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

Lesson 3: Half life and decay rate

# Exponential decay problem solving

Exponential decay refers to a process in which a quantity decreases over time, with the rate of decrease becoming proportionally smaller as the quantity gets smaller. Use the exponential decay formula to calculate k, calculating the mass of carbon-14 remaining after a given time, and calculating the time it takes to have a specific mass remaining. Created by Sal Khan.

## Want to join the conversation?

• There's some unstable atoms that have a very short half-life (in terms of seconds),how could we study them or even know that they exist if they are changing in such a rate?
• Usually we simply observe the things that they have decayed into. We know what the 'rules' of nuclear decay are, so we can work out what unstable atom was there. However, while this does tell us about the physical presence of such atoms, it doesn't tell us anything about their chemical properties, so for very unstable elements -- particularly the very heavy ones with atomic numbers over 100 -- we can only estimate the pure elements melting point, colour, density, etc, based on what we know about other, similar elements.
• what's an ion?
• An ion is an atom or molecule that has become positively charged (known as a "cation") or negatively charged (known as an "anion"). The charge is the result of the loss or gain of a valence electron. Losing an electron will cause a positive charge, gaining an electron will cause a negative charge.
• at sal says something 'natralog' ,what does it means?
• lt is the natural logarithm. You need to have mastered logarithms and exponential function before attempting to study chemistry, as they are used quite extensively.
This is an Algebra II topic -- and you must know this material before being able to study General Chemistry:
• What is the symbol for a natural log?
• ln is the way we denote natural log. ln(e) = 1.
• How come for some questions you make the constant k positive and for radioactive decay, the constant k becomes negative? Is it supposed to be like this for these two types of examples?
• If you want decay, you need to raise e to a negative power. If you want growth, the power needs to be positive.
• If radioactive atoms are constantly decaying, how do they exist in the first place?
(1 vote)
• They get created in stars, and then for some of them it takes a long time to decay. If they decay fast then they are really rare!
• Hi, at when you substituted the initial amount of C-14 into the equation, shouldn't it be converted into SI units (kg)? In addition to this, does time have an SI units of seconds in physics, meaning the values must be converted to the appropriate SI units before being substituted in? Thanks!
• Strictly speaking, kg ought to be used. If you have an instructor who is insistent upon strict adherence to SI units, then you must use kg.

However, kg is typically too big of a unit for practical use in laboratory chemistry. Thus, we often use g, mg, or µg even though they are not SI. For that matter, we use liter and mL when neither of those is SI because m³ is just to much volume for practical lab use. In fact, SI has officially listed the liter as a "non-SI unit acceptable within SI" because very few people are willing to try to use m³ instead of a liter.

So, yes, strictly speaking the only mass SI accepts is the kg. But, unless your instructor insists otherwise, in chemistry grams are much more likely to be used.
• Why is the formula `Ne^(-kt)`? Wouldn't it make more intuitive sense for the formula to be `N(1/2)^(t/h)` where "h" is the half life?
• The half-life formula is derived using 1st order kinetics since radioactive decay is a first order reaction.

A first order reaction has the general form of: A -> products. For radioactive decay problems you can imagine the reactant decaying into new nuclides where the rate of the reaction only depends on the original radioactive nuclide.

The rate law is written as: Rate=k[A], where 'k' is the rate constant and [A] is the concentration of the reactant in molarity. This can be rewritten as: -Δ[A]/Δt = k[A], where the rate is being expressed as the disappearance of reactant A per unit of time. If you use some calculus to figure out the integrated rate law (it's a separable differential equation) you arrive at: ln[A] = -kt + ln[A]o, where [A]o is the initial concentration of A.
This can be rearranged into: ln([A]/[A]o) = -kt
Which can be rearranged into: [A] = [A]o*e^(-kt) which is the form Sal is using which ultimately originated from the rate law of a first order reaction.

The other equation is derived from ln([A]/[A]o) = -kt. At the time of half life (h), half of the original sample has decayed which can be written as: ln((1/2*[A]o)/[A]o) = -kh.
Which simplifies into ln(1/2) = -kh. And if we solve for half life we get: h = -ln(1/2)/k, which is where Sal got his equation. So all the equations result from chemical kinetics of a first order reaction. Hope that clears things up.
• I understand that e is a mathematical number but whats its exact number or rounded number
(1 vote)
• It's like pi, you can't write all the digits of it.
You can easily use the internet to find it to however many decimal places you like.