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Chapter 10: Problem 109

Assume that a single cylinder of an automobile engine has a volume of \(524 \mathrm{~cm}^{3} .\) (a) If the cylinder is full of air at \(74^{\circ} \mathrm{C}\) and 0.980 atm, how many moles of \(\mathrm{O}_{2}\) are present? (The mole fraction of \(\mathrm{O}_{2}\) in dry air is \(0.2095 .\) ) (b) How many grams of \(\mathrm{C}_{8} \mathrm{H}_{18}\) could be combusted by this quantity of \(\mathrm{O}_{2}\), assuming complete combustion with formation of \(\mathrm{CO}_{2}\) and \(\mathrm{H}_{2} \mathrm{O}\) ?

Short Answer

Expert verified
(a) First, we'll calculate the moles of air in the cylinder using the Ideal Gas Law: \(n = \frac{PV}{RT}\). Then, we'll find the moles of \(\mathrm{O}_{2}\) by multiplying the moles of air by the given mole fraction of \(\mathrm{O}_{2}\): \(Moles\,of\,O2 = Moles\,of\,air \cdot 0.2095\). (b) We'll use the balanced equation \(C_8H_{18} + 12.5O_2 \longrightarrow 8CO_2 + 9H_2O\) to find the moles of \(C_8H_{18}\) that can be combusted: \(Moles\,of\,C_8H_{18} = \frac{Moles\,of\,O2}{12.5}\). Finally, we'll convert the moles of \(C_8H_{18}\) to grams using the molar mass: \(Grams\,of\,C_8H_{18} = Moles\,of\, C_8H_{18} \cdot 114.22\,g/mol\).

Step by step solution

01

Calculate the moles of air present in the cylinder using the Ideal Gas Law

The Ideal Gas Law can be written as: \(PV=nRT\) We are given the temperature, pressure, and volume of the cylinder. We can use these values to find the moles (n) of air present in the cylinder. First, convert the temperature from Celsius to Kelvin. \(Temperature = 74^{\circ}\mathrm{C} + 273.15 = 347.15\mathrm{K}\) Now, convert the pressure to atmospheres: \(Pressure = 0.980\mathrm{atm}\) The volume of the cylinder is already given in cm³, so the values are in the desired units. Now, plug in the values and the gas constant \(R = 0.0821\frac{L \cdot atm}{mol \cdot K}\) to solve for the moles of air in the cylinder: \(n = \frac{PV}{RT}\)
02

Calculate the moles of O2 in the cylinder

To calculate the moles of O2 in the cylinder, use the mole fraction of O2 in the air, which is given as 0.2095: \(Moles\,of\,O2 = Moles\,of\,air \cdot Mole\,fraction\,of\,O2\) Now that we have the moles of O2 in the cylinder, we can proceed to find how many grams of C8H18 can be combusted by this quantity of O2. (b) Finding the grams of C8H18 that could be combusted by this quantity of O2
03

Write the balanced chemical equation for the combustion reaction

The combustion reaction of C8H18 is: \(C_8H_{18} + O_2 \longrightarrow CO_2 + H_2O\) Balancing this equation, we get: \(C_8H_{18} + 12.5O_2 \longrightarrow 8CO_2 + 9H_2O\)
04

Calculate the moles of C8H18 that can be combusted

Using the stoichiometry from the balanced equation, we can determine the moles of C8H18 that can be combusted: \(Moles\,of\,C_8H_{18} = \frac{Moles\,of\,O2}{12.5}\)
05

Convert the moles of C8H18 to grams

Lastly, using the molar mass of C8H18, we can convert the moles of C8H18 to grams: \(Molar\,mass\,of\,C_8H_{18} = 8\cdot(12.01\,g/mol) + 18\cdot(1.01\,g/mol) = 114.22\,g/mol\) \(Grams\,of\,C_8H_{18} = Moles\,of\, C_8H_{18} \cdot Molar\,mass\,of\,C_8H_{18}\) Now we have determined the grams of C8H18 that can be combusted by the quantity of O2 available in the cylinder.

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Most popular questions from this chapter

Hurricane Wilma of 2005 is the most intense hurricane on record in the Atlantic basin, with a low-pressure reading of 882 mbar (millibars). Convert this reading into (a) atmospheres, (b) torr, and (c) inches of \(\mathrm{Hg}\).

In an experiment reported in the scientific literature, male cockroaches were made to run at different speeds on a miniature treadmill while their oxygen consumption was measured. In one hour the average cockroach running at \(0.08 \mathrm{~km} / \mathrm{hr}\) consumed \(0.8 \mathrm{~mL}\) of \(\mathrm{O}_{2}\) at 1 atm pressure and \(24{ }^{\circ} \mathrm{C}\) per gram of insect mass. (a) How many moles of \(\mathrm{O}_{2}\) would be consumed in 1 hr by a 5.2-g cockroach moving at this speed? (b) This same cockroach is caught by a child and placed in a 1 - qt fruit jar with a tight lid. Assuming the same level of continuous activity as in the research, will the cockroach consume more than \(20 \%\) of the available \(\mathrm{O}_{2}\) in a 48 -hr period? (Air is \(21 \mathrm{~mol}\) percent \(\left.\mathrm{O}_{2} .\right).\)

Newton had an incorrect theory of gases in which he assumed that all gas molecules repel one another and the walls of their container. Thus, the molecules of a gas are statically and uniformly distributed, trying to get as far apart as possible from one another and the vessel walls. This repulsion gives rise to pressure. Explain why Charles's law argues for the kineticmolecular theory and against Newton's model.

(a) What are the mole fractions of each component in a mixture of \(15.08 \mathrm{~g}\) of \(\mathrm{O}_{2}, 8.17 \mathrm{~g}\) of \(\mathrm{N}_{2}\), and \(2.64 \mathrm{~g}\) of \(\mathrm{H}_{2} ?\) (b) What is the partial pressure in atm of each component of this mixture if it is held in a 15.50-L vessel at \(15^{\circ} \mathrm{C}\) ?

Suppose you have a fixed amount of an ideal gas at a constant volume. If the pressure of the gas is doubled while the volume is held constant, what happens to its temperature? [Section 10.4]
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