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Chapter 5: Problem 8

Use the ideal-gas equation of state to estimate the molar volume in \(\mathrm{m}^{3} / \mathrm{mol}\) and the density of air in \(\mathrm{kg} / \mathrm{m}^{3}\) at \(40^{\circ} \mathrm{C}\) and a gauge pressure of \(3.0 \mathrm{atm}\)

Short Answer

Expert verified
The molar volume of the air at given conditions is 0.0305 \(\mathrm{m}^{3} / \mathrm{mol}\) and the density is about 0.948 \(\mathrm{kg} / \mathrm{m}^{3}\).

Step by step solution

01

Convert units

First, convert all units into SI units (standard international). The pressure needs to be converted from atm to Pa, temperature from C to K, and molar mass from g/mol to kg/mol. Conversion factors are: 1 atm = 1.013x\(10^5\) Pa, \(T(K) = T(C) + 273.15\), and 1 g/mol=1x\(10^{-3}\) kg/mol.
02

Calculation of Molar Volume

Using the ideal gas law \(V = \frac{nRT}{P}\) we can find the volume. Here we take \(n = 1\) mol, \(R = 8.314 \, (\mathrm{m^{3} \cdot Pa} / \mathrm{K \cdot mol})\), \(P\) is the converted pressure, and \(T\) is converted temperature. Calculate the value of \(V\), which represents the molar volume of gas.
03

Calculation of Density

Now, using the formula for density, \(蟻 = \frac{M}{V}\), we can find the density of the gas. Here, \(V\) is the molar volume obtained in the previous step and \(M\) is the Molar Mass of Dry Air, which is approximately 28.97 g/mol or 28.97x\(10^{-3}\) kg/mol. Calculate the value of \(蟻\), which represents the density of the gas.

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Key Concepts

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

Molar Volume
In chemistry, molar volume is defined as the volume occupied by one mole of a substance. For gases, this can be calculated using the ideal gas law formula: \[ V = \frac{nRT}{P} \]Where:
  • \( V \) is the volume in cubic meters per mole (\( \mathrm{m}^3/\mathrm{mol} \))
  • \( n \) is the amount of substance, usually 1 mole
  • \( R \) is the ideal gas constant (8.314 \( \mathrm{m}^3 \cdot \mathrm{Pa} / \mathrm{K} \cdot \mathrm{mol} \))
  • \( T \) is the temperature in Kelvin
  • \( P \) is the pressure in Pascals.
To find the molar volume from the ideal gas law, you need to properly convert the temperature and pressure into Kelvin and Pascals, respectively. This step ensures accuracy in calculating the space occupied by a mole of gas at certain conditions. Molar volume can tell you a lot about how gases behave under different conditions, such as changes in temperature and pressure.
Density of Air
The density of air is an important factor in various scientific calculations. It tells us how much mass is contained in a certain volume of gas. Mathematically, density \( \rho \) is expressed as:\[ \rho = \frac{M}{V} \]Where:
  • \( \rho \) is the density
  • \( M \) is the molar mass of the substance (for air, approximately 28.97 g/mol or 28.97 \( \times 10^{-3} \) kg/mol)
  • \( V \) is the volume
In case of gases, density depends on the molar volume, which is influenced by temperature and pressure conditions as indicated by the ideal gas law.If you know the molar mass and have calculated the molar volume, you can find the density by dividing the molar mass by the molar volume. Changes in temperature and pressure can cause the density of air to vary significantly.
Unit Conversion
Unit conversion is a fundamental step in scientific calculations. It ensures that all variables are represented in consistent units, making equations compatible and results accurate.
  • Temperature: Convert Celsius to Kelvin using the formula: \( T(K) = T(掳C) + 273.15 \)
  • Pressure: Convert atmospheres to Pascals using the factor: 1 atm = 1.013 \( \times 10^5 \) Pa
  • Molar Mass: Convert grams per mole to kilograms per mole by realizing that 1 g/mol = \( 1 \times 10^{-3} \) kg/mol
These conversions ensure that the ideal gas law constants and measurements such as pressure and temperature are accurately applied. Consistent unit systems prevent errors and allow you to focus solely on the calculations.
Pressure and Temperature Calculations
Calculations involving pressure and temperature are crucial when applying the ideal gas law. The conditions under which a gas is examined directly affect both the molar volume and density.For pressure, converting from atmospheres (atm) to pascals (Pa) involves multiplying the pressure by the conversion factor: 1 atm = 1.013 \( \times 10^5 \) Pa.This allows the pressure measurement to be compatible with the other variables in the ideal gas equation.For temperature, converting to Kelvin from degrees Celsius ensures absolute temperature values are used in calculations, as the Kelvin scale starts at absolute zero: \( T(K) = T(掳C) + 273.15 \).These conversions and calculations ensure the correct application of the ideal gas law, helping to determine accurate values for physical properties such as molar volume and density. Understanding how to accurately calculate these values empowers you to analyze how gases behave under different scenarios.

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Most popular questions from this chapter

Steam reforming is an important technology for converting refined natural gas, which we take here to be methane, into a synthesis gas that can be used to produce a varicty of other chemical compounds. For example, consider a reformer to which natural gas and steam are fed in a ratio of 3.5 moles of steam per mole of methane. The reformer operates at 18 atm, and the reaction products leave the reformer in chemical equilibrium at \(875^{\circ} \mathrm{C}\). The steam reforming reaction is $$\mathrm{CH}_{4}+\mathrm{H}_{2} \mathrm{O} \rightleftharpoons \mathrm{CO}+3 \mathrm{H}_{2}$$ and the water-gas shift reaction also occurs in the reformer. $$\mathrm{CO}+\mathrm{H}_{2} \mathrm{O} \rightleftharpoons \mathrm{CO}_{2}+\mathrm{H}_{2}$$ The equilibrium constants for these two reactions are given by the expressions At \(875^{\circ} \mathrm{C}, K_{\mathrm{R}}=872.9 \mathrm{atm}^{2}\) and \(K \mathrm{w} \mathrm{G}=0.2482 .\) The process is to produce \(100.0 \mathrm{kmol} / \mathrm{h}\) of hydrogen. Calculate the feed rates (kmol/h) of methane and steam and the volumetric flow rate \(\left(\mathrm{m}^{3} / \mathrm{min}\right)\) of gas leaving the reformer.

The current global reliance on fossil fuels for heating, transportation, and electric power generation raises concems regarding the release of \(\mathrm{CO}_{2}\) and \(\mathrm{CH}_{4},\) which are greenhouse gases thought to lead to climate change, and NO, which contributes to smog. One potential solution to these problems is to produce transportation fuels from renewable biomass. You have been asked to evaluate a proposed process for converting forest residues to alcohols that may be used as transportation fuels. In the first stage of the process, steam and dry wood from hybrid poplar trees (which grow between five and eight feet a year and can be harvested roughly every five years) are fed to a gasifier in which the biomass is converted to light gases in the following reactions: $$\begin{aligned} \mathrm{C}+\mathrm{H}_{2} \mathrm{O} & \rightarrow \mathrm{CO}+\mathrm{H}_{2} \\\ \mathrm{CO}+\mathrm{H}_{2} \mathrm{O} & \rightarrow \mathrm{CO}_{2}+\mathrm{H}_{2} \\ \mathrm{C}+\mathrm{CO}_{2} & \rightarrow 2 \mathrm{CO} \\ \mathrm{C}+2 \mathrm{H}_{2} & \rightarrow \mathrm{CH}_{4} \\ \mathrm{CH}_{4}+\mathrm{H}_{2} \mathrm{O} & \rightarrow \mathrm{CO}+3 \mathrm{H}_{2} \end{aligned}$$ The effluents from the reactor are a gas stream containing \(\mathrm{H}_{2}, \mathrm{CO}, \mathrm{CO}_{2}, \mathrm{CH}_{4},\) and \(\mathrm{H}_{2} \mathrm{O},\) and a solid char stream that contains only carbon and hydrogen. The char is discarded and the gases go through additional steps in which the hydrogen and carbon monoxide are converted to mixed alcohols. This problem only concerns the gasifier. \(\cdot\) Elemental composition of biomass: 51.9 mass \(\%\) C \(, 6.3 \%\) H, and \(41.8 \%\) O \(\cdot\) Pressure and temperature of entering steam: \(155^{\circ} \mathrm{C}, 4.4 \mathrm{atm}\) \(\cdot\) Feed ratio of steam to biomass: 1.1 kg steam/kg biomass \(\cdot\) Yield and dry-basis composition of product gas: 1.35 kg dry gas/kg biomass at \(700^{\circ} \mathrm{C}, 1.2\) atm; 50.7 mol\% \(\mathrm{H}_{2}, 23.8 \%\) CO, \(18.0 \% \mathrm{CO}_{2}, 7.5 \% \mathrm{CH}_{4}\) (a) Taking a basis of \(100 \mathrm{kg}\) of biomass fed, draw and completely label a flowchart for the gasifier incorporating the given data, labeling the volumes of the steam fed and the gases produced. Perform a degree-of-freedom analysis. (b) Calculate the mass and mass composition of the char and the volumes of the steam feed and product gas streams. (c) List advantages and possible drawbacks of using biomass rather than petroleum as a fuel source.

Ammonia is one of the chemical constituents of industrial waste that must be removed in a treatment plant before the waste can safely be discharged into a river or estuary. Ammonia is normally present in wastewater as aqueous ammonium hydroxide \(\left(\mathrm{NH}_{4}^{+} \mathrm{OH}^{-}\right) .\) A two- part process is frequently carried out to accomplish the removal. Lime (CaO) is first added to the wastewater, leading to the reaction $$\mathrm{CaO}+\mathrm{H}_{2} \mathrm{O} \rightarrow \mathrm{Ca}^{2+}+2\left(\mathrm{OH}^{-}\right)$$ The hydroxide ions produced in this reaction drive the following reaction to the right, resulting in the conversion of ammonium ions to dissolved ammonia: $$\mathrm{NH}_{4}^{+}+\mathrm{OH}^{-}=\mathrm{NH}_{3}(\mathrm{g})+\mathrm{H}_{2} \mathrm{O}(\mathrm{l})$$ Air is then contacted with the wastewater, stripping out the ammonia. (a) One million gallons per day of alkaline wastewater containing 0.03 mole \(\mathrm{NH}_{3} /\) mole ammoniafree \(\mathrm{H}_{2} \mathrm{O}\) is fed to a stripping tower that operates at \(68^{\circ} \mathrm{F}\). Air at \(68^{\circ} \mathrm{F}\) and 21.3 psia contacts the wastewater countercurrently as it passes through the tower. The feed ratio is \(300 \mathrm{ft}^{3}\) air/gal wastewater, and 93\% of the ammonia is stripped from the wastewater. Calculate the volumetric flow rate of the gas leaving the tower and the partial pressure of ammonia in this gas. (b) Briefly explain in terms a first-year chemistry student could understand how this process works. Include the equilibrium constant for the second reaction in your explanation. (c) This problem is an illustration of challenges associated with addressing undesirable releases into the environment; namely, in developing a process to prevent dumping ammonia into a waterway, the release is instead made to the atmosphere. Suppose you are to write an article for a newspaper on the installation of the process described in the beginning of this problem. Explain why the company is installing the two-part process, and then explain the ultimate fate of the ammonia. Take one of two positions - either that the release is harmless or that it jeopardizes the environment in the vicinity of the plant. since this is a newspaper article, it cannot be more than 800 words.

The oxidation of nitric oxide $$\mathrm{NO}+\frac{1}{2} \mathrm{O}_{2} \rightleftharpoons \mathrm{NO}_{2}$$ takes place in an isothermal batch reactor. The reactor is charged with a mixture containing 20.0 volume percent NO and the balance air at an initial pressure of \(380 \mathrm{kPa}\) (absolute). (a) Assuming ideal-gas behavior, determine the composition of the mixture (component mole fractions) and the final pressure (kPa) if the conversion of NO is 90\%. (b) Suppose the pressure in the reactor eventually equilibrates (levels out) at \(360 \mathrm{kPa}\). What is the equilibrium percent conversion of NO? Calculate the reaction equilibrium constant at the prevailing temperature, \(K_{p}\left[(\mathrm{atm})^{-0.5}\right]\), defined as $$K_{p}=\frac{\left(p_{\mathrm{NO}_{2}}\right)}{\left(p_{\mathrm{NO}}\right)\left(p_{\mathrm{O}_{2}}\right)^{0.5}}$$ where \(p_{i}(\mathrm{atm})\) is the partial pressure of species \(i\left(\mathrm{NO}_{2}, \mathrm{NO}, \mathrm{O}_{2}\right)\) at equilibrium. (c) Assuming that \(K_{\mathrm{p}}\) depends only on temperature, estimate the final pressure and composition in the reactor if the feed ratio of NO to \(\mathrm{O}_{2}\) and the initial pressure are the same as in \(\operatorname{Part}(\) a), but the feed to the reactor is pure \(\mathrm{O}_{2}\) instead of air. (d) Replace the partial pressures in the expression for \(K_{\mathrm{p}}\), and use the result to explain how reactor pressure influences the conversion of NO to \(\mathrm{NO}_{2}\)

Liquid hydrazine ( \(\mathrm{SG}=0.82\) ) undergoes a family of decomposition reactions that can be represented by the stoichiometric expression $$3 \mathrm{N}_{2} \mathrm{H}_{4} \rightarrow 6 x \mathrm{H}_{2}+(1+2 x) \mathrm{N}_{2}+(4-4 x) \mathrm{NH}_{3}$$ (a) For what range of values of \(x\) is this equation physically meaningful? (b) Plot the volume of product gas \([V(\mathrm{L})]\) at \(600^{\circ} \mathrm{C}\) and 10.0 bar absolute that would be formed from 50.0 liters of liquid hydrazine as a function of \(x,\) covering the range of \(x\) values determined in Part (b). (c) Speculate on what makes hydrazine a good propellant.
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