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 NO. Write the rate law. (b) If the rate constant for this reaction at \(1000 \mathrm{~K}\) is \(6.0 \times 10^{4} \mathrm{M}^{-2} \mathrm{~s}^{-1}\), what is the reaction rate when \([\mathrm{NO}]=0.035 \mathrm{M}\) and \(\left[\mathrm{H}_{2}\right]=0.015 \mathrm{M} ?(\mathrm{c})\) 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 \(0.010 \mathrm{M}\) ?

Short answer

(a) The rate law for this reaction is given by Rate = k [NO]^2 [H₂]. (b) When [NO]=0.035 M and [H₂]=0.015 M, the reaction rate is 2.205 × 10⁻³ M s⁻¹. (c) When [NO]=0.10 M and [H₂]=0.010 M, the reaction rate is 6.0 × 10⁻³ M s⁻¹.

Step-by-step solution

Part (a) - Writing the Rate Law

Given that the reaction is first order in H₂ and second order in NO, we can write the rate law by using the following equation: Rate = k [NO]^(order in NO) [H₂]^(order in H₂) where k is the rate constant, the order in NO is 2, and the order in H₂ is 1. So, the rate law is: Rate = k [NO]^2 [H₂]

Part (b) - Calculating the Reaction Rate for Given Concentrations

With the rate constant k = \(6.0 \times 10^4 M^{-2} s^{-1}\), and the given concentrations of NO and H₂, we can determine the rate using the rate law equation: Rate = k [NO]^2 [H₂] Rate = (6.0 × 10^4 M⁻² s⁻¹)(0.035 M)^2(0.015 M) Performing the calculation, we get: Rate = 2.205 × 10⁻³ M s⁻¹

Part (c) - Calculating the Reaction Rate for Different Concentrations

Now we need to calculate the reaction rate at 1000 K when the concentration of NO is increased to 0.10 M, and the concentration of H₂ is 0.010 M. We can use the rate law equation again: Rate = k [NO]^2 [H₂] Rate = (6.0 × 10^4 M⁻² s⁻¹)(0.10 M)^2(0.010 M) Performing the calculation, we get: Rate = 6.0 × 10⁻³ M s⁻¹

Key concepts

Understanding Reaction Order

In chemistry, the term 'reaction order' refers to the power to which the concentration of a reactant is raised in the rate law expression. It indicates how the rate of a chemical reaction is affected by the concentration of the reactants.

For instance, for a reaction where the rate law is expressed as \( Rate = k [A]^m [B]^n \), 'm' and 'n' represent the reaction orders with respect to reactants \( A \) and \( B \), respectively. The overall reaction order is the sum of these individual orders, in this case, \( m+n \).

Understanding reaction orders is essential for predicting how changes in concentrations impact the speed of the reaction. A first-order reaction implies that the rate is directly proportional to the concentration of one reactant. Meanwhile, a second-order reaction means that the rate is proportional to the square of the concentration of one reactant or the product of the concentrations of two reactants.

Determining the Rate Constant

The rate constant, symbolized by 'k', is a proportionality constant in the rate law of a chemical reaction. It is a crucial value that reveals the intrinsic speed of a reaction under certain conditions without any concentration effects factored in.

The units of the rate constant vary depending on the overall reaction order. For example, a second-order reaction will have a rate constant with the unit \( \text{M}^{-1}\text{s}^{-1} \), indicating that the rate is affected by the concentration squared. The exercise mentioned a rate constant \( k = 6.0 \times 10^{4} \text{M}^{-2}\text{s}^{-1} \) at 1000 K, signifying that it relates to a third-order reaction: second-order in NO and first-order in \( H_2 \).

The rate constant depends on temperature and the specific reaction but is independent of the reactant concentrations. Calculating or experimentally determining it is essential for understanding the kinetics of a reaction.

Calculating Reaction Rates

Reaction rate calculation is applying the rate law and the rate constant to figure out how fast a reaction occurs under certain concentrations of reactants. It's a matter of plugging values into the rate equation and computing the result.

In the given exercise, once we have the concentration of reactants NO and \( H_2 \), and the rate constant \( k \), we use the equation \( \text{Rate} = k [\text{NO}]^2 [\text{H}_2] \) to calculate the rate. The rate law is crucial because it quantifies the relationship between reactant concentrations and the reaction's velocity.

Remember that for accurate calculations, one must ensure that the units match and that concentration values are in molarity (M). By knowing the rate, we can predict how altering concentrations or changing the temperature (which affects 'k') will influence the speed of the reaction, allowing for better control and optimization in chemical processes and industries.