# What is meant by 'decay constant'?

• esmeralda4
In summary, the decay constant of a radioactive isotope refers to the constant rate at which the isotope decays, expressed in units of time. It is related to the half-life of the isotope and can be calculated using the exponential function. The decay constant is also known as the probability of decay and is always inverse time units. It can be greater than 1, indicating a high rate of decay per unit time.
esmeralda4
Hi,

My textbook states a decay constant of an isotope as 3.84 x 10 to the minus 12 - per second.

Surely decay constant can't be the number of decays per second because that wouldn't stay constant. As the isotope decays there are less atoms to decay and therefore the rate reduces.

So what does decay constant mean? and why are the units per second?

Any help would be appreciated!

Thanks

esmeralda4 said:
So what does decay constant mean? and why are the units per second?
Suppose at some point in time t0 a sample of some radioactive substance contains A0 atoms of that substance. How many will be left at some later time t>t0?

One way to express this is in terms of the half life T1/2:
$$A(t) = A_0 \, 2^{-\,\frac{t-t_0}{T_{1/2}}}$$
Note that the half life T1/2 has units of time.

An equivalent form is to use the exponential function $\exp(x) = e^x$ instead of $2^x$.
$$A(t) = A_0 \, 2^{-\,\frac{t-t_0}{T_{1/2}}} = A_0 \, e^{-\,\frac{\ln 2}{T_{1/2}}(t-t_0)} = A_0 \, e^{-\lambda (t-t_0)}$$
Here, the decay constant λ (which has units of 1/time) is related to the half life via
$$\lambda \equiv \frac{\ln 2}{T_{1/2}}$$

Basically it means that it is decaying at a constant rate, thus allowing its decay to be defined by an exponential function. Think of it like this you start out with a large quantity and then divide it in half. Each time you divide it it is getting smaller and smaller at the same constant rate. This will keep happening following the graph of an exponential function.

The decay constant is the fraction of the number of atoms that decay in 1 second.
It is the constant λ in the decay equation:
dN/dt = -λN The - sign indicates decay, dN/dt is the number of decays per second (also known as 'Activity') and N is the number of atoms present.
Eg if λ= 0.1 and we start with 1000 atoms then after 1 second 100 will have decayed and 900 will be left undecayed. After another second 90 will decay leaving 910. After another second 9.1 will have decayed (but you can't have 9.1 atoms!) leaving 900.9 (you can't have 900.9 !) this shows the pattern of the numbers in decay and plotting these numbers against time will give an exponential decay curve.
The decay constant is also known as the probability of decay, I hope you can see that in my example with λ = 0.1 the probability of an atom decaying is 1 in 10...0.1
In your question the decay canstant of 3.84x10^-12 is a very small probability but the principle is the same

Technician is mostly correct, but it should be noted that the probability of decay is compounded continuously, to use a banking term, rather than compounded once a second. That's where the e comes in.

I think it's important to point out that stating dN(t)/dt = -λN(t) is the same thing as stating N(t) = N0 e-λt, because for someone not familiar with diff eqs, that might not be obvious.

Radioactive isotopes are characterized by the following rule. The probability that a radioactive decays in unit time is a constant that depends on the nature of the isotope, but not on external conditions (pressure, temperature, external fields, etc). It is called a decay constant because the expected number of remaining isotopes is an exponentially decaying function:
$$N(t) = N_0 \, e^{-\lambda \, t}$$

Thanks for all the responses. I think I've got it... If I was to define Decay Constant using words only would it be acceptable to say the 'fraction of the number of atoms remaining to decay in one second'? Does decay constant have to be between 0 and 1? Thanks again.

I would say it is the "fraction of the atoms that do decay in 1 second "rather than the "fraction that remain to decay"
It is a fraction so a number between 0 and 1 or a percentage

esmeralda4 said:
Thanks for all the responses. I think I've got it... If I was to define Decay Constant using words only would it be acceptable to say the 'fraction of the number of atoms remaining to decay in one second'? Does decay constant have to be between 0 and 1? Thanks again.
Think of it as instantaneous velocity. It says miles per hour, but if the vehicle is accelerating for five minutes and then stopping, you can still tell how many miles per hour it's traveling at any given moment.

So you don't look at what fraction would actually decay in a second, because the amount that decays depends on amount that is present. You can only define instantaneous rate. So yes, if the same number of atoms per second was decaying for a full second, that's the fraction that would have decayed in a second.

But that's precisely why the decay constant can be higher than one. It just tells you that at a given rate, all of the material would decay before the second is out. Of course, that doesn't happen, because less material means fewer decay events. But the rate can still be greater than 1.

Does any of that make sense? Some calculus really does help with this one.

the decay constant is also a probability of decay...

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technician said:
the decay constant is also a probability of decay... this cannot be greater than 1
It's probability per unit time, and that can be greater than 1.

Correction

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A decay constant is always inverse time units. If you look at OP, he gives his in inverse seconds.

Edit: A good way to tell is that it's in the exponent of the formula. You cannot take an exponent of a unit. So arguments to exponents, logarithms, and trigonometric functions must be dimensionless. In this case, you multiply decay constant by time t. Time has units of time, and so decay constant must have units of inverse time for their product to be dimensionless.

K^2 said:
It's probability per unit time, and that can be greater than 1.

If the probability per unit time is 30 per second, what does that mean?

I'm not arguing with you, I just want to understand.

cepheid said:
If the probability per unit time is 30 per second, what does that mean?

I'm not arguing with you, I just want to understand.
Picture the graph that tells you how many particles you have left. It's a decaying exponent. At time zero, draw a tangent line through it and see where it intersects the time axis. At rate of 30/second, that tangent line will cross at 1/30th of a second. Notice that if you draw that tangent line anywhere else, it will still cross the time axis 1/30th of a second ahead of a point from which you've drawn it. That's the nature of exponential decay.

Edit: I added an attachment to demonstrate this. Blue curve is exponential decay with rate of 2/second. The red and yellow lines are tangent at 0 and 1 seconds respectively. Notice that they cross the time axis at 0.5 and 1.5 seconds respectively.

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K^2...You are quite correct...I have made correction...decay constant can be greater than 1.
Mental block cleared..cheers

A decay constant (probability of a decay per unit time) is a physical quantity with dimension {time-1}. Therefore it has a difgerent numerical value in different systems of units. The probability, however, that a nucleus would decay in a finite time interval is:
$$P_d \left\lbrace [0, t] \right\rbrace = 1 - e^{-\lambda \, t}$$
This is indeed a number between 0 and 1.

THE decay constant λ, =3.84x10^-12 mentioned in the original post does mean that the rate of decay of N nuclei is λN at that instant

esmeralda4 said:
Thanks for all the responses. I think I've got it... If I was to define Decay Constant using words only would it be acceptable to say the 'fraction of the number of atoms remaining to decay in one second'? Does decay constant have to be between 0 and 1? Thanks again.

Think about it this way: $\frac{1}{λ}$ is the average time it takes for a nucleus of that isotope to decay. So the the reciprocal, λ, is the average percent of atoms of a sample that decay per second. So if λ=0.1, than 10% of the atoms left of the sample will decay each second. If you started with 100, after 1 second you would lose 10% and have 90 left. Then after 2 seconds, you would lose 10% of 90 and have 81 left, and so forth.

## What is the definition of decay constant?

The decay constant, denoted by the symbol λ, is a measure of the rate at which a radioactive substance decays. It is the probability that a nucleus of an atom will decay per unit time.

## How is decay constant related to half-life?

The half-life of a radioactive substance is the amount of time it takes for half of the initial amount of the substance to decay. The relationship between half-life (T1/2) and decay constant (λ) is given by the equation T1/2 = ln(2)/λ.

## What factors affect the value of decay constant?

The value of decay constant is affected by the type of radioactive substance, the energy of the decay process, and the physical and chemical conditions of the substance's environment.

## What units is decay constant measured in?

Decay constant is typically measured in units of inverse time, such as seconds-1 or years-1.

## How is decay constant used in nuclear physics and chemistry?

Decay constant is an important parameter in understanding the behavior of radioactive substances and in predicting the decay of these substances over time. It is used in calculations related to nuclear reactions, radiometric dating, and radiation dosimetry.

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