The half-life of a reaction, t1/2, is the amount of time needed for a reactant concentration to decrease by half compared to its initial concentration. Its application is used in chemistry and medicine to predict the concentration of a substance over time. The concepts of half life plays a key role in the administration of drugs into the target, especially in the elimination phase, where half life is used to determine how quickly a drug decrease in the target after it has been absorbed in the unit of time (sec, minute, day,etc.) or elimination rate constant ke (minute-1, hour-1, day-1,etc.). It is important to note that the half-life is varied between different type of reactions. The following section will go over different type of reaction, as well as how its half-life reaction are derived. The last section will talk about the application of half-life in the elimination phase of pharmcokinetics.
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Zero order kinetics
In zero-order kinetics, the rate of a reaction does not depend on the substrate concentration. In other words, saturating the amount of substrate does not speed up the rate of the reaction. Below is a graph of time (t) vs. concentration [A] in a zero order reaction, several observation can be made: the slope of this plot is a straight line with negative slope equal negative k, the half-life of zero order reaction decrease as the concentration decrease.
We learn that the zero order kinetic rate law is as followed, where [A] is the current concentration, [A]0 is the initial concentration, and k is the reaction constant and t is time:
[ [A]= [A]_0 - kt label{1}]
In order to find the half life we need to isolate t on its own, and divide it by 2. We would end up with a formula as such depict how long it takes for the initial concentration to dwindle by half:
[t_{1/2} = dfrac{[A]_0}{2k} label{2}]
The t1/2 formula for a zero order reaction suggests the half-life depends on the amount of initial concentration and rate constant.
First Order Kinetics
In First order reactions, the graph represents the half-life is different from zero order reaction in a way that the slope continually decreases as time progresses until it reaches zero. We can also easily see that the length of half-life will be constant, independent of concentration. For example, it takes the same amount of time for the concentration to decrease from one point to another point.
In order to solve the half life of first order reactions, we recall that the rate law of a first order reaction was:
[[A]=[A]_0e^{-kt} label{4}]
To find the half life we need to isolate t and substitute [A] with [A]0/2, we end up with an equation looking like this:
[t_{1/2} = dfrac{ln 2}{k} approx dfrac{0.693}{k} label{5}]
The formula for t1/2 shows that for first order reactions, the half-life depends solely on the reaction rate constant, k. We can visually see this on the graph for first order reactions when we note that the amount of time between one half life and the next are the same. Another way to see it is that the half life of a first order reaction is independent of its initial concentration.
Second Order Reactions
Half-life of second order reactions shows concentration [A] vs. time (t), which is similar to first order plots in that their slopes decrease to zero with time. However, second order reactions decrease at a much faster rate as the graph shows. We can also note that the length of half-life increase while the concentration of substrate constantly decreases, unlike zero and first order reaction.
In order to solve for half life of second order reactions we need to remember that the rate law of a second order reaction is:
[dfrac{1}{[A]} = kt + dfrac{1}{[A]_0} label{6}]
As in zero and first order reactions, we need to isolate T on its own:
[t_{1/2} = dfrac{1}{k[A]_0} label{7}]
This replacement represents half the initial concentration at time, t (depicted as t1/2). We then insert the variables into the formula and solve for t1/2. The formula for t1/2 shows that for second order reactions, the half-life only depends on the initial concentration and the rate constant.
Example 1: Pharmacokinetics
A following example is given below to illustrate the role of half life in pharmacokinetics to determine the drugs dosage interval.
The therapeutic range of drug A is 20-30 mg/L. Its half life in the target in 5 hours. Once the drug is metabolized in the target, its concentration will decrease over time. To ensure its maximal effect of the drug in the target, the administration will be monitored so that the minimum serum concentration will never go lower than 20 mg/L and the maximum serum concentration will never exceed 30mg/L. As a result, it is important to administer drug A to the target every 5 hours to ensure its effective therapeutic range.
Another important application of half life in pharmacokinetics is that half-life tells how tightly drugs bind to each ligands before it is undergoing decay (ks). The smaller the value of ks, the higher the affinity binding of drug to its target ligand, which is an important aspect of drug design
Problems
1. Define the following term: therapeutic range, half-life, zero order reaction, first order reaction, second order reaction.
2. Examine the following graph and answer
What is the therapeutic range of drug B?
From the graph, estimate the dosage interval of drug B to ensure its maximum effect?
The patient forgot to take the drug at the end of the dosage interval, he decided to take double the amount of drug B at the end of the next dosage interval. Will the drug still be in its therapeutic range?
3. A patient is treating with 32P. How long does it takes for the radioactivity to decay by 90%? The half-life of the material is 15 days.
4. In first order half life, what is the best way to determine the rate constant k? Why?
5. In a first order reaction, A --> B. The half-life is 10 days.
Determine its rate constant k?
Half-life Game
How much time required for this reaction to be at least 50% and 60% complete?
Solutions
1. Therapeutic range: the range that is between maximum drug concentration and a minimum drug concentration in which it is capable of fully exhibit its effective activity
Half-life: the amount of time it required for a reaction to undergoing decay by half.
Zero order reaction, First order reaction, Second order reaction : see module to understand its definition
2. Looking at the graph, we can see the therapeutic range is the amplitude of the graph, which is 5-15 mg/L
The dosage interval is the half-life of the drug, looking at the graph, the half-life is 10 hours.
Even though it will get in the therapeutic range, such practice is not recommended.
3. Using k=ln2/t1/2, plug in half-life we will find k = 4.62x10-2 day-1
If we want the product to decay by 90%, that means 10% is left non-decayed, so [A]t/[A]o = .1
From ln([A]t/[A]o) = -kt, plug in value of k and [A]t/[A]o we then have t = 50 days
4. The best way to determine rate constant k in half-life of first order is to determine half-life by experimental data. The reason is half-life in first order order doesnt depend on initial concentration.
5. Rate constant, k, will be equal to k=ln2/t1/2 , so k = 0.0693 day-1
For the reaction to be 50% complete, that will be exactly the half-life of the reaction at 10 days
For the reaction to be 60% complete, using the similar equation derived from question 4, we have [A]t/[A]o = .4 --> t = 13.2 days
References
Half-life 2
- Chang, Raymond. Physical Chemistry for the Biosciences. Sausalito,CA: University Science Books. Pages 312-319. 2005
- Bauer, Larry. Applied Clinical Pharmacokinetics. New York City, New York: McGraw-Hill. 2008
- Mozayani, Ashraf and Raymond, Lionel. Handbook of Drug Interactions: A Clinical and Forensic Guide. Totowa, New Jersy: Humana Press. 2003
Contributors and Attributions
- David Macias, Samuel Fu, Sinh Le
The Half-Life calculator can be used to understand the radioactive decay principles. It can be used to calculate the half-life of a radioactive element, the time elapsed, initial quantity, and remaining quantity of an element. Half-life is a concept widely used in chemistry, physics, biology, and pharmacology.
What is half life?
There are stable and unstable nuclei in each radioactive element. Unstable nuclei are radioactive decay and emit alpha, beta, or gamma-rays that eventually decay to stable nuclei while stable nuclei of a radioactive don’t change. Half-life is defined as the time needed to undergo its decay process for half of the unstable nuclei.
Each radioactive element has a different half life decay time. The half-life of carbon-10, for example, is only 19 seconds, so it is impossible to find this isotope in nature. Uranium-233 has a half-life of about 160000 years, on the other hand. This shows the variation in the half-life of different elements.
The concept if half-life can also be used to characterize some exponential decay. For instance, the biological half-life of metabolites.
Half-life is more like a probability measure. It doesn't mean that half of the radioactive element would have decayed after the half-life is over. However, it is a highly accurate estimate when enough nuclei are available in a radioactive element.
![Half-life element Half-life element](/uploads/1/3/7/4/137470486/395919258.jpg)
Half life formula
By using the following decay formula, the number of unstable nuclei in a radioactive element left after t can be calculated:
(N(t) = N_0 times 0.5^{(t/T)})
In this equation:
![Half-Life Half-Life](/uploads/1/3/7/4/137470486/800853415.jpg)
N(t) refers to the quantity of a radioactive element that exists after time t has elapsed.
N(0) refers to the initial amount of the element.
T refers to the half-life of an element.
The remaining amount of a material can also be calculated using a variety of other parameters:
(N (t) = N_0 times e^{-dfrac{t}{tau}})
(N (t) = N_0 times e^{(-lambda t)})
λ refers to the decay constant, which is the rate of decay of an element.
τ refers to the mean lifetime of an element. The average time a nucleus has remained unchanged.
The foregoing are all three equations that characterize the radioactivity of material and are linked to each other, which can be expressed as follow:
How to calculate half life?
As of now, you have been through the formula for half-life, and you may be wondering how to find half-life by using that half-life equation. Calculating half-life is somewhat complicated, but we will simplify the process for your understanding. Let’s calculate the half-life of an element by assuming a few things for the sake of calculations.
- Suppose the original amount of a radioactive element is:
N (0) = 200 g
- Now let’s assume the final quantity of that element is:
Peggle pack download free. N (t) = 50 g
- If it took 120 seconds to decay from 3 kg to 1 kg, the time elapsed would be:
t = 120 seconds
- Write the half-life equation and place the above values in that equation:
T = 60 seconds
So, if an element with the initial value of 200 grams decayed to 50 grams in 120 seconds, its half-life will 60 seconds.
Similarly, you can also calculate other parameters such as initial quantity, remaining quantity, and time by using the above equations. If you don’t want to get yourself into these complex calculations, just put the values in the above calculator. Our calculator will simplify the whole process for you.
Check out a few more calculators by us, designed specifically for you.
Example
How many grams of an isotope will remain in 30 years if the half-life of 500 grams of a radioactive isotope is 6 years?
Solution:
In this half-life problem, we already have a half-life, time, and initial quantity of a radioactive isotope. We need to calculate the remaining quantity of that isotope. Let’s find the remaining quantity step by step:
Half-life Equation
- Identify the values from the above problem description.
Darkest dungeon free download pc. N (t) =?
N (0) = 500 g
T = 6 years
t = 30 years
- Write the equation of half-life and substitute the values.
- Solve the equation for the remaining quantity N (t). After simplifying these values, we will get:
N (t) = 15.625 g
A radioactive isotope will remain 15.625 grams after 30 years if its half-life is 6 years, and initial values are 500 grams. Similarly, the elapsed time t and the initial quantity N (0) of a radioactive isotope can also be calculated by following the same process.
How to use our half life calculator?
Half-life Definition Chemistry
Calculating half-life using the above calculator is very simple because you just have to input values to get half-life of any element. This calculator not only calculates the half-life, but it can also be used to calculate the other parameters of the half-life equation such as time elapsed, initial and remaining value. You can find the different tabs for calculating each parameter.
To calculate the half-life of an element, go to the half-life tab:
- Enter the initial and remaining quantity of the element in the corresponding input boxes.
- Enter the total time it took to decay. You can select the unit of time from seconds, minutes, hours, months, year, etc.
- Press the Calculate It will instantly show you the half-life of the element.
Similarly, you can calculate initial and remaining values as well as the time elapsed by clicking on the respective tabs and entering values in the input boxes. You don’t need to put any effort into calculating half life because this calculator does all the complex calculations by only taking values and give the results in a blink of an eye. Moreover, you can use this calculator to solve any type of half-life problems in school or college.