mass of CO in the tank, take \(30\%\) of the total amount of gas obtained in step 2. Convert from lb-mol CO to lb_CO using the molecular weight of CO.
04

Calculation of Final Temperature for Given Pressure

For part (c), we want to know at what temperature the gas pressure would increase to 3000 psig. Given that the ideal gas law is slightly inaccurate at high pressures, we should use our ideal gas law equation with \(P\) set to 3000 psig, and solve for \(T\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Unit Conversion
Understanding unit conversion is crucial when working with the Ideal Gas Law, especially in exercises where variables need to be expressed in different units. The Ideal Gas Law is often provided using metric units, such as atmospheres (atm) for pressure, liters for volume, and Kelvin for temperature.

In some scenarios, you might need to convert these units to the Imperial system or vice versa. Here are some common conversions you might encounter:
  • Pressure: 1 atm equals 14.696 psi (pounds per square inch).
  • Volume: 1 liter is about 0.0353 cubic feet (ft鲁).
  • Moles: The unit lb-mol (pound-mole) can be used when converting the number of moles, with 1 lb-mol equal to 453.59237 moles.
  • Temperature: Convert from Kelvin to Fahrenheit by using the formula: \[ T(掳F) = T(K) \times \frac{9}{5} - 459.67 \]
This understanding is foundational for solving problems where data isn't initially provided in your preferred unit system. Careful and accurate unit conversion ensures that the resulting calculations are valid.
Gas Mixtures
In the context of the Ideal Gas Law, a gas mixture refers to a combination of different gases. For instance, if you have a tank that holds a mixture of 30% carbon monoxide (CO) and 70% nitrogen (N鈧�) by mole percentage, the characteristics and behavior of the mixture must be considered in calculations.
Gas mixtures can be treated as a single gas when working with the Ideal Gas Law, provided that percentages by moles are correctly applied. To determine the total number of moles of a gas in a mixture, you multiply the total moles by the percentage of each gas. For example:
  • Find the total gas moles first, then apply the percentage to each component.
  • If you need the mass of a component, like CO, convert the moles to mass using its molecular weight (in this case, the molecular weight of CO is approximately 28.01 g/mol).
This allows for accurate calculations of gas quantities in different conditions and mixtures. Managing and understanding these percentages is essential when dealing with gases in mixed states.
Pressure Calculation
Pressure plays a significant role when applying the Ideal Gas Law, especially when conditions change, such as temperature or container volume. Pressure calculations often involve changes from standard atmospheric pressure (atm) to pounds per square inch (psig) or other units in the Imperial system.
The formula derived from the Ideal Gas Law, expressed in given units, usually looks like this: \[ P(\mathrm{psig}) \times V(\mathrm{ft}^3) = n(\mathrm{lb-mol}) \times R \times (460+T(掳F)) \]In this equation, note:
  • The addition of 460 to the temperature represents the Rankine scale conversion, ensuring temperature calculations align correctly.
  • R refers to the gas constant, typically with a value of 10.73 psi ft鲁/lb-mol掳F when using these units.
Pressure calculations become essential when working fluctuations in pressure, such as determining the pressure inside a tank at a given temperature. Incorrect pressure calculations could lead to erroneous interpretations of gas behavior.
Temperature Calculation
Temperature is a core aspect of the Ideal Gas Law, as it directly affects pressure and volume. Temperature must often be converted from one scale to another to be consistent with unit requirements, such as degrees Fahrenheit or Kelvin.
In tasks involving temperature, it's common to see changes explained through concepts like heating or cooling a gas to change pressures. If you need to increase the pressure in a container to a specified safety limit, you must calculate the necessary temperature to reach that pressure using:\[ P(\mathrm{psig}) \times V(\mathrm{ft}^3) = n(\mathrm{lb-mol}) \times R \times (460+T(掳F)) \]For example, to find the temperature needed to increase the pressure from an initial value to a high-pressure safety limit:
  • Set P to the desired pressure (like 3000 psig).
  • Keep V, n, and R consistent with the initial conditions.
  • Solve for T (掳F) to find the new temperature needed.
This computation is vital for ensuring safe operations in pressure vessels and other applications where temperature and pressure are interdependent.

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Most popular questions from this chapter

Things were going smoothly at the Breaux Bridge Drug Co. pilot plant during the midnight to 8 a.m. shift until Ther猫se Lagniappe, the reactor operator, let the run instruction sheet get too close to the Coleman stove that was being used to heat water to prepare Lagniappe's bihourly cup of Community Coffee. What followed ended in a total loss of the run sheet, the coffee, and a substantial portion of the novel Lagniappe was writing. Remembering the less than enthusiastic reaction she got the last time she telephoned her supervisor in the middle of the night, Lagniappe decided to rely on her memory of the required flow-rate settings. The two liquids being fed to a stirred-tank reactor were circulostoic acid (CSA: \(M W=75, S G=0.90\) ) and flubitol (FB: \(M W=90, S G=0.75\) ). The product from the system was a popular over-the-counter drug that simultaneously cures high blood pressure and clumsiness. The molar ratio of the two feed streams had to be between 1.05 and 1.10 mol CSA/mol FB to keep the contents of the reactor from forming a solid plug. At the time of the accident, the flow rate of CSA was 45.8 L min. Lagniappe set the flow of flubitol to the value she thought had been in the run sheet: 55.2 L/min. Was she right? If not, how would she have been likely to learn of her mistake? (Note: The reactor was stainless steel, so she could not see the contents.)

In the manufacture of pharmaceuticals, most active pharmaceutical ingredients (APIs) are made in solution and then recovered by separation. Acetaminophen, a pain-killing drug commercially marketed as Tylenol", is synthesized in an aqueous solution and subsequently crystallized. The slurry of crystals is sent to a centrifuge from which two effluent streams emerge: ( 1 ) a wet cake containing 90.0 wt\% solid acetaminophen \((\mathrm{MW}=\) 151 g/mol) and 10.0 wt\% water (plus some acetaminophen and other dissolved substances, which we will neglect), and (2) a highly dilute aqueous solution of acetaminophen that is discharged from the process. The wet cake is fed to a dryer where the water is completely evaporated, leaving the residual acetaminophen solids bone dry. If the evaporated water were condensed, its volumetric flow rate would be \(50.0 \mathrm{Lh}\). Following is a flowchart of the process, which runs 24 h/day, 320 days/yr. A denotes acetaminophen. (a) Calculate the yearly production rate of solid acetaminophen (tonne/yr), using as few dimensional equations as possible. (b) A proposal has been made to subject the liquid solution leaving the centrifuge to further processing to recover more of the dissolved acetaminophen instead of disposing of the solution. On what would the decision depend?

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Drop-on-demand (DoD) technology is an emerging form of drug delivery in which a reservoir is filled with a solution of an active pharmaceutical ingredient (API) dissolved in a volatile liquid, and a device sprays nanometer-scale drops of the solution onto an edible substrate, such as a small strip the size of a stick of chewing gum. The liquid evaporates very rapidly, causing the API to crystallize on the substrate. The exact dose required by a patient can be administered based on the known concentration of the API in the reservoir and the volume of solution deposited on the sa1) For a man of \(245 \mathrm{lb}_{m}\) dosage is \(2.2 \mathrm{mL}=5.3 \cdot 10^{20}\) drops a2) For a child of \(65 ~ l b_{m}\) dosage is \(0.60 \mathrm{mL}=1.4 \cdot 10^{20}\) drops b) \(V_{\text {dose}}=\frac{D \cdot W_{p} \cdot 0.4536 \mathrm{kg} / \mathrm{lb}_{f}}{M W_{A P I} \cdot m_{s}} \cdot \frac{1000 \mathrm{mL}}{1 \mathrm{L}}\) c) \(\frac{A}{V}=\frac{3}{\pi},\) the nanometric drops have larger area, through which the water can evaporate faster.ubstrate, enabling greater dosage accuracy than can be provided by administering fractions of tablets. (a) A DoD device is charged with a 1.20 molar solution of ibuprofen (the API) in \(n\) -hexane. The molecular weight of ibuprofen is \(206.3 \mathrm{g} / \mathrm{mol} .\) If a prescribed dosage is \(5.0 \mathrm{mg}\) ibuprofen/kg patient weight, how many milliliters of solution should be sprayed for a 245 -pound man and a 65 -pound child? How many drops are in each dose, assuming that each drop is a sphere with a radius of \(1 \mathrm{nm} ?\) (b) The DoD device is to be automated, so that the operator enters a patient's body weight into a computer that determines the required solution volume and causes that volume to be sprayed on the substrate. Derive a formula for the volume, \(V_{\text {dare }}(\mathrm{mL}),\) in terms of the following variables: \(M_{\mathrm{s}}(\operatorname{mol} \mathrm{API} / \mathrm{L})=\) molarity of reservoir solution \(\mathrm{SG}_{\mathrm{s}}=\) specific gravity of reservoir solution \(\mathrm{MW}_{\mathrm{API}}(\mathrm{g} / \mathrm{mol})=\) molecular weight of \(\mathrm{API}\) \(D(\mathrm{mg} \mathrm{APl} / \mathrm{kg} \text { body weight })=\) prescribed dosage \(W_{\mathrm{p}}\left(\mathrm{b}_{\mathrm{f}}\right)=\) patient's weight Check your formula by verifying your solution to Part (a). (c) Calculate the surface-to-volume ratio of a sphere of radius \(r .\) Then calculate the total drop surface area of \(1 \mathrm{mL}\left(=1 \mathrm{cm}^{3}\right)\) of the solution if it were sprayed as drops of (i) radius \(1 \mathrm{nm}\) and (ii) \(1 \mathrm{mm}\). Speculate on the likely reason for spraying nanoscale drops instead of much larger drops.

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