Question

Consider the following reaction: $$2 \mathrm{NO}(g)+2 \mathrm{H}_{2}(g) \longrightarrow \mathrm{N}_{2}(g)+2 \mathrm{H}_{2} \mathrm{O}(g)$$ (a) The rate law for this reaction is first order in \(\mathrm{H}_{2}\) and second order in \(\mathrm{NO}\) . Write the rate law. (b) If the rate constant for this reaction at 1000 \(\mathrm{K}\) is \(6.0 \times 10^{4} M^{-2} \mathrm{s}^{-1}\) what is the reaction rate when \([\mathrm{NO}]=0.035 M\) and \(\left[\mathrm{H}_{2}\right]=0.015 M ?\) (c) What is the reaction rate at 1000 \(\mathrm{K}\) when the concentration of \(\mathrm{NO}\) is increased to 0.10 \(\mathrm{M}\)while the concentration of \(\mathrm{H}_{2}\) is 0.010\(M ?\) (d) What is the reaction rate at 1000 \(\mathrm{K}\) if \([\mathrm{NO}]\) is decreased to 0.010 \(\mathrm{M}\) and \(\left[\mathrm{H}_{2}\right]\) is increased to 0.030 \(\mathrm{M}\) ?

Short answer

(a) The rate law for this reaction is: \[Rate = k[\mathrm{NO}]^2[\mathrm{H}_2]\] (b) The reaction rate when [NO] = 0.035 M and [H₂] = 0.015 M is 0.0525 M.s⁻¹. (c) The reaction rate when [NO] = 0.10 M and [H₂] = 0.010 M is 0.060 M.s⁻¹. (d) The reaction rate when [NO] = 0.010 M and [H₂] = 0.030 M is 0.0018 M.s⁻¹.

Step-by-step solution

Part A: Writing the rate law

The rate law is a mathematical equation that expresses the reaction rate as a function of the reactant concentrations. The given reaction has a first order dependence on H₂ concentration and a second order dependence on NO concentration. Therefore, the rate law can be written as: \[Rate = k[\mathrm{NO}]^2[\mathrm{H}_2]\]

Part B: Determining the reaction rate

We are given the values of the rate constant(k), NO, and H₂ concentrations. Now, we can substitute these values into the rate law equation and calculate the reaction rate: \[Rate = (6.0 \times 10^4 M^{-2}s^{-1})(0.035 M)^2(0.015 M)\] \[Rate = 0.0525 M.s^{-1}\]So, the reaction rate when [NO] = 0.035 M and [H₂] = 0.015 M is 0.0525 M.s⁻¹.

Part C: Reaction rate with increased NO concentration

We are given new concentrations for NO and H₂. We can use the same rate law equation and substitute these new values to calculate the reaction rate: \[Rate = (6.0 \times 10^4 M^{-2}s^{-1})(0.10 M)^2(0.010 M)\] \[Rate = 0.060 M.s^{-1}\]Therefore, the reaction rate when [NO] = 0.10 M and [H₂] = 0.010 M is 0.060 M.s⁻¹.

Part D: Reaction rate with decreased NO concentration and increased H₂ concentration

We are given new concentrations for NO and H₂. Again, we can use the same rate law equation and substitute these new values to calculate the reaction rate: \[Rate = (6.0 \times 10^4 M^{-2}s^{-1})(0.010 M)^2(0.030 M)\] \[Rate = 0.0018 M.s^{-1}\]Hence, the reaction rate when [NO] = 0.010 M and [H₂] = 0.030 M is 0.0018 M.s⁻¹.

Key concepts

Chemical Kinetics

Chemical kinetics is the branch of physical chemistry that concerns itself with understanding the rates of chemical reactions. It's a fascinating area of study because it helps scientists understand how quickly a reaction occurs and what can affect this speed.

At the heart of chemical kinetics is the reaction rate, which is the speed at which reactants are transformed into products. This rate can vary greatly, from the near-instantaneous nature of some ionic reactions to the painstakingly slow rusting of iron. Factors that affect reaction rates include the concentration of reactants, temperature, presence of a catalyst, and surface area of solid reactants or catalysts.

An integral part of studying chemical kinetics is to conduct experiments to measure reaction rates and then use the data to deduce the rate law, which provides a mathematical relationship between the reaction rate and the concentration of reactants. Understanding chemical kinetics is essential not just for academic research, but also has practical applications in designing industrial processes, pharmaceuticals, and even in environmental science for pollution control.

Rate Constant

The rate constant, symbolized by the letter 'k' in kinetic equations, is a proportionality factor that provides the relationship between the reaction rate and the concentrations of reactants as dictated by the rate law. Its value is determined experimentally and is a quantification of how rapidly a reaction occurs.

Importantly, the rate constant isn't constant in all scenarios—it changes with temperature, and its units vary depending on the overall order of the reaction. For instance, a reaction with a second-order rate law would have a rate constant with units of M^-1s^-1. Understanding the rate constant is essential when comparing different reactions or the same reaction under different temperatures, as it is an intrinsic value that indicates the inherent speed at which a process happens under specific conditions.

Reactant Concentration

Reactant concentration often plays a starring role in the dynamics of a chemical reaction. Reflected in the reaction's rate law, it tells us how the rate of a reaction will change when we alter the amount of one or more of the reactants.

For a given reaction, if the rate law indicates a first-order dependence on a reactant, it means that the rate is directly proportional to that reactant's concentration. Double the amount of reactant, and you double the reaction rate. Conversely, a second-order dependence implies that the rate changes with the square of the reactant's concentration. An increase in the reactant concentration results in a reaction rate that speeds up quadratically, making these reactions highly sensitive to changes in concentration. This concept is fundamental in controlling reaction rates in various applications, from industrial synthesis to pharmacokinetics in medicine.