Question

This reaction is first order in \(\mathrm{N}_{2} \mathrm{O}_{5}\) $$ \mathrm{N}_{2} \mathrm{O}_{5}(g) \longrightarrow \mathrm{NO}_{3}(g)+\mathrm{NO}_{2}(g) $$ The rate constant for the reaction at a certain temperature is \(0.053 / \mathrm{s}\) a. Calculate the rate of the reaction when \(\left[\mathrm{N}_{2} \mathrm{O}_{5}\right]=0.055 \mathrm{M}\). b. What would the rate of the reaction be at the concentration indicated in part a if the reaction were second order? Zero order? (Assume the same numerical value for the rate con- stant with the appropriate units.)

Short answer

a. The rate for the first order reaction is 0.00291 M/s. b. For second order, the rate would be 0.00015835 M^{-1}s^{-1}. For zero order, the rate would be 0.053 M/s.

Step-by-step solution

Understanding First Order Kinetics

For a first-order reaction, the rate of the reaction is directly proportional to the concentration of the reactant. The rate law can be expressed as Rate = k[A], where k is the rate constant and [A] is the concentration of the reactant.

Calculate Rate for First Order Reaction

Using the rate law for a first-order reaction, we calculate the rate by multiplying the rate constant, k, given as 0.053 /s, by the concentration of N2O5, 0.055 M. The calculation will be as follows: Rate = k[N2O5] = 0.053 /s * 0.055 M.

Understanding Second Order Kinetics

For a second-order reaction, the rate is proportional to the square of the concentration of the reactant. The rate law can be expressed as Rate = k[A]^2.

Calculate Rate for Second Order Reaction

Presuming the same numerical value for the rate constant with appropriate units for second-order kinetics (M^{-1}s^{-1}), we calculate the rate by squaring the concentration of N2O5 and multiplying by the rate constant: Rate = k[N2O5]^2 = 0.053 M^{-1}s^{-1} * (0.055 M)^2.

Understanding Zero Order Kinetics

For a zero-order reaction, the rate is independent of the concentration of the reactant and is equal to the rate constant k. The rate law can be expressed as Rate = k.

Calculate Rate for Zero Order Reaction

Since the rate is equal to the rate constant for a zero-order reaction, and the units appropriate for k in zero-order kinetics are M/s, we use the given numerical value: Rate = k = 0.053 M/s.

Key concepts

First-Order Reaction

In the realm of chemical kinetics, a first-order reaction is fundamental and straightforward to understand. It reflects a scenario where the reaction rate is directly proportional to the concentration of a single reactant. Mathematically, the rate law for such a reaction is expressed as Rate = k[A], where 'k' represents the rate constant and [A] is the molar concentration of the reactant in question.

To conceptualize this, imagine a crowd of people leaving a building through a single exit door. The rate at which people exit (analogous to the reaction rate) is directly linked to how many are inside (the concentration of the reactant). If the room is full, people exit quickly, but as the room empties, the rate slows down. Similarly, as the reactant is consumed in a first-order reaction, the reaction rate decreases correspondingly.

Second-Order Reaction

A second-order reaction is one whose rate is proportional to the square of the concentration of one reactant or the product of the concentrations of two reactants. The general form of the rate law for such a reaction is Rate = k[A]^2 or Rate = k[A][B], depending on whether the reaction involves one or two reactants, respectively.

Consider the analogy of a dance hall where pairs of dancers are needed for a dance to occur. If few people are present, it becomes difficult to find a partner, and the dance initiation rate is low. As more people enter the room, pairing up becomes easier, and the rate of dancing starts ramping up. Translate this to chemistry: when the reactant concentrations are higher, they collide more frequently, which can dramatically increase the rate of reaction.

Zero-Order Reaction

Unlike first or second-order reactions, a zero-order reaction exhibits a constant rate that is independent of the concentration of the reactant. This is characterized by the rate law Rate = k, where 'k' is the rate constant. Because the rate is constant, it is only affected by changes in the rate constant which can occur due to variations in factors such as temperature or catalyst presence.

Visualize a conveyor belt in a factory that moves at a set speed regardless of the number of items on it. Similarly, in a zero-order reaction, the reaction proceeds at a steady rate regardless of the concentration of reactants present, akin to a conveyor belt that doesn't accelerate or decelerate based on the load it carries.

Rate Constant

Central to the study of chemical kinetics is the rate constant 'k', an intrinsic factor that reveals the speed at which a reaction occurs under given conditions. It is determined experimentally and varies with temperature, catalysts, and the inherent nature of the reacting substances. The units of 'k' differ depending on the order of reaction—being s^{-1} for first-order, M^{-1}s^{-1} for second-order, and M/s for zero-order kinetics.

Imagine the rate constant as a 'speed rating' for a reaction, much like a speed limit sign on a highway sets the maximum allowable vehicle velocity. It determines how fast the reaction can go under ideal conditions but does not factor in the actual concentration of reactants at any given time.

Reaction Rate

The reaction rate is the speed at which reactants are converted into products in a chemical reaction. It can be affected by various factors such as reactant concentration, temperature, surface area, and the presence of a catalyst. In general, an increase in the concentration of reactants or temperature or the addition of a catalyst will result in a faster reaction rate.

One can draw a parallel with baking cookies. If you increase the temperature of the oven (akin to raising the temperature in a reaction), the cookies bake faster (reactants convert to products more rapidly). If more trays of cookies (representative of higher reactant concentration) are put into the oven, the overall 'reaction' of baking is happening more visibly. The essence here is that the reaction rate reflects how swiftly a chemical change is occurring, a critical aspect for chemists to control and utilize for desired outcomes in various processes.