Enter the three you know out of the initial quantity N₀, remaining quantity N(t), elapsed time t and half-life t½. The one left blank is calculated, along with the decay constant λ and the mean lifetime τ. The equation below is linked to the fields, so you can also type the numbers right into it.
Leave only the field you want to find blank (fill in exactly three). All values are numbers greater than 0. Any units are fine, but use the same unit for both time fields and for both quantity fields (for example, years for both times). The remaining quantity must be smaller than the initial quantity.
Result and graph
Fill in three of the fields on the left and press "Calculate". The result and a graph will appear here.
Table of Contents
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What you can do on this page
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What is this calculation used for?
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How to Use
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Formulas and graphs
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Symbols and terms
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Good to know before you start
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How to calculate it in Excel
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How to calculate it in Google Sheets
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How to calculate it in Python
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How to write it in LaTeX and other math languages (copy and paste)
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How to have ChatGPT do the calculation
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DataChef Features
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Related Features
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NumberChef Calculators List
What you can do on this page
- Enter three of the initial quantity \(N_0\), the remaining quantity \(N(t)\), the elapsed time \(t\) and the half-life \(t_{1/2}\), and the one left blank is calculated
- The decay constant \(\lambda\) and the mean lifetime \(\tau\) are shown at the same time
- A graph with a marker at each half-life shows how the quantity keeps halving (exponential decay)
- A plain-language explanation of the formulas and copy-and-paste formulas for Excel, Google Sheets and Python are all on this page
This works for quantities that drop by the same fraction over each equal stretch of time (exponential decay). Radioactive decay is the classic example, and the same formula is used to model drug levels in the blood. Any units are fine, but use the same unit for the two time fields (elapsed time and half-life) and for the two quantity fields (initial and remaining quantity).
What is this calculation used for?
Carbon-14 dating (archaeology)
Living things take in radioactive carbon-14. After death, the intake stops, and the carbon-14 decreases with a half-life of about 5730 years. If a piece of wood from an ancient site has 25% of the carbon-14 it had when alive, it is estimated to be \(5730 \times \ln 4 \div \ln 2 = 11460\) years old (2 half-lives).
Many of the "thousands of years ago" dates for ancient sites and artifacts in textbooks are based on this calculation.
Estimating how medical radioisotopes decrease (a medical example)
Technetium-99m, used in hospital nuclear medicine scans, has a half-life of about 6 hours. As an example, 24 hours after a scan it has dropped to \((1/2)^{24/6} = (1/2)^4 = 1/16\), which is 6.25%.
Short half-life isotopes are chosen in medicine because they decrease quickly in the body like this (in reality the body also clears them, so safety is judged by the standards of expert agencies).
How long medicines and caffeine stay in the body
How fast a drug leaves the blood is also described by its half-life. For example, if caffeine has a half-life of 4 hours, the caffeine in about two cups of coffee (200 mg) drops to \(200 \times (1/2)^{3} = 25\) mg after 12 hours.
Half-lives vary a lot from person to person, and decisions about medicine should follow your doctor or pharmacist. Still, the numbers show why coffee late in the day can affect your sleep.
Reading news about radioactive materials (emergency preparedness)
Iodine-131, which often appears in news about nuclear accidents, has a half-life of about 8 days. After 80 days (10 half-lives) it drops to \((1/2)^{10} = 1/1024\), about 0.1%. Cesium-137, with a half-life of about 30 years, still has \(1/8\) (12.5%) left after 90 years.
Once you see that "short half-life = fades quickly, long half-life = stays a long time", you can check the numbers in the news yourself.
Estimating how long radioactive waste must be managed
For radioactive waste from nuclear power and other sources, how long it must be isolated and managed is planned from the half-lives of the isotopes it contains. For example, this formula shows that cesium-137, with a 30-year half-life, needs about 10 half-lives (about 300 years) to drop to nearly 1/1000.
The same calculation as on this page is the basis of management plans that span many generations.
Formulas and graphs
Formula for the remaining quantity (the basic half-life formula)
Graph
Standard notation
(the usual math form)
\(N(t)\)
\(=\)
\(N_0\)
\(\times\)
\(\left(\dfrac{1}{2}\right)\)
\(t / t_{1/2}\)
In words
(symbols replaced with words)
④
\(N(t)\): quantity left after time \(t\)
\(=\)
①
\(N_0\): initial quantity
\(\times\)
②
\(\dfrac{1}{2}\): half
③
\(t / t_{1/2}\): elapsed time ÷ half-life
The formula in words
①
Multiply the
\(N_0\): initial quantity
②
by
\(\dfrac{1}{2}\): half
③
over and over, as many times as
\(t / t_{1/2}\): elapsed time ÷ half-life
(the number of half-lives in the elapsed time)
④
and you get the
\(N(t)\): remaining quantity
Quick example
If there are 8 mg of a drug with a half-life of 2 days in the body, the amount left after 6 days (6 days ÷ 2 days = halved 3 times) is
amount left after 6 days \(N(t)\)
\(=\)
initial quantity (8 mg)
\(\times\)
half \(\dfrac{1}{2}\)
6 days ÷ 2 days (3 times)
\(8 \times \left(\dfrac{1}{2}\right)^{3} = 8 \times \dfrac{1}{8} = 1\)
Key idea
The exponent at the upper right, \(t / t_{1/2}\), is "how many half-lives fit into the elapsed time", that is, the number of times the quantity halves. With 6 days ÷ 2 days = 3, it halves 3 times: 8 mg → 4 mg → 2 mg → 1 mg. Even when the count is not a whole number (for example, 2.5 half-lives), the extended idea of exponents lets you use this formula as is.
Formula for the half-life
Graph
Standard notation
(the usual math form)
\(t_{1/2}\)
\(=\)
\(t\)
\(\times\)
\(\ln 2\)
\(\div\)
\(\ln\left(\dfrac{N_0}{N(t)}\right)\)
In words
(symbols replaced with words)
④
\(t_{1/2}\): half-life
\(=\)
①
\(t\): elapsed time
\(\times\)
②
\(\ln 2\): natural log of 2 (about 0.693)
\(\div\)
③
\(\ln(N_0 / N(t))\): natural log of "initial ÷ remaining"
The formula in words
①
Multiply the
\(t\): elapsed time
②
by the
\(\ln 2\): natural log of 2 (about 0.693)
, then divide by the
③
\(\ln(N_0 / N(t))\): natural log of "initial ÷ remaining"
④
and you get the
\(t_{1/2}\): half-life
Quick example
If 100 g of a substance drops to 25 g in 10 years, its half-life is
half-life \(t_{1/2}\)
\(=\)
elapsed time (10 years)
\(\times\)
\(\ln 2\) (about 0.693)
\(\div\)
\(\ln(100 \div 25)\) (about 1.386)
\(\ln(100 \div 25) = \ln 4 \approx 1.386\)
\(t_{1/2} = 10 \times 0.693 \div 1.386 = 5\)
Key idea
The part \(\ln 2 \div \ln(N_0 / N(t))\) is actually 1 over "the number of times it halved". 100 g → 25 g is a drop by a factor of \(100 \div 25 = 4\), which is 2 halvings, so the half-life is 10 years ÷ 2 = 5 years. The formula uses logarithms to count how many times it halved.
Formula for the decay constant
Standard notation
(the usual math form)
\(\lambda\)
\(=\)
\(\ln 2\)
\(\div\)
\(t_{1/2}\)
In words
(symbols replaced with words)
③
\(\lambda\): decay constant
\(=\)
①
\(\ln 2\): natural log of 2 (about 0.693)
\(\div\)
②
\(t_{1/2}\): half-life
The formula in words
①
Take the
\(\ln 2\): natural log of 2 (about 0.693)
②
divide it by the
\(t_{1/2}\): half-life
③
and you get the
\(\lambda\): decay constant
Quick example
The decay constant of a substance with a 5-year half-life is
decay constant \(\lambda\)
\(=\)
\(\ln 2\) (about 0.693)
\(\div\)
half-life (5 years)
\(\lambda = 0.693 \div 5 \approx 0.139\)
Key idea
The decay constant \(\lambda\) tells what fraction of the quantity still left decays per very short unit of time. The shorter the half-life, the larger \(\lambda\) (faster decay); the longer the half-life, the smaller \(\lambda\) (slower decay).
Formula for the mean lifetime
Standard notation
(the usual math form)
\(\tau\)
\(=\)
\(t_{1/2}\)
\(\div\)
\(\ln 2\)
In words
(symbols replaced with words)
③
\(\tau\): mean lifetime
\(=\)
①
\(t_{1/2}\): half-life
\(\div\)
②
\(\ln 2\): natural log of 2 (about 0.693)
The formula in words
①
Take the
\(t_{1/2}\): half-life
②
divide it by the
\(\ln 2\): natural log of 2 (about 0.693)
③
and you get the
\(\tau\): mean lifetime
Quick example
The mean lifetime of a substance with a 5-year half-life is
mean lifetime \(\tau\)
\(=\)
half-life (5 years)
\(\div\)
\(\ln 2\) (about 0.693)
\(\tau = 5 \div \ln 2 \approx 7.213\)
Key idea
The mean lifetime \(\tau\) is the time until one atom decays, averaged over all the atoms. It is also the reciprocal of the decay constant (\(\tau = 1 / \lambda\)). It is a bit longer than the half-life, about 1.44 times, because "long-lived" atoms that take a long time to decay pull the average up.
Half-life calculations start from one idea, "elapsed time ÷ half-life = number of halvings". The remaining quantity is the initial quantity multiplied by \(\dfrac{1}{2}\) that many times. To find the half-life, you count the number of halvings with a logarithm.
Symbols and terms
Symbols
| \(N_0\) | N naught (N zero) | The initial quantity, the amount at the moment you start counting time (\(t = 0\)). (Example - if there are 100 g at first, \(N_0 = 100\)) |
| \(N(t)\) | N of t | The remaining quantity, the amount left after time \(t\) has passed. It is written this way because it is a function of \(t\) (its value depends on \(t\)). |
| \(t\) | t | The elapsed time, the time it took to drop from the initial to the remaining quantity. Use the same unit as the half-life. |
| \(t_{1/2}\) | t one-half (half-life) | The half-life, the time it takes for the quantity to fall to exactly half. Each substance has its own value, and it is the same no matter when you start measuring. |
| \(\lambda\) | lambda | The decay constant, the fraction that decays per very short unit of time. It is found by \(\lambda = \ln 2 \div t_{1/2}\). |
| \(\tau\) | tau | The mean lifetime, the average time until one atom decays. It is found by \(\tau = 1 \div \lambda = t_{1/2} \div \ln 2\). |
| \(\ln x\) | natural log of x | The natural logarithm, the logarithm with base \(e\) (about 2.718). It tells "\(e\) to what power gives \(x\)". On a scientific calculator use the ln key, and in Excel the LN function. |
Terms
| half-life | The time it takes for a quantity to fall to exactly half. For a radioactive substance it is the time for the number of atoms (the strength of the radioactivity) to halve, and for a drug the time for its blood level to halve. The quantity keeps halving over each equal stretch of time (half, a quarter, an eighth, and so on). |
| exponential decay | Decrease at a rate proportional to the amount still left. It is fast at first and slows down as less remains. Only quantities that decrease this way have a constant half-life. |
| radioactive decay | The process in which an unstable nucleus gives off radiation and turns into a different nucleus. When each single atom decays is random, but a large number of atoms decreases steadily according to the half-life. |
| decay constant | A constant that tells what fraction of the remaining atoms decays per very short unit of time. It goes hand in hand with the half-life; if you know one, you can calculate the other. |
| mean lifetime | The time until one atom decays, averaged over many atoms. It is about 1.44 times the half-life. |
| natural logarithm (natural log) | The logarithm with base \(e\) (about 2.718, Euler's number). It suits calculations about rates of growth and decay, and appears in the formulas for the half-life and the decay constant. |
| exponent | A number that tells how many times to multiply the same number, like the 3 at the upper right of \(2^3\). In the half-life formula, the number of halvings \(t / t_{1/2}\) is the exponent. |
Good to know before you start
Here is what helps you use the calculation on this page with real understanding, not just by pressing the button.
| Fractions and repeated halving (Grades 4–5) |
|
| Powers and exponents (Grade 6) |
|
| Extending exponents (Algebra 2) |
|
| Logarithms (Algebra 2) |
|
How to calculate it in Excel
Copy the whole table below and paste it into cell A1 in Excel. It works as is.
Table to find the remaining quantity
| Initial quantity N0 | 100 |
| Elapsed time t | 10 |
| Half-life t1/2 | 5 |
| Remaining quantity N(t) | =B1*POWER(1/2,B2/B3) |
Table to find the half-life
| Initial quantity N0 | 100 |
| Remaining quantity N(t) | 25 |
| Elapsed time t | 10 |
| Half-life t1/2 | =B3*LN(2)/LN(B1/B2) |
Table to find the elapsed time
| Initial quantity N0 | 100 |
| Remaining quantity N(t) | 25 |
| Half-life t1/2 | 5 |
| Elapsed time t | =B3*LN(B1/B2)/LN(2) |
Table to find the decay constant and mean lifetime
| Half-life t1/2 | 5 |
| Decay constant λ | =LN(2)/B1 |
| Mean lifetime τ | =B1/LN(2) |
After pasting, the upper rows in column B are your inputs, and the last row shows the result calculated automatically.
"POWER(1/2, B2/B3)" is the power function that multiplies by 1/2 (elapsed time ÷ half-life) times, and "LN" gives the natural logarithm (ln).
For example, B4 shows 25 in the first table (100 g becomes 25 g in 10 years) and 5 in the second table (a 5-year half-life). Just replace the numbers in column B with your own.
"POWER(1/2, B2/B3)" is the power function that multiplies by 1/2 (elapsed time ÷ half-life) times, and "LN" gives the natural logarithm (ln).
For example, B4 shows 25 in the first table (100 g becomes 25 g in 10 years) and 5 in the second table (a 5-year half-life). Just replace the numbers in column B with your own.
How to calculate it in Google Sheets
Copy the whole table below and paste it into cell A1 in Google Sheets. It works as is.
Table to find the remaining quantity
| Initial quantity N0 | 100 |
| Elapsed time t | 10 |
| Half-life t1/2 | 5 |
| Remaining quantity N(t) | =B1*POWER(1/2,B2/B3) |
Table to find the half-life
| Initial quantity N0 | 100 |
| Remaining quantity N(t) | 25 |
| Elapsed time t | 10 |
| Half-life t1/2 | =B3*LN(2)/LN(B1/B2) |
Table to find the elapsed time
| Initial quantity N0 | 100 |
| Remaining quantity N(t) | 25 |
| Half-life t1/2 | 5 |
| Elapsed time t | =B3*LN(B1/B2)/LN(2) |
Table to find the decay constant and mean lifetime
| Half-life t1/2 | 5 |
| Decay constant λ | =LN(2)/B1 |
| Mean lifetime τ | =B1/LN(2) |
The same formulas as in Excel (POWER and LN) work in Google Sheets as is. Copy the whole table, paste it into cell A1, and replace the numbers in column B with your own.
How to calculate it in Python
import math
initial_quantity = 100.0 # initial quantity N0
remaining_quantity = 25.0 # remaining quantity N(t)
elapsed_time = 10.0 # elapsed time t
# half-life t1/2 = t × ln2 ÷ ln(N0 ÷ N(t))
half_life = elapsed_time * math.log(2) / math.log(initial_quantity / remaining_quantity)
decay_constant = math.log(2) / half_life # decay constant λ
mean_lifetime = half_life / math.log(2) # mean lifetime τ
print(f"Half-life: {half_life}")
print(f"Decay constant: {decay_constant}")
print(f"Mean lifetime: {mean_lifetime}")
# Check: remaining quantity N(t) = N0 × (1/2)^(t/t1/2)
remaining = initial_quantity * 0.5 ** (elapsed_time / half_life)
print(f"Check (remaining quantity): {remaining}")
Runs with only the math module from the standard library. math.log() is the natural logarithm (ln), and "**" is the power operator. In this example, the half-life is 5.0, the decay constant is about 0.1386, the mean lifetime is about 7.213, and the check gives a remaining quantity of 25.0. Change the three values at the top and run it.
How to write it in LaTeX and other math languages (copy and paste)
Formula for the remaining quantity (the basic half-life formula)
N(t) = N₀ × (1/2)^(t/t½)
N(t) = N_0 \left(\frac{1}{2}\right)^{t/t_{1/2}}
<math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
<mrow>
<mi>N</mi><mo>(</mo><mi>t</mi><mo>)</mo>
<mo>=</mo>
<msub><mi>N</mi><mn>0</mn></msub>
<msup>
<mrow><mo>(</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mo>)</mo></mrow>
<mrow><mi>t</mi><mo>/</mo><msub><mi>t</mi><mrow><mn>1</mn><mo>/</mo><mn>2</mn></mrow></msub></mrow>
</msup>
</mrow>
</math>
N(t) = N_0 (1/2)^(t/t_(1/2))
n0*(1/2)^(t/thalf)
Nt := n0*(1/2)^(t/thalf);
Nt = n0*(1/2)^(t/thalf);
N(t) = N_0 (1/2)^(t/t_(1/2))
Formula for the half-life
t½ = t × ln 2 ÷ ln(N₀/N(t))
t_{1/2} = \dfrac{t \ln 2}{\ln(N_0 / N(t))}
<math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
<mrow>
<msub><mi>t</mi><mrow><mn>1</mn><mo>/</mo><mn>2</mn></mrow></msub>
<mo>=</mo>
<mfrac>
<mrow><mi>t</mi><mo>⁢</mo><mi>ln</mi><mo>⁡</mo><mn>2</mn></mrow>
<mrow><mi>ln</mi><mo>⁡</mo><mo>(</mo><msub><mi>N</mi><mn>0</mn></msub><mo>/</mo><mi>N</mi><mo>(</mo><mi>t</mi><mo>)</mo><mo>)</mo></mrow>
</mfrac>
</mrow>
</math>
t_(1/2) = (t ln 2)/(ln(N_0//N(t)))
t*Log[2]/Log[n0/nt]
thalf := t*ln(2)/ln(n0/nt);
thalf = t*log(2)/log(n0/nt);
t_(1/2) = t ln(2)/ln(N_0/N(t))
Formula for the decay constant
λ = ln 2 ÷ t½
\lambda = \dfrac{\ln 2}{t_{1/2}}
<math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
<mrow>
<mi>λ</mi>
<mo>=</mo>
<mfrac>
<mrow><mi>ln</mi><mo>⁡</mo><mn>2</mn></mrow>
<msub><mi>t</mi><mrow><mn>1</mn><mo>/</mo><mn>2</mn></mrow></msub>
</mfrac>
</mrow>
</math>
lambda = (ln 2)/t_(1/2)
Log[2]/thalf
lambda := ln(2)/thalf;
lambda = log(2)/thalf;
λ = ln(2)/t_(1/2)
Formula for the mean lifetime
τ = t½ ÷ ln 2
\tau = \dfrac{1}{\lambda} = \dfrac{t_{1/2}}{\ln 2}
<math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
<mrow>
<mi>τ</mi>
<mo>=</mo>
<mfrac>
<msub><mi>t</mi><mrow><mn>1</mn><mo>/</mo><mn>2</mn></mrow></msub>
<mrow><mi>ln</mi><mo>⁡</mo><mn>2</mn></mrow>
</mfrac>
</mrow>
</math>
tau = t_(1/2)/(ln 2)
thalf/Log[2]
tau := thalf/ln(2);
tau = thalf/log(2);
τ = t_(1/2)/ln(2)
How to have ChatGPT do the calculation
You are a calculation assistant for half-life problems. Do the following calculations by actually running Python code, and base your answer only on the numbers from the execution result (do not answer by mental math or guessing). A radioactive substance starts at 100 g and has dropped to 25 g after 10 years. Assume it follows exponential decay, N(t) = N0 × (1/2)^(t / half-life). Find each of the following: 1. The half-life of this substance 2. The decay constant λ (= ln2 ÷ half-life) 3. The mean lifetime τ (= half-life ÷ ln2) 4. The amount left 20 years after the start Show the formulas you used and the numbers from the execution result.
How to Use
-
1Enter your numbersType the numbers you want to calculate with into the input fields
-
2CalculatePress the "Calculate" button
-
3Check the resultThe result appears on the spot. The same page also explains the idea behind the calculation and the formula
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Related Features
Exponent Calculator (Powers, a to the nth Power)
Calculates a to the nth power. The "(1/2) to the nth power" in the half-life formula is one of these
Logarithm Calculator (Log Calculator)
Finds the base, the argument or the log value from the other two. It is the basis of the formula for the half-life
Geometric Sequence Calculator (nth Term and Sum)
Works with sequences where you keep multiplying by the same number. The amounts after each half-life form a geometric sequence with common ratio 1/2
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