Question

In the combustion of hexane (a low-boiling component of gasoline),$$2 \mathrm{C}_{6} \mathrm{H}_{14}(g)+19 \mathrm{O}_{2}(g) \longrightarrow 12 \mathrm{CO}_{2}(g)+14 \mathrm{H}_{2} \mathrm{O}(g)$$ it was found that the rate of decrease of \(\mathrm{C}_{6} \mathrm{H}_{14}\) was \(1.20 \mathrm{~mol} \mathrm{~L}^{-1} \mathrm{~s}^{-1}\) (a) What was the rate of reaction with respect to \(\mathrm{O}_{2} ?\) (b) What was the rate of formation of \(\mathrm{CO}_{2}\) ? (c) What was the rate of formation of \(\mathrm{H}_{2} \mathrm{O}\) ?

Short answer

The rate of reaction with respect to \(O_2\) is \(-11.4\ mol\cdot L^{-1}\cdot s^{-1}\), the rate of formation of \(CO_2\) is \(7.20\ mol\cdot L^{-1}\cdot s^{-1}\), and the rate of formation of \(H_2O\) is \(8.40\ mol\cdot L^{-1}\cdot s^{-1}\).

Step-by-step solution

Understanding the Given Chemical Equation

Firstly, identify the balanced chemical equation for the combustion of hexane: \(2 C_6H_{14}(g) + 19 O_2(g) \rightarrow 12 CO_2(g) + 14 H_2O(g)\). This tells us the stoichiometry of the reactants and products: for every 2 moles of hexane that react, 19 moles of oxygen are consumed, 12 moles of carbon dioxide, and 14 moles of water are produced.

Determine the Rate of Reaction with Respect to \(O_2\)

Using the stoichiometry and the given rate of decrease of hexane, \(-1.20 mol\cdot L^{-1}\cdot s^{-1}\), calculate the rate of reaction for \(O_2\) by multiplying the rate of hexane by the molar ratio of \(O_2\) to \(C_6H_{14}\), which is \(\frac{19}{2}\). The negative sign indicates the consumption of the reactant.

Calculate the Rate of Formation of \(CO_2\)

To find the rate of formation of \(CO_2\), take the given rate of decrease of hexane and multiply by the molar ratio of \(CO_2\) to \(C_6H_{14}\), which is \(\frac{12}{2}\).

Compute the Rate of Formation of \(H_2O\)

Finally, calculate the rate of formation of water by multiplying the rate of hexane by the molar ratio of \(H_2O\) to \(C_6H_{14}\), which is \(\frac{14}{2}\).

Key concepts

Combustion of Hexane

Combustion reactions are a staple in chemical education, particularly due to their relevance in everyday life, such as the burning of fuels. Let's delve into the combustion of hexane, which plays a key role in powering engines since it is a significant component of gasoline. In a combustion reaction, a hydrocarbon like hexane reacts with oxygen to produce carbon dioxide and water. The chemical equation for the combustion of hexane can be expressed as: \[\begin{equation}2 C_6H_{14}(g) + 19 O_2(g) \rightarrow 12 CO_2(g) + 14 H_2O(g)\rightarrow 12 CO_2(g) + 14 H_2O(g)\rightarrow 12 CO_2(g) + 14 H_2O(g)\rightarrow 12 CO_2(g) + 14 H_2O(g)\end{equation}\]Balancing this equation is crucial as it reflects the conservation of mass — one of the fundamental principles in chemistry. The coefficients indicate the ratios in which the substances react and form. Here, we can see for every two moles of hexane that burn, nineteen moles of oxygen are consumed, and this combustion yields twelve moles of carbon dioxide and fourteen moles of water vapor.

When we explore reaction rates, understanding the quantitative aspect of these balanced equations becomes even more essential, as it allows us to calculate how fast reactants are consumed and products are formed during the reaction.

Chemical Equation Stoichiometry

Stoichiometry is the mathematical relationship between the amounts of reactants and products in a chemical reaction, based on the balanced chemical equation. It serves as the recipe for the reaction, providing the exact proportions needed to ensure that all reactants are used efficiently, with minimal leftover.

For instance, the combustion of hexane involves a stoichiometric relationship between hexane and oxygen: two moles of hexane react with nineteen moles of oxygen. Applying stoichiometry allows us to figure out how much of each product is formed and how much of each reactant is needed. It aids in understanding how the number of particles and the masses of reactants and products are conserved and transformed.

Educationally, mastering stoichiometry fortifies the concept of mole ratios, which becomes vital when we attempt to compute reaction rates. Without a balanced equation and a thorough understanding of stoichiometry, it would be nearly impossible to proceed to evaluate how the rates of individual substances are interrelated.

Reaction Rate Calculation

Calculating reaction rates involves determining the speed at which reactants are converted into products in a chemical reaction. The rate can be indicated in terms of the concentration of a reactant decreasing over time or the formation of a product over time. For the combustion of hexane, knowing the rate at which hexane decreases, we can calculate the rates for oxygen consumption and the formation rates for carbon dioxide and water.

The rates for other substances involved in the reaction are obtained using stoichiometry. By considering the ratios from the balanced equation, you can calculate the rate of oxygen consumption by multiplying the rate of hexane disappearance by the stoichiometric ratio of oxygen to hexane, \[\begin{equation}Rate_{O_2} = Rate_{C_6H_{14}} \times \frac{19}{2} = -1.20 \times \frac{19}{2} \ mol\cdot L^{-1}\cdot s^{-1}\end{equation}\]Similarly, the rate of carbon dioxide and water formation is linked to the same stoichiometric principles and can be calculated accordingly.

In summary, whether it's for educational purposes or industrial applications, accurate reaction rate calculations are indispensable. They allow for proper control of the reaction conditions, optimization of the reaction pathways, and ensuring the safety and efficiency of chemical processes.