Approximately how long did it take for 75 percent of the initial amount of

The half-life calculator is a tool that helps you understand the principles of radioactive decay. You can use it to not only learn how to calculate half-life, but also as a way of finding the initial and final quantity of a substance or its decay constant. This article will also present you with the half-life definition and the most common half-life formula.

Each radioactive material contains a stable and an unstable nuclei. Stable nuclei don't change, but unstable nuclei undergo radioactive decay, emitting alpha particles, beta particles or gamma rays and eventually decaying into a stable nuclei. Half-life is defined as the time required for half of the unstable nuclei to undergo their decay process.

Each substance has a different half-life. For example, carbon-10 has a half-life of only 19 seconds, making it impossible for this isotope to be encountered in nature. Uranium-233, on the other hand, has the half-life of about 160 000 years.

This term can also be used more generally to describe any kind of exponential decay - for example, the biological half-life of metabolites.

Half-life is a probabilistic measure - it doesn't mean that exactly half of the substance will have decayed after the time of the half-life has elapsed. Nevertheless, it is an approximation that gets very accurate when a sufficient number of nuclei are present.

The number of unstable nuclei remaining after time t can be determined according to this equation:

N(t) = N(0) * 0.5(t/T)

where:

• N(t) is the remaining quantity of a substance after time t has elapsed.
• N(0) is the initial quantity of this substance.
• T is the half-life.

It is also possible to determine the remaining quantity of a substance using a few other parameters:

N(t) = N(0) * e(-t/τ)

N(t) = N(0) * e(-λt)

• τ is the mean lifetime - the average amount of time a nucleus remains intact.
• λ is the decay constant (rate of decay).

All three of the parameters characterizing a substance's radioactivity are related in the following way:

T = ln(2)/λ = ln(2)*τ

1. Determine the initial amount of a substance. For example, N(0) = 2.5 kg.
2. Determine the final amount of a substance - for instance, N(t) = 2.1 kg.
3. Measure how long it took for that amount of material to decay. In our experiment, we observed that it took 5 minutes.
4. Input these values into our half-life calculator. It will compute a result for you instantaneously - in this case, the half-life is equal to 19.88 minutes.
5. If you are not certain that our calculator returned the correct result, you can always check it using the half-life formula.

Confused by exponential formulas? Try our exponent calculator.

Half life is a similar concept to the doubling time in biology. Check our generation time calculator to learn how exponential growth is both useful and a problem in laboratories!

Half-life is defined as the time taken by a substance to lose half of its quantity.

To find half-life:

1. Find the substance's decay constant.
2. Divide ln 2 by the decay constant of the substance.

Half-life of radium-218 is 25.2 x 10-6 seconds.

Half-life of carbon-20 is 16 x 10-3 seconds.

Half-life of uranium-235 is 22.21 x 1015 seconds.

As you know, the rate of a first-order reaction depends linearly on the concentration of a single reactant. The rate of a first-order reaction that takes the form

#color(blue)(A -> "products")#

can thus be written as

#color(blue)("rate" = - (d["A"])/dt = k * ["A"])" "#, where

#k# - the rate constant for the reaction

Now, in Integral form (I won't go into the derivation here), the rate law for a first-order reaction is equal to

#color(blue)(ln( (["A"])/(["A"_0])) = - k * t)" "#, where

#["A"]# - the concentration of the reactant after the passing of time #t#
#["A"_0]# - the initial concentration of the reactant

Now, the half-life of the reaction is the time needed for the concentration of the reactant to reach half of its initial value.

You can say that

#t = t_"1/2" implies ["A"] = 1/2 * ["A"_0]#

Plug this into the equation for the rate law to get

#ln( (1/2 * color(red)(cancel(color(black)(["A"_0]))))/(color(red)(cancel(color(black)(["A"_0]))))) = -k * t_"1/2"#

This means that you have

#t_"1/2" = ln(1/2)/(-k) = (ln(1) - ln(2))/(-k) = ln(2)/k#

In your case, you want to figure out how many half0lives must pass in order for #75%# of the reactant to be consumed. In other words, you need #25%# of the reactant to remain after a time #t_x#

#t = t_"x" implies ["A"] = 25/100 * ["A"_0] = 1/4 * ["A"_0]#

Once again, plug this into the rate law equation to get

#ln( (1/4 * color(red)(cancel(color(black)(["A"_0]))))/(color(red)(cancel(color(black)(["A"_0]))))) = - k * t_"x"#

This means that you have

#t_"x" = ln(1/4)/(-k) = (ln(1) - ln(4))/(-k) = ln(4)/k#

But since

#ln(4) = ln(2^2) = 2 * ln(2)#

you can say that

#t_"x" = 2 * overbrace(ln(2)/k)^(color(red)(t_text(1/2))) = color(green)(2 * t_"1/2")#

Therefore,

#X = 2#

Simply put, the reaction will be #75%# completed after the passing of two half-lives.