Kinetics: Initial Rates and Integrated Rate Laws
- Hey, it's Professor Dave. Let's talk about kinetics.
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- Kinetics is the study of reaction rates, or how fast a reaction goes. There are a lot of reasons why one chemical reaction might happen in the snap of a finger. And another might take a whole day. So let's learn about what those reasons might be.
Reaction rates are generally measured as an increase in the concentration of products per unit time, or molarity per second. So for the following reaction, we can discuss the rate as the change in the concentration of each product over the change in time. Or the change in concentration of reactant over change in time, though that one will be negative since we are using up the reactant to make products.
Remember, the triangle is a capital delta, and it means change in. And brackets mean concentration. Rates of change will obey stoichiometry. So for example, if we look at this reaction, we have to understand that oxygen appears at 1/4 the rate of NO2, and at 1/2 the rate at which N2O5 disappears.
We can measure rates of reaction in different ways, but any method will involve measuring the changing concentration of a substance. For example, if the product of a reaction is a gas, we can measure the changing pressure of the gas being produced.
If a reaction goes from clear to a colored solution, we can monitor the light absorbance using a spectra photometer. But whatever we do, we can use the data to plot concentration versus time. When looking at this data, we can calculate the instantaneous rate, which is the rate at any given moment.
We do this by looking at the slope of the tangent line at a point, or the line that just touches an individual point on the line. This is more precise than taking an average value over a range of points. The rate will always depend on the concentration of one or more reactants in some way.
The relationship between the rate and a particular concentration is illustrated by the reaction order with respect to a particular substance. For example, let's say we run a reaction several times with different initial concentrations to see what it does to the rate. If we keep everything else the same but we double the concentration of one reactant, and as a result the rate doubles, then the reaction is first order with respect to that reactant.
Instead, if we double the concentration and the rate quadruples, the reaction is second order with respect to the reactant. And if a change doesn't affect the rate, it's zero order. The overall reaction order is just the sum of the orders from the individual reactants.
So if a reaction is first order with respect to each of two reactants, it would be second order overall. We can describe the kinetics by using a rate law. Let's say we have the following generic reaction.
We would write the rate law by representing the concentration of each reactant raised to an exponent that reflects the reaction order. These exponents are not related to the stoichiometric coefficient from the chemical equation and must be determined experimentally. There is also a rate constant k, which is a proportionality constant between rate and concentration.
Now that we have the terminology down, how do we experimentally determine the order of a reaction with respect to each given reactant? We do so by using initial rate data. We can run several trials of a reaction and vary the initial concentration of one reactant at a time.
By doing this, we can see the effect that one reactant's concentration has on the rate. Let's look at the reaction from before. We can clearly see that doubling the initial concentration of the reactant makes it disappear twice as fast. So the reaction is first order with respect to N2O5, and therefore first order overall.
We can determine the reaction order for each react in this way. If we double a concentration, the impact on the rate tells us the order with respect to that substance. If the rate doesn't change, it's zero order. If it doubles, it's first order, quadruples, second order, and if it's cut in half, then it's an order of negative one.
So let's look at some sample data and try to decipher the reaction order for each substance. For this reaction, let's perform trials where one substance's concentration stays the same, but the other one changes. We can see for the first two trials, O2 concentration stays the same, but NO doubles. As a result, the rate quadruples, so the reaction must be second order with respect to NO. If we compare trials one and three, NO stays the same, but O2 doubles. There, the rate doubles, so the reaction must be first order with respect to O2. So this must be the rate law. The reaction orders happened to match the stoichiometric coefficients, but this is a coincidence. It won't always be the case.
Also, from the rate data, we can calculate the rate constant. Just plug in the rate and concentrations from any one of the trials, and solve for k. While we're discussing the rate constant, we should understand that the units on it will be specific to the overall reaction order. This is because they must cancel out the concentration units to give molarity per second, which are units that makes sense for the rate.
So for zero order, they will be molarity clarity per second, since a zero order reaction doesn't depend on concentration. For first order, it's inverse seconds, so that when combined with molarity, we get molarity per second. For second order, it's 1 over molarity times seconds, so that when combined with molar squared, we get molarity per second.
Looking at the previous reaction, the units on k must be 1 over molar squared times seconds, since there will be molar cubed in the numerator. Likewise, this means that if you know the rate constant, the overall reaction order. Just see how many powers of molarity have to be in the numerator to result in a rate of molarity per second.
If we want to discern the concentration at any given time, we can use the integrated rate law. We will use different integrated rate laws, depending on the overall reaction order. Here are the different rate laws and resulting integrated rate laws that correspond to each reaction order.
The great thing about these integrated rate laws is that they can be plotted in a y equals mx plus b format to give us a linear plot. See how in each case, there is a distinct y, m, x, and b. And x is always time.
We can see that for a zero order reaction, concentration versus time gives us a straight line. For a first order reaction, natural log of concentration versus time gives us a straight line. And for a second order reaction, the inverse of concentration versus time gives us a straight line.
So when we record kinetic data, we can try to plot it according to these different relationships and the one that gives us a straight line will tell us the overall reaction order, simply by graphical analysis. Let's check comprehension.
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