# Radioactive decay and finding half life

• SUchica10
In summary: Then solve for T. It's just that when you solve for T, it will be in units of inverse time (ie t=-kt). So if you wanted to calculate the half-life in years, you'd have to take the logarithm of both sides, and then solve for t. In summary, the tritium 3 sample decayed to 94.5% of it's original amount after a year. It would take the sample to decay to 20% of its original amount after a year if the initial mass is kept at 0.
SUchica10
A sample of tritium 3 decayed to 94.5% of its original amount after a year.
(a) What is the half-life of tritium-3?
(b) How long would it take the sample to decay to 20% of its original
amount?

The only equations given in my book are dm/dt = km and m(t) = m(0)e^kt but the mass is not given in the problem. Is there another equation I should be using or am I just not seeing what exactly I should be using from the problem?

It doesn't matter what mass it is because you know what ratios they have to decay to.

Hint: Use the second eqn. Let initial mass be at t=0, ie m(0) in th expression. So at t = 1 year, m(t) becomes 94.5% of m(0).
Can you proceed now ?
Of course as Kurdt said you will find the final expression independent of mass, since only the ratio that remains is what matters.

"The sample decayed to 94.5% of it's original amout."

So if the initial mass is M, what is the mass after 1 year?

$$m(1)=0,945m(0)$$. Do you see why?

So that m(0) now cancel out and you have:

$$0,945=e^{-k}$$

You can now find k.

I understand it now and I'm pretty sure I got it I just have one question... why is it e^(-k)? I understand t = 1 so that just leaves me with k but why is it negative?

Because it describes the decay of a radioactive substance thus the power to which an exponential should be raised to describe decay must be negative. A positive power would lead to growth.

You don't really need the exponential form. Since the question is ask for "half-life", write
$$M(t)= M_0(\frac{1}{2})^\frac{t}{T}$$
Since for t= T, that multiplies M_0 by 1/2, T is the half-life.

When t= 1,
$$M(1)= .954M_0= M_0(\frac{1}{2})^\frac{1}{T}$$
so
$$.954= (\frac{1}{2})^\frac{1}{T}$$
Take logarythms of both sides and solve for T.

I'm really sorry but now I'm confused.

Your book didn't really do you any favors on the k thing. Since the amount of radioactive material remaining decreases as time increases, the first equation should properly be written as:

$$\frac{dm}{dt}=-kt$$

The minus sign on the kt represents the fact that m is decreasing as time increases (and vice-versa.)

When you solve that equation, you'll get

$$m=m_{0}e^{-kt}$$

That's good. As time increases, m decreases. k will be the decay constant, which is just ln(2)/(half-life). With the way your equations are written, k is actually the negative decay constant. It should be pretty obvious when you try to solve it, since the half-life has to be a positive value.

Anyway, remember that k isn't actually the value you're looking for in part a. If has units of inverse time. half-life = ln(2)/k is what you want.

SUchica10 said:
I'm really sorry but now I'm confused.

What is it that confuses you? Do you understand that If the half life is T, then the amount is multiplied by 1/2 every T years? Do you know how to solve $$.954= (\frac{1}{2})^\frac{1}{T}$$?

As I said, take logarithms of both sides.

## 1. What is radioactive decay and how does it work?

Radioactive decay is the process by which an unstable atom releases energy in the form of radiation, in order to become more stable. This can occur through various types of decay, such as alpha, beta, and gamma decay. The decay process is random and cannot be influenced by external factors.

## 2. What is half-life and how is it related to radioactive decay?

Half-life is the amount of time it takes for half of the atoms in a radioactive substance to decay. This is a constant value for a specific substance and is not affected by external factors. Half-life is directly related to the rate of decay, meaning that substances with a shorter half-life will decay faster than those with a longer half-life.

## 3. How is half-life used to measure the age of a substance?

By measuring the amount of radioactive material in a substance and knowing its half-life, scientists can calculate the age of the substance. This is based on the principle that the amount of radioactive material decreases over time as it decays, and the rate of decay is constant according to the substance's half-life.

## 4. Can half-life be used to predict when a substance will completely decay?

No, half-life cannot be used to predict when a substance will completely decay. While it can provide an estimate of when half of the substance will decay, the remaining half will continue to decay at a constant rate, making it impossible to determine exactly when all of the substance will decay.

## 5. Are there any factors that can affect the half-life of a substance?

No, the half-life of a substance is constant and cannot be affected by external factors. However, certain factors such as temperature and pressure can affect the rate of decay, which in turn can affect the amount of time it takes for half of the substance to decay. But the half-life itself remains constant.

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