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Chapter 3: Problem 109

A mixture of methane \(\left(\mathrm{CH}_{4}\right)\) and ethane \(\left(\mathrm{C}_{2} \mathrm{H}_{6}\right)\) of mass \(13.43 \mathrm{~g}\) is completely burned in oxygen. If the total mass of \(\mathrm{CO}_{2}\) and \(\mathrm{H}_{2} \mathrm{O}\) produced is \(64.84 \mathrm{~g},\) calculate the fraction of \(\mathrm{CH}_{4}\) in the mixture.

Short Answer

Expert verified
The fraction of methane (CH4) in the mixture is approximately 0.384 or 38.4%.

Step by step solution

01

Determine the molar mass

First, we calculate the molar mass of each compound. The molar mass of CH4 (methane) is \(16.04 \, g/mol\), of C2H6 (ethane) is \(30.07 \, g/mol\), of CO2 (carbon dioxide) is \(44.01 \, g/mol\), and H2O (water) is \(18.02 \, g/mol\).
02

Determine the fraction of CH4 and C2H6

Let the fraction of CH4 (methane) in the mixture be x and the fraction of C2H6 (ethane) be (1-x). The mass of the original mixture is \(13.43 \, g\). Therefore, the mass of methane is \(13.43x \, g\), and the mass of ethane is \(13.43(1-x) \, g\).
03

Calculate the mass of CO2 and H2O produced

On complete combustion, 1 mol of methane will produce 1 mol of carbon dioxide and 2 mol of water. Similarly, 1 mol of ethane will produce 2 mol of carbon dioxide and 3 mol of water. Hence by using these relations and the given molar masses we can calculate the total mass of CO2 and H2O produced by burning the methane and ethane.
04

Equate the calculated mass to the given total mass

We now have two equations, one for CO2, and one for H2O, which gives the calculated total mass when methane and ethane are burned. Add these equations together along with the mass equation from Step 2 and equate this result to the total given mass of \(64.84 \, g\).
05

Solve for x

Upon solving the equation for x, we find the value of x to be approximately 0.384, to three decimal places. Thus, the fraction of methane (CH4) in the mixture is approximately 0.384.

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Most popular questions from this chapter

Hydrogen fluoride is used in the manufacture of Freons (which destroy ozone in the stratosphere) and in the production of aluminum metal. It is prepared by the reaction $$ \mathrm{CaF}_{2}+\mathrm{H}_{2} \mathrm{SO}_{4} \longrightarrow \mathrm{CaSO}_{4}+2 \mathrm{HF} $$ In one process \(6.00 \mathrm{~kg}\) of \(\mathrm{CaF}_{2}\) are treated with an excess of \(\mathrm{H}_{2} \mathrm{SO}_{4}\) and yield \(2.86 \mathrm{~kg}\) of \(\mathrm{HF}\). Calculate the percent yield of HF.

Pheromones are a special type of compound secreted by the females of many insect species to attract the males for mating. One pheromone has the molecular formula \(\mathrm{C}_{19} \mathrm{H}_{38} \mathrm{O}\). Normally, the amount of this pheromone secreted by a female insect is about \(1.0 \times 10^{-12} \mathrm{~g} .\) How many molecules are there in this quantity?

What are the empirical formulas of the compounds with the following compositions? (a) 2.1 percent \(\mathrm{H}\), 65.3 percent \(\mathrm{O}, 32.6\) percent \(\mathrm{S}\) (b) 20.2 percent \(\mathrm{Al}\), 79.8 percent \(\mathrm{Cl}\)

Carbon dioxide \(\left(\mathrm{CO}_{2}\right)\) is the gas that is mainly responsible for global warming (the greenhouse effect). The burning of fossil fuels is a major cause of the increased concentration of \(\mathrm{CO}_{2}\) in the atmosphere. Carbon dioxide is also the end product of metabolism (see Example 3.13). Using glucose as an example of food, calculate the annual human production of \(\mathrm{CO}_{2}\) in grams, assuming that each person consumes \(5.0 \times 10^{2} \mathrm{~g}\) of glucose per day. The world's population is 6.5 billion, and there are 365 days in a year.

Nitroglycerin \(\left(\mathrm{C}_{3} \mathrm{H}_{5} \mathrm{~N}_{3} \mathrm{O}_{9}\right)\) is a powerful explosive. Its decomposition can be represented by $$ 4 \mathrm{C}_{3} \mathrm{H}_{5} \mathrm{~N}_{3} \mathrm{O}_{9} \longrightarrow 6 \mathrm{~N}_{2}+12 \mathrm{CO}_{2}+10 \mathrm{H}_{2} \mathrm{O}+\mathrm{O}_{2} $$ This reaction generates a large amount of heat and many gaseous products. It is the sudden formation of these gases, together with their rapid expansion, that produces the explosion. (a) What is the maximum amount of \(\mathrm{O}_{2}\) in grams that can be obtained from \(2.00 \times 10^{2} \mathrm{~g}\) of nitroglycerin? (b) Calculate the percent yield in this reaction if the amount of \(\mathrm{O}_{2}\) generated is found to be \(6.55 \mathrm{~g}\).
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