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## Class 12 Physics (India)

### Course: Class 12 Physics (India)>Unit 13

Lesson 3: Half life and decay rate

# Exponential decay formula proof (can skip, involves calculus)

We can use half-lifes of radioactive substances to determine how old something is. Using your calculus skills, learn how to use the N(t)=Ne^(-kt) exponential decay equation to measure the amount of a radioactive substance we have at time T. Created by Sal Khan.

## Want to join the conversation?

• what is mean by lambda
• λ = 1.21 x 10^-4
• How did the scientists calculate the half life of an element ?
• We measure the decay constant, which can be done in a lab fairly easily. This is the constant we would normally use in computations, not the half-life. However, the half-life can be calculated from the decay constant as follows:
half-life = ln (2) / (decay constant).

To measure the decay constant, we take a sample of known mass and measure the number of radioactive decays per second as a function of time. Then we do a little bit of math to get the decay constant.
• How do people really determine the half life of elements or compounds which have have great half life periods? Like C-14, how is it really done?
• how did we know that there is such a thing as a half-life in the first place? as compared to say, a quartet-life or a sixtet-life? how did we realise in the very beginning that radioactive decay always conforms to such an intriguing exponential pattern that is dependent on its half-life?
(1 vote)
• I did not get why the rate of decay is dependent on the amount of substance we already have. How does a nucleus "know" that it is surrounded by more nuclei and it is supposed to decay faster than another nucleus that is surrounded by lesser number of nuclei?
• The disintegration process is completely random. A given nucleus Doesn't know when its supposed to decay, but the rate evens out when you have billions of billions of atoms.
If you take a piece of uranium, it will decay at a certain rate.
If you divide it into two pieces but keep them together. you will still get the same number of disintegrations per second.
If you take only one piece (half the size), you will get half the disintegrations per second (the other piece is giving off the other half, but you aren't counting them).
Thus, the observed rate of decay depends on the amount of substance you have.
• How do you get to know which Carbon to use @ time.. C12 or C14 ?
• C14 is a radioactive isotope of carbon, while C12 is just a stable isotope. Only radioactive isotopes can undergo radioactive decay.
(1 vote)
• why carbon 14 changed with nitrogen 14??
• Carbon-14 is a radioactive isotope of carbon. It has 6 protons and 8 neutrons. One of the neutrons turns into a proton by emitting an electron. The atom is now Nitrogen-14 because it has 7 protons and 7 neutrons.
• Is activity in Becquerels the same as the decay constant? Say I was given a radioactive compound with an activity of 10 Bq, could I just slot '10' in for lambda?
• No, they are not the same. The decay constant is mass independent, but the activity (which is measured in Becquerels) is not.

So, the equation is:
A (in Becquerels) = λN
where N is the number of atoms of the substance
• At , I don't understand how Sal interpreted the answer as, 1.2 * 10^-4. Why isn't is 1.2 * e^-4?
(1 vote)
• You misunderstood a common notation: 1.2e-4 means 1.2 x 10⁻⁴. You will sometimes (not commonly) see this written as 1.2EE-4
To indicate something to the power of e, you would write 1.2exp(-4) which does mean 1.2 * e⁻⁴
• why is e to the ln(N) equal to N?
(1 vote)
• because the natural log, by definition, is the inverse of raising e to a power.
Sal has vids about logarithms that you can watch if you need to review the properties of logs.