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The solubility coefficient of a gas may be defined as the number of cubic centimeters (STP) of the gas that dissolves in \(1 \mathrm{cm}^{3}\) of a solvent under a partial pressure of 1 atm. The solubility coefficient of \(\mathrm{CO}_{2}\) in water at \(20^{\circ} \mathrm{C}\) is \(0.0901 \mathrm{cm}^{3} \mathrm{CO}_{2}(\mathrm{STP}) / \mathrm{cm}^{3} \mathrm{H}_{2} \mathrm{O}(\mathrm{l})\). (a) Calculate the Henry's law constant in atm/mole fraction for \(\mathrm{CO}_{2}\) in \(\mathrm{H}_{2} \mathrm{O}\) at \(20^{\circ} \mathrm{C}\) from the given solubility coefficient. (b) How many grams of \(\mathrm{CO}_{2}\) can be dissolved in a \(12-\mathrm{oz}\) bottle of soda at \(20^{\circ} \mathrm{C}\) if the gas above the soda is pure \(\mathrm{CO}_{2}\) at a gauge pressure of 2.5 atm ( 1 liter \(=33.8\) fluid ounces)? Assume the liquid properties are those of water. (c) What volume would the dissolved \(C O_{2}\) occupy if it were released from solution at body temperature and pressure \(-37^{\circ} \mathrm{C}\) and 1 atm?

Short Answer

Expert verified
a) The Henry's Law constant will be calculated as per the conversion explained above. b) To determine the amount of \(CO_2\) dissolved, moles of \(CO_2\) will be calculated using Henry's law and then will be converted into mass with the molar mass. c) Finally, the volume that the dissolved \(CO_2\) would occupy if it were released, is to be calculated using the ideal gas law.

Step by step solution

01

Convert Solubility Coefficient to Henry's Constant

To convert the solubility coefficient \(0.0901 \, \mathrm{cm}^{3} \mathrm{CO}_{2} \, (\mathrm{STP}) / \mathrm{cm}^{3} \mathrm{H}_{2} \mathrm{O} (\mathrm{l})\) to Henry's constant in atm/mole fraction we first need to convert the volume of CO2 to moles at STP. At STP, 1 mole of a gas occupies 22.4 liters. \nSo, \(1 \, \mathrm{cm}^{3}=10^{-3} \, \mathrm{liters}\), therefore, \(0.0901 \, \mathrm{cm}^{3} = 0.0901 \times 10^{-3} \, \mathrm{moles}\) \nThen, Henry's law constant \(K_{H} = \frac{P}{X}\), where P is the partial pressure of the gas and X is the mole fraction. Here P is 1 atm and hence the mole fraction X becomes \(X= \frac{n_{\mathrm{CO2}}}{n_{\mathrm{CO2}} + n_{\mathrm{H2O}}}\) . Since, n_{\mathrm{H2O}} >> n_{\mathrm{CO2}} , X ~= n_{\mathrm{CO2}} and thus, \(K_{H} = 1 \, \mathrm{atm} / 0.0901 \times 10^{-3} \, \mathrm{moles}\)
02

Calculate the amount of dissolved CO2

For a 12-ounce bottle, the volume of water = 12/33.8 liters. At 2.5 atm, we can calculate the number of moles of CO2. Since the Henry's Law states that the amount of dissolved gas is directly proportional to its partial pressure in the gas phase, the amount of CO2 dissolved would be: \n\(n_{\mathrm{CO2}} = P/K_{H} = 2.5 \, \mathrm{atm} / K_{H}\) and the mass of CO2 would be \(m_{\mathrm{CO2}} = n_{\mathrm{CO2}} \times M_{\mathrm{CO2}}\) where \(M_{\mathrm{CO2}} = 44 \, \mathrm{g}\, / \, \mathrm{mole}\)
03

Calculate the volume of released CO2

To find the volume that the CO2 would occupy when released, we’d use the Ideal Gas Law equation \(PV = nRT\). We need to convert the temperature to Kelvin by adding 273 to the Celsius temperature. Thus, the volume is : \(V = nRT / P = n_{\mathrm{CO2}} \times R \times (37 + 273) \, K / 1 \, atm\), where R = 0.08206 L-atm/mol-K.

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Key Concepts

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

Solubility Coefficient
The solubility coefficient tells us how much gas can dissolve in a liquid at a specific pressure. It represents the volume of gas (in cubic centimeters, STP) that dissolves in one cubic centimeter of solvent at a partial pressure of 1 atm. In simpler terms, it measures how easily a gas mixes with a liquid. For example, at 20°C, the solubility coefficient of CO₂ in water is 0.0901 cm³ of CO₂ (STP) per cm³ of water.
Understanding this concept is important because it helps us predict the behavior of gases in liquids. When a gas has a high solubility coefficient, it means the gas can dissolve significantly in the liquid. This concept is used frequently in fields like chemistry and environmental science.
Ideal Gas Law
The Ideal Gas Law is a fundamental equation in chemistry that relates pressure, volume, temperature, and moles of a gas. It is expressed as: \(PV = nRT\), where \(P\) is the pressure, \(V\) is the volume, \(n\) is the number of moles, \(R\) is the gas constant (0.08206 L-atm/mol-K), and \(T\) is the temperature in Kelvin.
This equation helps predict how a gas will behave under different conditions or how it will change when any of these variables are altered. For instance, when calculating the volume that dissolved COâ‚‚ would occupy if released, the Ideal Gas Law allows us to account for the new conditions at body temperature by adjusting these variables to reflect the properties of gases.
Mole Fraction
The mole fraction is a way of expressing the concentration of a component in a mixture. It is defined as the ratio of the moles of one component to the total moles in the mixture. For a gas such as COâ‚‚ dissolving in water, the mole fraction \(X\) can be calculated using the formula: \(X = \frac{n_{\mathrm{CO2}}}{n_{\mathrm{CO2}} + n_{\mathrm{H2O}}}\).
Here, \(n_{\mathrm{CO2}}\) represents the moles of CO₂ and \(n_{\mathrm{H2O}}\) represents the moles of water. In most cases, because the moles of water are much larger compared to the moles of dissolved gas, the mole fraction simplifies to \(X \approx n_{\mathrm{CO2}}\). Understanding the mole fraction is crucial in calculating Henry’s law constant and predicting how much gas will dissolve under certain conditions.

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Most popular questions from this chapter

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.

A gas mixture containing 85.0 mole \(\% \mathrm{N}_{2}\) and the balance \(n\) -hexane flows through a pipe at a rate of \(100.0 \mathrm{m}^{3} / \mathrm{h} .\) The pressure is 2.00 atm absolute and the temperature is \(100^{\circ} \mathrm{C}\). (a) What is the molar flow rate of the gas in \(\mathrm{kmol} / \mathrm{h}\) ? (b) Is the gas saturated? If not, to what temperature ( \(^{C} C\) ) would it have to be cooled at constant pressure in order to begin condensing hexane? (c) To what temperature ( \(C\) ) would the gas have to be cooled at constant pressure in order to condense \(80 \%\) of the hexane?

Nitric acid is a chemical intermediate primarily used in the synthesis of ammonium nitrate, which is used in the manufacture of fertilizers. The acid also is important in the production of other nitrates and in the separation of metals from ores. Nitric acid may be produced by oxidizing ammonia to nitric oxide over a platinum-rhodium catalyst, then oxidizing the nitric oxide to nitrogen dioxide in a separate unit where it is absorbed in water to form an aqueous solution of nitric acid.The reaction sequence is as follows:$$\begin{aligned} 4 \mathrm{NH}_{3}+5 \mathrm{O}_{2} & \rightarrow 4 \mathrm{NO}+6 \mathrm{H}_{2} \mathrm{O} \\\4 \mathrm{NO}+2 \mathrm{O}_{2} & \rightarrow 4 \mathrm{NO}_{2} \\\4 \mathrm{NO}_{2}+2 \mathrm{H}_{2} \mathrm{O}(\mathrm{l})+\mathrm{O}_{2} & \rightarrow 4 \mathrm{HNO}_{3}(\mathrm{aq}) \end{aligned}$$.Ammonia vapor produced by vaporizing pure liquid ammonia at 820 kPa absolute is mixed with air, and the combined stream enters the ammonia oxidation unit. Air at \(30^{\circ} \mathrm{C}, 1\) atm absolute, and \(50 \%\) relative humidity is compressed and fed to the process. A fraction of the air is sent to the cooling and hydration units, while the remainder is passed through a heat exchanger and mixed with the ammonia. The total oxygen fed to the process is the amount stoichiometrically required to convert all of the ammonia to HNO \(_{3},\) while the fraction sent to the ammonia oxidizer corresponds to the stoichiometric amount required to convert ammonia to NO.The ammonia reacts completely in the oxidizer, with \(97 \%\) forming NO and the rest forming \(\mathrm{N}_{2}\). Only a negligible amount of \(\mathrm{NO}_{2}\) is formed in the oxidizer. However, the gas leaving the oxidizer is subjected to a series of cooling and hydration steps in which the NO is completely oxidized to \(\mathrm{NO}_{2}\) which in turn combines with water (some of which is present in the gas from the oxidizer and the rest is added) to form a 55 wt\% aqueous solution of nitric acid. The product gas from the process may be taken to contain only \(\mathrm{N}_{2}\) and \(\mathrm{O}_{2}\). (a) Taking a basis of \(100 \mathrm{kmol}\) of ammonia fed to the process, calculate (i) the volumes \(\left(\mathrm{m}^{3}\right)\) of the ammonia vapor and air fed to the process using the compressibility-factor equation of state; (ii) the amount (kmol) and composition (in mole fractions) of the gas leaving the oxidation unit; (iii) the required volume of liquid water \(\left(\mathrm{m}^{3}\right)\) that must be fed to the cooling and hydration units; and (iv) the fraction of the air fed to the ammonia oxidizer. (b) Scale the results from Part (a) to a new basis of 100 metric tons per hour of 55\% nitric acid solution.(c) Nitrogen oxides (collectively referred to as \(\mathrm{NO}_{x}\) ) are a category of pollutants that are formed in many ways, including processes like that described in this problem. List the annual emission rates of the three largest sources of \(\mathrm{NO}_{x}\) emissions in your home region. What are the effects of exposure to excessive concentrations of \(\mathrm{NO}_{x} ?\) (d) A platinum-rhodium catalyst is used in ammonia oxidation. Fxplain the function of the catalyst, describe its structure, and explain the relationship of the structure to the function.

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?

A fuel gas containing methane and ethane is burned with air in a furnace, producing a stack gas at \(300^{\circ} \mathrm{C}\) and \(105 \mathrm{kPa}\) (absolute). You analyze the stack gas and find that it contains no unburned hydrocarbons, oxygen, or carbon monoxide. You also determine the dew-point temperature.(a) Estimate the range of possible dew-point temperatures by determining the dew points when the feed is either pure methane or pure ethane. (b) Estimate the fraction of the feed that is methane if the measured dew- point temperature is \(59.5^{\circ} \mathrm{C}\). (c) What range of measured dew point temperatures would lead to calculated methane mole fractions within 5\% of the value determined in Part (b)?

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