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Chapter 14: Problem 25

(a) Consider the combustion of \(\mathrm{H}_{2}(g): 2 \mathrm{H}_{2}(g)+\mathrm{O}_{2}(g)\) \(\longrightarrow 2 \mathrm{H}_{2} \mathrm{O}(g) .\) If hydrogen is burning at the rate of 0.48 \(\mathrm{mol} / \mathrm{s}\), what is the rate of consumption of oxygen? What is the rate of formation of water vapor? (b) The reaction \(2 \mathrm{NO}(g)+\mathrm{Cl}_{2}(g) \longrightarrow 2 \mathrm{NOCl}(g)\) is carried out in a closed vessel. If the partial pressure of \(\mathrm{NO}\) is decreasing at the rate of 56 torr \(/ \mathrm{min}\), what is the rate of change of the total pressure of the vessel?

Short Answer

Expert verified
The rate of consumption of oxygen is \(0.24 \, mol/s\) and the rate of formation of water vapor is \(0.48 \, mol/s\). The rate of change of total pressure of the vessel is \(-28 \, torr/min\).

Step by step solution

01

Identify the stoichiometric coefficients

In the balanced combustion of hydrogen, we have the equation as: \(2 H_{2}(g) + O_{2}(g) \longrightarrow 2 H_{2}O(g)\) Here, the stoichiometric coefficients are 2 for hydrogen, 1 for oxygen, and 2 for water vapor.
02

Relate the rates using stoichiometry

We can relate the rate of consumption of hydrogen, oxygen, and formation of water vapor using their stoichiometric coefficients: \(\frac{d[H_2]}{dt} : \frac{d[O_2]}{dt} : \frac{d[H_2O]}{dt} = -2 : -1 : 2\)
03

Calculate the rate of consumption of oxygen

The given rate of consumption of hydrogen is 0.48 mol/s: \(\frac{d[H_2]}{dt} = -0.48 \, mol/s\) Using the ratio from Step 2, we can calculate the rate of consumption of oxygen: \(\frac{d[O_2]}{dt} = (-0.48 * (-1))/(-2) = 0.24 \, mol/s\)
04

Calculate the rate of formation of water vapor

Similarly, using the ratio from Step 2, we can calculate the rate of formation of water vapor: \(\frac{d[H_2O]}{dt} = (-0.48 * 2)/(-2) = 0.48 \, mol/s\) b) Rate of change of the total pressure of the vessel
05

Identify the stoichiometric coefficients

In the balanced reaction, we have the equation as: \(2 NO(g) + Cl_2(g) \longrightarrow 2 NOCl(g)\) Here, the stoichiometric coefficients are 2 for NO, 1 for Cl鈧�, and 2 for NOCl.
06

Relate the rates using stoichiometry

We can relate the rate of consumption of NO, Cl鈧�, and formation of NOCl using their stoichiometric coefficients: \(\frac{d[NO]}{dt} : \frac{d[Cl_2]}{dt} : \frac{d[NOCl]}{dt} = -2 : -1 : 2\)
07

Calculate the rate of consumption of Cl鈧� and formation of NOCl

The given rate of decrease in partial pressure of NO is 56 torr/min: \(\frac{d[P_{NO}]}{dt} = -56 \, torr/min\) Using the ratio from Step 2, we can calculate the rate of consumption of Cl鈧� and formation of NOCl: \(\frac{d[P_{Cl_2}]}{dt} = (-56 * (-1))/(-2) = -28 \, torr/min\) \(\frac{d[P_{NOCl}]}{dt} = (-56 * 2)/(-2) = 56 \, torr/min\)
08

Calculate the rate of change of the total pressure

Now, we can calculate the rate of change of the total pressure of the vessel by adding the rates of change in partial pressures: \(\frac{d[P_{total}]}{dt} = \frac{d[P_{NO}]}{dt} + \frac{d[P_{Cl_2}]}{dt} + \frac{d[P_{NOCl}]}{dt} = -56 \, torr/min - 28 \, torr/min + 56 \, torr/min = -28 \, torr/min\) So, the rate of change of the total pressure of the vessel is -28 torr/min.

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Key Concepts

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

Combustion Reactions
Combustion reactions are chemical processes where a substance reacts quickly with oxygen, releasing energy in the form of light or heat. They are commonly known as burning and are fundamental in everyday life, from engines to candles. In these reactions, oxygen (O鈧�) combines with a fuel (like hydrogen in the given exercise), to form combustion products.
For the exercise, we look at hydrogen combustion:
  • Reactants: 2H鈧�(g) + O鈧�(g)
  • Products: 2H鈧侽(g)
The stoichiometric coefficients tell us the proportion of reactants and products: 2 moles of H鈧� react with 1 mole of O鈧� to produce 2 moles of water vapor (H鈧侽). Understanding this balanced equation is key as it shows how substances interact in fixed ratios, which helps predict how much of each substance is consumed or produced during the reaction.
These equations also allow for the calculation of unknown rates. For example, if H鈧� is consumed at 0.48 mol/s, you can determine the rate at which O鈧� is consumed (0.24 mol/s) and the rate of water vapor formation (0.48 mol/s). This is essential for engineering and scientific calculations related to energy release and resource consumption.
Stoichiometry
Stoichiometry is the study of the quantitative relationships between reactants and products in a chemical reaction. In simple terms, it鈥檚 all about measuring the correct amount of ingredients in chemistry, much like a recipe in cooking. Stoichiometry is crucial for understanding and balancing chemical equations.
In the context of the provided exercise, stoichiometry helps us determine the rates of consumption of reactants and formation of products. Let's go through the steps using the given reaction:
  • The balanced equation: 2H鈧�(g) + O鈧�(g) 鈫� 2H鈧侽(g)
  • The stoichiometry of the equation is 2:1:2, meaning every 2 moles of hydrogen will react with 1 mole of oxygen to produce 2 moles of water vapor.
By using these ratios, we can relate the rates of consumption and formation. If hydrogen is consumed at a rate of 0.48 mol/s, using stoichiometry, we calculate that oxygen is consumed at a rate of 0.24 mol/s and water is produced at 0.48 mol/s.
Stoichiometry not only aids in calculating chemical quantities but also ensures efficient use of resources and predictability in chemical processes.
Reaction Rate Calculations
Reaction rate calculations are used to determine the speed of a reaction, that is, how quickly reactants are converted into products. In chemical kinetics, the reaction rate can provide insights into the dynamics of a chemical process, crucial for industries and research.
For example, let's consider the exercise with nitric oxide (NO) and chlorine (Cl鈧�):
  • The reaction: 2NO(g) + Cl鈧�(g) 鈫� 2NOCl(g)
  • Here, if the partial pressure of NO decreases by 56 torr/min, we use stoichiometry to find the rate of pressure change for Cl鈧� and NOCl.
By applying the stoichiometric ratios (2:1:2), we calculate:
  • The rate of pressure decrease for Cl鈧� is 28 torr/min.
  • The rate of pressure increase for NOCl is 56 torr/min.
Next, we determine the overall rate of pressure change in the vessel:
Calculate it by summing the rates: i.e., i.e., -56 (NO) + -28 (Cl鈧�) + 56 (NOCl) which equals -28 torr/min.
Reaction rate calculations allow prediction and control of reaction speeds, essential for both safety and efficiency in chemical manufacturing.

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Most popular questions from this chapter

(a) Two reactions have identical values for \(E_{a} .\) Does this ensure that they will have the same rate constant if run at the same temperature? Explain. (b) Two similar reactions have the same rate constant at \(25^{\circ} \mathrm{C}\), but at \(35^{\circ} \mathrm{C}\) one of the reactions has a larger rate constant than the other. Account for these observations.

A colored dye compound decomposes to give a colorless product. The original dye absorbs at \(608 \mathrm{nm}\) and has an extinction coefficient of \(4.7 \times 10^{4} \mathrm{M}^{-1} \mathrm{~cm}^{-1}\) at that wavelength. You perform the decomposition reaction in a \(1-\mathrm{cm}\) cuvette in a spectrometer and obtain the following data: $$ \begin{array}{rl} \hline \text { Time (min) } & \text { Absorbance at } 608 \mathrm{nm} \\ \hline 0 & 1.254 \\ 30 & 0.941 \\ 60 & 0.752 \\ 90 & 0.672 \\ 120 & 0.545 \end{array} $$ From these data, determine the rate law for the reaction "dye \(\longrightarrow\) product" and determine the rate constant.

The \(\mathrm{NO}_{x}\) waste stream from automobile exhaust includes species such as \(\mathrm{NO}\) and \(\mathrm{NO}_{2}\). Catalysts that convert these species to \(\mathrm{N}_{2}\) are desirable to reduce air pollution. (a) Draw the Lewis dot and VSEPR structures of \(\mathrm{NO}, \mathrm{NO}_{2},\) and \(\mathrm{N}_{2} .(\mathbf{b})\) Using a resource such as Table 8.4 , look up the energies of the bonds in these molecules. In what region of the electromagnetic spectrum are these energies? (c) Design a spectroscopic experiment to monitor the conversion of \(\mathrm{NO}_{x}\) into \(\mathrm{N}_{2}\), describing what wavelengths of light need to be monitored as a function of time.

The first-order rate constant for reaction of a particular organic compound with water varies with temperature as follows: $$ \begin{array}{ll} \hline \text { Temperature (K) } & \text { Rate Constant }\left(\mathbf{s}^{-1}\right) \\ \hline 300 & 3.2 \times 10^{-11} \\ 320 & 1.0 \times 10^{-9} \\ 340 & 3.0 \times 10^{-8} \\ 355 & 2.4 \times 10^{-7} \\ \hline \end{array} $$ From these data, calculate the activation energy in units of \(\mathrm{kJ} / \mathrm{mol}\)

(a) For a generic second-order reaction \(\mathrm{A} \longrightarrow \mathrm{B},\) what quantity, when graphed versus time, will yield a straight line? (b) What is the slope of the straight line from part (a)? (c) How do the half-lives of first-order and second-order reactions differ?
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