Question

The citizens of the world burn the fossil fuel equivalent of \(7 \times 10^{12} \mathrm{~kg}\) of petroleum per year. Assume that all of this petroleum is in the form of octane \(\left(\mathrm{C}_{8} \mathrm{H}_{18}\right)\) and calculate how much \(\mathrm{CO}_{2}\) (in kg) the world produces from fossil fuel combustion per year. (Hint: Begin by writing a balanced equation for the combustion of octane.) If the atmosphere currently contains approximately \(3 \times 10^{15} \mathrm{~kg}\) of \(\mathrm{CO}_{2}\), how long will it take for the world's fossil fuel combustion to double the amount of atmo- spheric carbon dioxide?

Short answer

It will take a number of years equal to the additional CO2 needed to double atmospheric levels (3 x 10^15 kg) divided by the annual CO2 production, calculated after Step 4, to double the amount of atmospheric CO2.

Step-by-step solution

Write and Balance the Chemical Equation

Begin by writing the chemical reaction for the combustion of octane (C8H18). The combustion reaction involves octane reacting with oxygen (O2) to produce carbon dioxide (CO2) and water (H2O). The balanced chemical equation for this reaction is: 2 C8H18 + 25 O2 -> 16 CO2 + 18 H2O. This equation shows that 2 moles of octane produce 16 moles of CO2.

Calculate Moles of Octane

Calculate the moles of octane combusted per year by using the weight of octane burned. The molar mass of octane (C8H18) is approximately 114 g/mol. Convert the weight of octane burned to kilograms (7 x 10^12 kg) into grams (7 x 10^15 g) and then divide by the molar mass of octane to get the moles of octane (7 x 10^15 g / 114 g/mol).

Calculate Moles of CO2 Produced

Using the balanced chemical equation, calculate the moles of CO2 produced from the moles of octane burned. For every 2 moles of octane, 16 moles of CO2 are produced. So, the number of moles of CO2 produced is 8 times the number of moles of octane.

Calculate Mass of CO2 Produced

Find the mass of CO2 produced by multiplying the moles of CO2 by the molar mass of CO2 (approximately 44 g/mol). Convert this mass back to kilograms to get the total mass of CO2 produced per year.

Calculate Time to Double Atmospheric CO2

With the current mass of atmospheric CO2 given as 3 x 10^15 kg, doubling this amount would yield 6 x 10^15 kg. Divide the additional amount needed to double (3 x 10^15 kg) by the annual production of CO2 obtained in Step 4 to find the time in years it would take to double the atmospheric CO2.

Key concepts

Chemical Reaction Balancing

Before diving into calculations and numbers, it's crucial to understand what balancing a chemical reaction entails. Balancing a chemical reaction is like putting together a precise recipe where the amount of each ingredient needs to be exactly right to get the desired product. In the context of our octane combustion exercise, the 'ingredients' are the reactants: octane \(\mathrm{C}_8\mathrm{H}_{18}\) and oxygen \(\mathrm{O}_2\), and the products: carbon dioxide \(\mathrm{CO}_2\) and water \(\mathrm{H}_2\mathrm{O}\).

In the balanced equation, every atom that goes into the reaction needs to be accounted for in the products that come out of it. For our octane combustion, this is represented by the equation 2 \(\mathrm{C}_8\mathrm{H}_{18}\) + 25 \(\mathrm{O}_2\) \rightarrow 16 \(\mathrm{CO}_2\) + 18 \(\mathrm{H}_2\mathrm{O}\). Notice that the number of carbon and hydrogen atoms is the same on both sides of the equation, fulfilling the Law of Conservation of Mass. Each coefficient (like the '2' in front of octane) indicates the quantity needed to maintain this balance—something vital in understanding the quantitative aspects of our reaction.

Mole Concept

The mole concept is a bridge between the microscopic world of atoms and the macroscopic world we live in. One mole is Avogadro's number (\(6.022 \times 10^{23}\)) of particles, whether they're atoms, molecules, or ions. In this exercise, we use this concept to connect the weight of octane burned to the amount of \(\mathrm{CO}_2\) produced.

The key is to convert the mass of octane, given in kilograms, to moles by using the molar mass of octane (\(114 \text{g/mol}\)). This conversion lets us understand exactly how many particles (or moles) of octane we are starting with. We can then use these moles to figure out how many moles of \(\mathrm{CO}_2\) are produced with stoichiometric calculations. Essentially, the mole concept allows us to count particles by weighing them, which is exceptionally useful in chemistry.

Stoichiometry

Stoichiometry is like the GPS of chemistry—it guides us from reactants to products using balanced equations as the map. It tells us how much of each reactant is needed to produce a certain amount of product. In the context of burning octane, we apply stoichiometry to link moles of fuel to moles of \(\mathrm{CO}_2\) produced.

By understanding that 2 moles of octane yield 16 moles of \(\mathrm{CO}_2\), we set up a ratio that can be used to find out the total amount of \(\mathrm{CO}_2\) generated from a given amount of octane. Once we have the moles of \(\mathrm{CO}_2\), we multiply by its molar mass to translate back to a weight that makes sense in our macroscopic world. This step-by-step journey from reactant to product is dictated by stoichiometric principles, crucial for predicting the outcomes of chemical reactions.

Environmental Impact of Fossil Fuels

When it comes to the environmental impact of burning fossil fuels like octane, the primary concern is the emission of greenhouse gases, particularly \(\mathrm{CO}_2\). This compound is a significant contributor to global warming and the subsequent climate change.

The exercise we've explored paints a stark picture: the massive amount of \(\mathrm{CO}_2\) produced from just a year's worth of fossil fuel consumption could contribute substantially to the atmospheric \(\mathrm{CO}_2\) levels. Taking into consideration the current mass of atmospheric \(\mathrm{CO}_2\) and the rate at which we're adding to it, we can project how long it will take to double the amount of this greenhouse gas in the atmosphere—a figure with genuine implications for environmental policy and our understanding of long-term climatic effects. It's this understanding of the broader consequences of chemical reactions that underscores the importance of green chemistry and the search for more sustainable energy sources.