Stephanie has a master's degree in Physical Chemistry and teaches college level chemistry and physics.
How can scientists use uranium deposits to date the age of the earth? No one was alive back then - there weren't even bacteria! Well, it turns out the radioactive decay of Uranium-238 is very predictable and can be described mathematically, using first-order reactions.
The rate of reaction, or reaction rate, is the speed at which a reaction progresses. We define this mathematically by measuring the rate at which reactants disappear or products appear, where rate is defined as a derivative with respect to time. In this lesson, we're concentrating on first-order reactions. Because first-order reaction rates only depend on the concentration of one reactant, we can define the rate of these reactions as the rate of disappearance of this reactant.
Rate laws are equations that mathematically describe this rate. Knowing which variables determine the rate of reaction allows us to draw useful conclusions about that reaction. In this lesson, we'll look at first-order reactions, which depend only on the concentration of one reactant. We'll then use this rate law to derive an equation for the half-life of the reaction.
In this expression, the brackets denote concentration, so A in brackets is the concentration of reactant A. The negative sign indicates that the concentration is going down over time, and t is time.
Differential Rate Law
First-order reactions are only dependent on the concentration of one reactant raised to the power of one. In other words, in first-order reactions, the rate is proportional to the concentration of reactant A. If it were proportional to the concentration of two different reactants or to the concentration of reactant A-squared, it would be a second-order reaction. We will leave that discussion for another lesson.
We can turn a proportion into an equation by multiplying by a constant. This constant, k, is the reaction rate coefficient.
Then, plugging the definition of rate from the previous section into our rate law gives us the differential rate law.
Integrated Rate Law
Starting with the differential rate law,
we want to move all the concentrations to one side of the equation and move the constant and time to the other side of the equation. It can now be read as:
Then we can integrate each side of the equation:
Once we do that, it will give us the integral representation of the rate law:
With a bit of algebra, this can be represented in another way as well:
Half-Life of First-Order Reactions
The half-life of a reaction is the time that it takes to reduce the concentration of a reactant by half. In other words, at this point, the concentration of the reactant is half its initial concentration:
If we plug this into our integrated rate law we would get the following equation for the half-life:
We can see from this equation that the half-life is only dependent on the reaction rate coefficient, k, which is a constant. This means that the amount of time it takes to reduce the reactant by half is not concentration-dependent. If we start out with 100% of reactant A, it takes the same time to go from 100% to 50% as it does to go from 50% to 25% and from 25% to 12.5%. We can plot this behavior for carbon-14, which has a half-life of 5,730 years:
Notice that the amount of carbon-14 never quite reaches zero - every 5,730 years, it halves again! Uranium-238 has a half-life of nearly 4.5 billion years! This is why it can be used to date the earth. Looking at how much Uranium deposits on Earth have decayed gives us a good idea about the age of the earth.
Let's look at how this information might be used. If a tree sample has 1.0x10-6 g of carbon -14 when it dies (i.e. when it stops exchanging carbon-14 with the earth), then how much carbon-14 will be left in the sample after 9,000 years?
When solving a problem like this, we begin by looking at what is given or known:
- The sample is carbon-14, which has a half-life of 5,730 years
- The time, t, in which carbon-14 decays, is 9,000 years
- The initial concentration is 1.0x10-6 g per sample
If the half-life of carbon-14 is 5,730 years, then we can determine k for carbon-14 from our half-life equation we just looked at:
We can then insert the known quantities into the integral representation of the rate law. When solving problems, it's best to choose the version of the equation which is easiest to solve. In this example, we want to use the equation without a natural log. Note that we can cancel out the units of time in the exponent:
Thus, after 9,000 years, there would be 3.4x10-7 g of carbon-14 left in the sample!
Let's briefly go over what we've learned one at a time.
Rates of reaction, or reaction rates, are the speeds at which a reaction progresses. First-order reaction rates are only dependent on the concentration of one reactant.
This relationship can be expressed in:
A differential rate law:
An integrated rate law:
This rate law also results in a half-life, or the time that it takes to reduce the concentration of a reactant by half, that isn't concentration-dependent:
Half-lives can be a powerful predictive tool for first order reactions. They're most often used in radioactive decay applications which can range from short-term medical use to long-term geological application.
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