Half-Life Formula Derivation with Examples | Definition | Calculator

Half-life denotes the time it takes for half of a particular sample to react. Furthermore, it represents the time required for a certain quantity to decrease to half its initial value.

It is also used to understand how long an atom survives radioactive decay. Also, half-life makes it easy to characterize any type of decay, whether exponential or non-exponential. A good example is that medical sciences refer to the half-life of drugs in the human body, which are biological.

Half-Life Formula

The Half-Life formula helps us find the time it takes a substance to decay to half its initial value. The half-life formula is used to calculate the half-life of radioactive substances and is essential for determining when a sample of radioactive material is safe to handle.

Half-life is a specific phenomenon that occurs daily in various chemical and nuclear reactions. An idea of ​​half-life can be obtained by imagining a situation where a person watches a movie in a theater. This guy is eating from a tub of popcorn. After about 15 minutes, half the popcorn is gone. Leftover popcorn lasts for the rest of the movie. In particular, it shows that the rate of eating popcorn is not at a constant speed and that popcorn has a half-life of 15 minutes.

The Half-Life formula is used to find the time it takes a substance to decay to half its initial value. The half-life formula is used to calculate the half-life of radioactive substances and is essential for determining when a radioactive material sample is safe to handle. The decay of a substance is proportional to the square of the time it takes to decay.

half life formula ncert

Formula:

t1/2 = 0.693 / λ

Where,

λ = rate constant of the decay

half-life calculator with steps

Generalized formula:

Nt = No(1/2)t/t1/2

Proof:

From equation, 

log_e(N/N_o) = – λt

N/N_o = e^-λt

N = N_oe^-λt

Since, t_1/2 = log_e2 / λ, 

λ = log₂2 / t_1/2

Substituting this value, 

N = N_oe^{-loge2 / t1/2× t}

N = N_o(e^-loge2 ) ^{t / t1/2}

N = N_o(1/2)^t/t1/2

Half-Life Formula Derivation

N(t) = N0e−λt

Furthermore, one must set t = T12 and N(T12) = ½ N0.

N(T12 ) = 12N0 = N0e−λT12

Now divide through by N0 and take the logarithm,

½ = e−λt, this leads to In(1/2) = −λT12

Now solving for T12,

T12 = −1λIn(12)

Following the laws of logarithms, one can take the “-1” up s an exponent of the logarithm. Finally, this gives

T12 = In(2))λ

half-life problems and answers examples

How to calculate the half-life formula easily?

The equation for decay is: A = A0 * exp(-Dt), where A0 is the material present at the beginning of the measurement, A is the material present at time t, and D is the decay constant. This equation is related to radioactive decay and some chemical reactions.

So solving using simple algebra: D = -ln(A/A0) / t

Once D is known, the half-life is found by making the concentration A0/2 (half the material) and re-arranging the same equation:

t = – ln( (A0/2)/A0 ) / D = -ln(1/2) / D = 0.693 / D

The “amount” units cancel out, so it doesn’t matter whether you count using atoms or tons, so long as the amount measurement uses the same units at the start and end of the experiment. The time units are preserved, so if your experiment clock is in hours, the half-life is reported in hours.

Why is radiation measured in half-life instead of a full life?

Because radioactivity, for all practical purposes, does not decay entirely to zero. Using full life as a metric requires setting some arbitrary level by which activity can be evaluated to zero. Half-life accurately indicates the time it takes for the radioactivity of a given isotope to decay to half its previous value.

Remember that the time takes for a “full life” in your words, to stop is not twice as long as half-life. It is infinite. At the end of each half-life period, there is half the activity of the previous measure; thus, the statement is indefinite.

How to define Half Life biologically?

A biological half-life is when a substance (drug, radioactive nuclide, or any other biological substance) loses half of its pharmacologic, physiologic, or radiological activity.

Does a shorter half-life mean more radioactive?

A shorter half-life means more decay/unit time for a given amount of sample. It means that it deteriorates quickly.

Yes, you will have more breakdowns in a shorter half-life model for a given time. You might call it ‘more radioactivity’.

Conclusion:

Half-life is the time taken by an element to disintegrate into its half. It is essential to pull out some crucial processes like carbon dating and to identify the rate and order of a particular reaction.

Formula:

t1/2 = 0.693 / λ

Where,

λ = rate constant of the decay

Generalized formula:

Nt = No(1/2)t/t1/2