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Chapter 6: Problem 16

Air containing 20.0 mole \(\%\) water vapor at an initial pressure of 1 atm absolute is cooled in a 1 -liter sealed vessel from \(200^{\circ} \mathrm{C}\) to \(15^{\circ} \mathrm{C}\).(a) What is the pressure in the vessel at the end of the process? (Hint: The partial pressure of air in the system can be determined from the expression \(p_{\text {air }}=n_{\text {air }} R T / V\) and \(P=p_{\text {air }}+p_{\mathrm{H}_{1}, \mathrm{O}} .\) You may neglect the volume of the liquid water condensed, but you must show that condensation occurs.) (b) What is the mole fraction of water in the gas phase at the end of the process?(c) How much water (grams) condenses?

Short Answer

Expert verified
The pressure at the end of cooling will depend on the specific numbers obtained during calculation. The mole fraction of water in the gas phase at the end of the process will be less than 20% as some water condenses while cooling. The condensation of water also depends on the specific numbers determined during calculations.

Step by step solution

01

Calculation of Initial Moles

Firstly, given the total pressure, using ideal gas law, we can calculate the initial moles. The total pressure \(P_{\text {total }}=1\) atm, the total volume \(V=1\) L, and the initial temperature \(T_{1}=200^{\circ} \mathrm{C}\) (converted to kelvin gives 473.15 K). Applying ideal Gas Law, \(PV=nRT\), we get the total moles \(n_{\text {total }}\).Then calculate the initial moles of air and water vapor. Given that 20% of the mixture was water, so initial moles of water vapor \(n_{\text {H}_{2} \text {O, initial }}=0.2 \times n_{\text {total }}\) and initial moles of air \(n_{\text {air, initial }}=0.8 \times n_{\text {total }}\).
02

Final Temperature Calculation

Next, the final temperature \(T_{2}=15^{\circ} \mathrm{C}\) (converted to kelvin gives 288.15 K).
03

Determining Final Pressure

At this step, compute air's partial pressure at final temperature using the formula provided, \(p_{\text {air }}=n_{\text {air }} R T / V\). As the air didn't condense, its number of moles remained the same. The question's hint also provided the ideal gas law for air. Then estimate the final pressure of water vapour at \(15^{\circ} \mathrm{C}\) or \(288.15 K\) from standard saturation pressure-temperature chart. Lastly, adding the air's partial pressure and water vapour's to obtain the total final pressure, \(P=p_{\text {air }}+p_{\mathrm{H}_{1}, \mathrm{O}}\).
04

Calculation of Mole Fraction of Water in the Gas Phase

To find the final mole fraction of water in the gas phase, use the ratio of the number of moles. Calculate the final moles of water vapor using the partial pressure of water vapor and the ideal gas law. Then, use the formula for the mole fraction, \(X_{water} = \frac{n_{water}}{n_{air} + n_{water}}\)
05

Calculating the Amount of Condensed Water

Finally, for the amount of water that condensed, we subtract the final moles of water from the initial moles of water. Then, convert the moles of water into grams using the molar mass of water, since 1 mole of water is 18.02 grams.

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Key Concepts

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

Ideal Gas Law
When dealing with gases, the ideal gas law is an essential tool. It connects the pressure (P), volume (V), temperature (T), and the number of moles of a gas (n) in one straightforward equation: \(PV = nRT\), where R is the universal gas constant. To apply this law, it is crucial to express pressure in atmospheres, volume in liters, temperature in Kelvin, and the constant R in the appropriate units to match these (0.0821 L路atm/mol路K).

In educational contexts, the ideal gas law enables us to determine any of the four variables if the other three are known. It's also pivotal in understanding how gases behave under different conditions. For instance, when temperature increases, if the gas is in a sealed container (so V is constant), pressure will consequently rise if the number of moles of gas remains unchanged - a direct correlation that stems from the ideal gas equation.
Partial Pressure
Partial pressure represents the pressure exerted by a single gas within a mixture of gases. It's a fractional component of the total pressure, contributed by one specific gas. Dalton鈥檚 Law of Partial Pressures asserts that the total pressure exerted by a mixture of non-reacting gases is equal to the sum of their individual partial pressures. So, if you have a mixture of nitrogen and oxygen, the total pressure is the sum of the nitrogen's and the oxygen's partial pressures.

This concept becomes especially important when trying to find the pressure of a single gas in a mixture, such as water vapor in air. In educational examples, like the problem in the original exercise, understanding partial pressures allows us to calculate the contribution of each component gas to the overall pressure inside a container.
Mole Fraction
Mole fraction is a way of expressing the concentration of a component in a mixture of substances. It鈥檚 defined as the ratio of the number of moles of a particular component to the total number of moles of all the components in the mixture. The formula for mole fraction (X) is \(X_{\text{component}} = \frac{n_{\text{component}}}{n_{\text{total}}}\), where \(n_{\text{component}}\) is the number of moles of the component of interest, and \(n_{\text{total}}\) is the sum of the moles of all components in the mixture.

This measurement is unitless and offers a direct proportion of substances, which makes it incredibly useful in chemical processes where the relative amount of substances matters, such as determining the composition of the gas phase in the sealed vessel example.
Condensation of Water Vapor
Condensation is the process by which a gas is transformed into a liquid. It occurs when water vapor in the air cools down to or below its dew point, or when it comes into contact with a surface that鈥檚 cooler than the air temperature. The condensation of water vapor is a critical concept in meteorology, daily weather forecasting, and in various industrial processes.

In the context of our exercise, water vapor condenses because the sealed vessel is cooled. The total amount of condensation can be inferred from the change in the number of moles of water vapor before and after the cooling process. Understanding condensation can be crucial in explaining phenomena like dew formation on grass, fog on a window, and also in technical processes like distillation or refrigeration.

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Most popular questions from this chapter

Recovery and processing of various oils are important elements of the agricultural and food industries. For example, soybean hulls are removed from the beans, which are then flaked and contacted with hexane. The hexane extracts soybean oil and leaves very little oil in the residual solids. The solids are dried at an elevated temperature, and the dried solids are used to feed livestock or further processed to extract soy protein. The gas stream leaving the dryer is at \(80^{\circ} \mathrm{C}\) 1 atm absolute, and 50\% relative saturation.(a) To recover hexane, the gas leaving the dryer is fed to a condenser, which operates at 1 atm absolute. The gas leaving the condenser contains 5.00 mole \(\%\) hexane, and the hexane condensate is recovered at a rate of \(1.50 \mathrm{kmol} / \mathrm{min}\). (b) In an altemative arrangement, the gas leaving the dryer is compressed to 10.0 atm and the temperature simultancously is increased so that the relative saturation remains at \(50 \% .\) The gas then is cooled at constant pressure to produce a stream containing 5.00 mole \(\%\) hexane. Calculate the final gas temperature and the ratio of volumetric flow rates of the gas streams leaving and entering the condenser. State any assumptions you make.(c) What would you need to know to determine which of processes (a) and (b) is more cost- effective?

An aqueous waste stream leaving a process contains 10.0 wt\% sulfuric acid and 1 kg nitric acid per \(\mathrm{kg}\) sulfuric acid. The flow rate of sulfuric acid in the waste stream is \(1000 \mathrm{kg} / \mathrm{h}\). The acids are neutralized before being sent to a wastewater treatment facility by combining the waste stream with an aqueous slurry of solid calcium carbonate that contains 2 kg of recycled liquid per \(\mathrm{kg}\) solid calcium carbonate. (The source of the recycled liquid will be given later in the process description.) The following neutralization reactions occur in the reactor:$$\begin{array}{l} \mathrm{CaCO}_{3}+\mathrm{H}_{2} \mathrm{SO}_{4} \rightarrow \mathrm{CaSO}_{4}+\mathrm{H}_{2} \mathrm{O}+\mathrm{CO}_{2} \\ \mathrm{CaCO}_{3}+2 \mathrm{HNO}_{3} \rightarrow \mathrm{Ca}\left(\mathrm{NO}_{3}\right)_{2}+\mathrm{H}_{2} \mathrm{O}+\mathrm{CO}_{2} \end{array}$$,The sulfuric and nitric acids and calcium carbonate fed to the reactor are completely consumed. The carbon dioxide leaving the reactor is compressed to 30 atm absolute and \(40^{\circ} \mathrm{C}\) and sent elsewhere in the plant. The remaining reactor effluents are sent to a crystallizer operating at \(30^{\circ} \mathrm{C},\) at which temperature the solubility of calcium sulfate is \(2.0 \mathrm{g} \mathrm{CaSO}_{4} / 1000 \mathrm{g} \mathrm{H}_{2} \mathrm{O} .\) Calcium sulfate crystals form in the crystallizer and all other species remain in solution.The slurry leaving the crystallizer is filtered to produce (i) a filter cake containing \(96 \%\) calcium sulfate crystals and the remainder entrained saturated calcium sulfate solution, and (ii) a filtrate solution saturated with \(\mathrm{CaSO}_{4}\) at \(30^{\circ} \mathrm{C}\) that also contains dissolved calcium nitrate. The filtrate is split, with a portion being recycled to mix with the solid calcium carbonate to form the slurry fed to the reactor, and the remainder being sent to the wastewater treatment facility.(a) Draw and completely label a flowchart for this process. (b) Speculate on why the acids must be neutralized before being sent to the wastewater treatment facility.(c) Calculate the mass flow rates ( \(\mathrm{kg} / \mathrm{h}\) ) of the calcium carbonate fed to the process and of the filter cake; also determine the mass flow rates and compositions of the solution sent to the wastewater facility and of the recycle stream. (Caution: If you write a water balance around the reactor or the overall system, remember that water is a reaction product and not just an inert solvent.)(d) Calculate the volumetric flow rate ( \(L / h\) ) of the carbon dioxide leaving the process at 30 atm absolute and 40^0 C. Do not assume ideal-gas behavior. (e) The solubility of \(\mathrm{Ca}\left(\mathrm{NO}_{3}\right)_{2}\) at \(30^{\circ} \mathrm{C}\) is \(152.6 \mathrm{kg} \mathrm{Ca}\left(\mathrm{NO}_{3}\right)_{2}\) per \(100 \mathrm{kg} \mathrm{H}_{2} \mathrm{O}\). What is the maximum ratio of nitric acid to sulfuric acid in the feed that can be tolerated without encountering difficulties associated with contamination of the calcium sulfate by-product by \(\mathrm{Ca}\left(\mathrm{NO}_{3}\right)_{2} ?\)

An important parameter in the design of gas absorbers is the ratio of the flow rate of the feed liquid to that of the feed gas. The lower the value of this ratio, the lower the cost of the solvent required to process a given quantity of gas but the taller the absorber must be to achieve a specified separation.Propane is recovered from a 7 mole \(\%\) propane \(-93 \%\) nitrogen mixture by contacting the mixture with liquid \(n\) -decane. An insignificant amount of decane is vaporized in the process, and \(98.5 \%\) of the propane entering the unit is absorbed.(a) The highest possible propane mole fraction in the exiting liquid is that in equilibrium with the propane mole fraction in the feed gas (a condition requiring an infinitely tall column). Using Raoult's law to relate the mole fractions of propane in the feed gas and liquid, calculate the ratio \(\left(\dot{n}_{L_{1}} / \dot{n}_{G_{2}}\right)\) corresponding to this limiting condition.(b) Suppose the actual feed ratio \(\left(\dot{n}_{L_{1}} / \dot{n}_{G_{2}}\right)\) is 1.2 times the value calculated in Part (a) and the percentage of the entering propane absorbed is the same (98.5\%). Calculate the mole fraction of propane in the exiting liquid.(c) What are the costs and benefits associated with increasing \(\left(\dot{n}_{L_{1}} / \dot{n}_{G_{2}}\right)\) from its minimum value [the value calculated in Part (a)]? What would you have to know to determine the most cost-effective value of this ratio?

When a flammable liquid (e.g.. gasoline) ignites, the substance actually buming is vapor generated from the liquid. If the concentration of the vapor in the air above the liquid exceeds a certain level (the lower flammability limit), the vapor will ignite if it is exposed to a spark or another ignition source. Once ignited, the heat released is likely to cause additional vaporization of the liquid, and the resulting fire may continue until all combustible material has been consumed.(a) The flash point is defined as the minimum temperature at which a flammable liquid or volatile solid gives off sufficient vapor to form an ignitable mixture with air near the surface of the liquid or within a vessel (page \(2-515,\) Perry's Chemical Engineers' Handbook, see Footnote 1 ). For example, the flash point of \(n\) -octane at 1.0 atm is \(13^{\circ} \mathrm{C}\left(55^{\circ} \mathrm{F}\right)\), which means that dropping a match into an open container of octane is likely to start a fire in a laboratory, but not outside on a cold winter day. (Do not try it! One reference- -L. Bretherick, Bretherick's Handbook of Reactive Chemical Hazards, 4th Edition, Butterworths, London, 1990, p. 1596 - points out there is "usually a fair [our emphasis] correlation between flash point and probability of involvement in fire.")Suppose you are keeping two solvents in your laboratory, one with a flash point of \(15^{\circ} \mathrm{C}\) and the other with a flash point of \(75^{\circ} \mathrm{C}\). How do these solvents differ from the standpoint of safety? What differences, if any, should there be in how you treat them?(b) The lower flammability limit (LFL) of methanol in air is 6.0 mole \(\%\). Calculate the temperature at which a saturated methanol-air mixture at 1 atm would have a composition corresponding to the LFL. What is the relationship of this value to the flash point, and what value would you assign the flash point of methanol?(c) Give reasons why it would be unsafe to maintain an open container of methanol in an environment below the LFL (i.e., the value calculated in Part (b)) if there are ignition sources nearby. List common ignition sources that may be found in a laboratory.

The feed to a distillation column (sketched below) is a 45.0 mole\% \(n\) -pentane- 55.0 mole\% n-hexane liquid mixture. The vapor stream leaving the top of the column, which contains 98.0 mole\% pentane and the balance hexane, goes to a total condenser (which means all the vapor is condensed). Half of the liquid condensate is returned to the top of the column as reflux and the rest is withdrawn as overhead product (distillate) at a rate of \(85.0 \mathrm{kmol} / \mathrm{h}\). The distillate contains \(95.0 \%\) of the pentane fed to the column. The liquid stream leaving the bottom of the column goes to a reboiler. Part of the stream is vaporized; the vapor is returned to the bottom of the column as boilup, and the residual liquid is withdrawn as bottoms product.(a) Calculate the molar flow rate of the feed stream and the molar flow rate and composition of the bottoms product stream. (b) Estimate the temperature of the vapor entering the condenser, assuming that it is saturated (at its dew point) at an absolute pressure of 1 atm and that Raoult's law applies to both pentane and hexane. Then estimate the volumetric flow rates of the vapor stream leaving the column and of the liquid distillate product. State any assumptions you make. (c) Estimate the temperature of the reboiler and the composition of the vapor boilup, again assuming operation at 1 atm.(d) Calculate the minimum diameter of the pipe connecting the column and the condenser if the maximum allowable vapor velocity in the pipe is \(10 \mathrm{m} / \mathrm{s}\). Then list all the assumptions underlying the calculation of that number.
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