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Chapter 5: Problem 24

Many references give the specific gravity of gases with reference to air. For example, the specific gravity of carbon dioxide is 1.53 relative to air at the same temperature and pressure. Show that this value is correct as long as the ideal-gas equation of state applies.

Short Answer

Expert verified
The calculated specific gravity of Carbon Dioxide relative to air under ideal conditions is approximately 1.52, which is close to the given figure of 1.53. Thus, within the assumptions of the ideal gas law, the specific gravity of 1.53 is substantially correct.

Step by step solution

01

Understanding Molar Mass

First, we need to calculate the molar masses of both CO2 and air. As per the periodic table, the molar mass of Carbon (C) is roughly 12 g/mole and Oxygen (O) is about 16 g/mole. Therefore, as CO2 has one atom of Carbon and two of Oxygen, the molar mass of CO2 is \(12g/mole + 2 * 16g/mole = 44g/mole\). Air is primarily composed of Nitrogen (N2) about 78%, Oxygen (O2) about 21%, and Argon (Ar) about 1%. The molar mass of these gases are approximately 28, 32 and 40 g/mole respectively, weighted by their percentages, we find the average molar mass of air to be roughly \( (0.78*28g/mole) + (0.21*32g/mole) + (0.01*40g/mole) = 28.97g/mole \).
02

Calculating Specific Gravity

Knowing that the specific gravity is the ratio of the two densities, but under ideal gas conditions, density is proportional to the molar mass, we could thus compare the molar masses instead since the volume, temperature, and pressure are constants. Therefore, we have the specific gravity as the ratio of the molar mass of gas to the molar mass of air. Our equation is thus: Specific Gravity \(= \frac{Molar mass of CO2}{Molar mass of air} = \frac{44g/mole}{28.97g/mole}\).
03

Checking the Result

Now, it's just a simple division problem. Our answer after calculation approximates close to 1.52, which is close to 1.53. You may point out that our value does not exactly match the provided specific gravity (1.53). This small discrepancy can be attributed to approximations in both our calculation and the provided value. The processes a gas undertakes in real world might not strictly adhere to the ideal gas law, and air is not an ideal gas. Therefore the answer provided may be a rounded-off laboratory value. We can thus say however, that the statement that the specific gravity of CO2 is 1.53 relative to air with reference to the ideal gas law, is substantially correct.

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Key Concepts

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

Molar Mass Calculation
To grasp the concept of molar mass calculation, think of it as finding the weight of one mole of a substance, similar to counting pennies in a dollar. A mole is simply a way of expressing a quantity of atoms or molecules in a specific volume. Let's break down how we calculate it using carbon dioxide (CO2) as our example.
  • Carbon (C) has a molar mass of approximately 12 grams per mole (g/mol).
  • Oxygen (O) has a molar mass of roughly 16 g/mol.
Carbon dioxide contains one carbon atom and two oxygen atoms, hence its total molar mass is calculated as follows:
\[ 12 \ g/mol + 2 \times 16 \ g/mol = 44 \ g/mol \]
This means every mole of CO2 weighs about 44 grams. Similarly, air is composed mainly of Nitrogen (N2), Oxygen (O2), and Argon (Ar). To find air's average molar mass, we weigh the contribution of each gas:
  • Nitrogen, 78% present, has a molar mass of 28 g/mol.
  • Oxygen, 21% present, has a molar mass of 32 g/mol.
  • Argon, 1% present, has a molar mass of 40 g/mol.
The average molar mass of air is approximately:
\[ (0.78 \times 28) + (0.21 \times 32) + (0.01 \times 40) = 28.97 \ g/mol \]
This calculation helps us when comparing the gas densities.
Ideal Gas Law
The Ideal Gas Law is central to understanding how gases behave under various conditions. It combines several gas laws to create a simple equation that provides an insight into the state of a gas. The formula is represented as:
\[ PV = nRT \]
where:
  • \( P \) is the pressure of the gas,
  • \( V \) is the volume,
  • \( n \) is the amount of substance in moles,
  • \( R \) is the ideal gas constant,
  • \( T \) is the temperature in Kelvin.
This equation tells us how changing one aspect, like pressure or temperature, affects the others when the gas behaves ideally. Understanding this is crucial when comparing the specific gravity of gases because it implies that under ideal conditions, the volume, temperature, and pressure are consistent. This consistency allows us to use the ratio of molar masses as a proxy for comparing densities, simplifying the calculation of specific gravity.
Density Comparison
In the realm of gases, density plays a key role in determining how substances will interact with one another under given conditions. Specific gravity is essentially a density comparison between a gas of interest and a reference gas, which is often air.
Since the state of a gas is influenced by temperature and pressure, these elements must remain constant when performing comparisons. What's fascinating is that under ideal gas conditions, a gas's density is directly proportional to its molar mass. This brings us to a neat trick: instead of measuring density, we can simply use the molar masses when these conditions are stable.
Thus, the specific gravity of carbon dioxide relative to air can be calculated by:
  • Taking the molar mass of CO2: 44 g/mol,
  • Dividing by the molar mass of air: 28.97 g/mol.
This gives us a specific gravity value close to 1.52. The closeness of this number to 1.53 demonstrates the practicality of this approach with slight deviations due to real-world approximations. By understanding these concepts, you can confidently compare the behaviors of various gases under similar conditions.

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Most popular questions from this chapter

Bread is typically made by first dissolving preserved yeast (a microscopic biological organism that consumes sugars and emits \(\mathrm{CO}_{2}\) as a waste product) in water, then adding other ingredients, including flour, sugar, fat (usually butter or shortening), and salt. After the ingredients are combined, the dough is "kneaded," or mixed to promote the formation of a protein network from two proteins (gliadin and glutenin) \(^{15}\) present in wheat flour. This network is what strengthens the dough and allows it to stretch elastically without breaking. The dough is then allowed to rise in a process called "proofing," in which the yeast consumes sugar and releases \(\mathrm{CO}_{2}\), which inflates air pockets in the dough that are subsequently filled with air. Finally the dough is baked; the gas pockets expand due to the temperature rise and evaporation of water, the starches from the flour are dehydrated (dried), and the yeast dies. A good French bread has an open, porous structure. The pores must be stabilized by the protein network until the bread is dried sufficiently to hold its shape. The bread collapses if the protein network fails prematurely. (a) Rouille et al. \(^{16}\) investigated the influence of ingredients and mixing conditions on the quality of frozen French bread dough. Each loaf was initially formed roughly as a cylinder with a mass of 150 g (including essentially no \(\mathrm{CO}_{2}\) ), a diameter of \(2.0 \mathrm{cm},\) and a length of \(25.0 \mathrm{cm}\). Determine the specific volume of a bread dough proofed for two hours at \(28^{\circ} \mathrm{C}\) from which \(1.20 \mathrm{cm}^{3}\) gas/min per 100 g dough evolves as bubbles within the dough. State your assumptions. (b) During proofing, the increases in volume of a series of control loaves were monitored along with the mass of \(\mathrm{CO}_{2}\) evolved. Rupture of the protein network during proofing can be detected when the volume of the dough no longer increases at the same rate as the production of \(\mathrm{CO}_{2}\) from the yeast. Data from one of these experiments are shown in the table below. Plot the specific volumes of \(\mathrm{CO}_{2}\) (per \(100 \mathrm{g}\) dough) and dough as a function of time. If the preferred proofing time is such that the dough achieves \(70 \%\) of its total volume before collapse, specify the proper proofing time for this formula. (c) The referenced study found that the parameter with the most significant influence on dough quality was mixing time, with an extended mixing time producing a stronger protein network. Why might extended mixing times not be desirable in commercial production of bread? (d) Suggest causes for the following undesirable bread-baking outcomes: (i) a flat, dense loaf; (ii) an overly large loaf. (e) Suggest why the period during which the dough rises is called "proofing." Remember that yeast is a biological organism. $$\begin{array}{|l|c|c|c|c|c|c|c|c|c|c|c|c|c|} \hline t \text { (min) } & 0 & 20 & 40 & 60 & 80 & 100 & 120 & 140 & 160 & 180 & 200 & 220 & 240 \\ \hline \Delta V \text { (cm }^{3} \text { dough) } & 0 & 0 & 20 & 60 & 80 & 115 & 155 & 198 & 247 & 305 & 322 & 334 & 336 \\ \hline \text { gas evolved }\left(\mathrm{g} \mathrm{CO}_{2}\right) & 0.0 & 37.2 & 63.2 & 68.8 & 126.3 & 192.7 & 234.8 & 315.8 & 385.4 & 515.0 & 578.1 & 657.5 & 745.0 \\ \hline \end{array}$$

A natural gas contains 95 wt\% \(\mathrm{CH}_{4}\) and the balance \(\mathrm{C}_{2} \mathrm{H}_{6}\). Five hundred cubic meters per hour of this gas at \(40^{\circ} \mathrm{C}\) and 1.1 bar is to be burned with \(25 \%\) excess air. The air flowmeter is calibrated to read the volumetric flow rate at standard temperature and pressure. What should the meter read (in SCMH) when the flow rate is set to the desired value?

A gas turbine power plant receives a shipment of hydrocarbon fuel whose composition is uncertain but may be represented by the expression \(\mathrm{C}_{x} \mathrm{H}_{y}\). The fuel is burned with excess air. An analysis of the product gas gives the following results on a moisture-free basis: \(10.5 \%(\mathrm{v} / \mathrm{v}) \mathrm{CO}_{2}, 5.3 \% \mathrm{O}_{2},\) and \(84.2 \% \mathrm{N}_{2}\) (a) Determine the molar ratio of hydrogen to carbon in the fuel ( \(r\) ), where \(r=y / x\), and the percentage excess air used in the combustion. (b) What is the air-to-fuel ratio ( \(m^{3}\) air/kg of fuel) if the air is fed to the power plant at \(30^{\circ} \mathrm{C}\) and \(98 \mathrm{kPa} ?\) (c) The specific gravity of the fuel (a petroleum product) is \(0.85 .\) Estimate the ratio standard cubic feet of gas fed to the turbine per barrel of fuel. (d) What are the issues associated with using oil as a fuel as opposed to natural gas? Consider two factors: (i) the complete composition of typical fuel oils and their resulting emissions, and (ii) the availability and global distribution of the two fuel sources.

The lower flammability limit (LFL) and the upper flammability limit (UFL) of propane in air at 1 atm are, respectively, 2.3 mole \(\% \mathrm{C}_{3} \mathrm{H}_{8}\) and 9.5 mole \(\% \mathrm{C}_{3} \mathrm{H}_{8} .^{17}\) If the mole percent of propane in a propane-air mixture is between \(2.3 \%\) and \(9.5 \%,\) the gas mixture will burn explosively if exposed to a flame or spark; if the percentage is outside these limits, the mixture is safe-a match may burn in it but the flame will not spread. If the percentage of propane is below the LFL, the mixture is said to be too lean to ignite; if it is above the UFL, the mixture is too rich to ignite. (a) Which would be safer to release into the atmosphere- -a fuel-air mixture that is too lean or too rich to ignite? Explain. (b) A mixture of propane in air containing 4.03 mole \(\% \mathrm{C}_{3} \mathrm{H}_{8}\) is fed to a combustion furnace. If there is a problem in the furnace, the mixture is diluted with a stream of pure air to make sure that it cannot accidentally ignite. If propane enters the furnace at a rate of \(150 \mathrm{mol} \mathrm{C}_{3} \mathrm{H}_{8} / \mathrm{s}\) in the original fuel- air mixture, what is the minimum molar flow rate of the diluting air? (c) The actual diluting air molar flow rate is specified to be \(130 \%\) of the minimum value. Assuming the fuel mixture (4.03 mole\% \(\mathrm{C}_{3} \mathrm{H}_{8}\) ) enters the furnace at the same rate as in Part (b) at \(125^{\circ} \mathrm{C}\) and 131 kPa and the diluting air enters at \(25^{\circ} \mathrm{C}\) and \(110 \mathrm{kPa}\), calculate the ratio \(\left(\mathrm{m}^{3} \text { diluting air) } /\right.\) (m \(^{3}\) fuel gas) and the mole percent of propane in the diluted mixture. (d) Give several possible reasons for feeding air at a value greater than the calculated minimum rate.

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.
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