Exponential Decay Calculator

About Exponential Decay Calculator

Exponential decay: quantity decreases by a constant fraction per time period. After one half-life, 50% remains.

How to use this calculator

1. Enter the starting amount, or initial value, as the amount at time 0.
2. Enter the decay rate or half-life, depending on what you know.
3. Enter the time elapsed in the same time unit used for the rate.
4. Click calculate to find the remaining amount, then check the formula and steps to see how the result was found.

The formula explained

The formula \(N(t) = N_0 \cdot e^{-\lambda t}\) computes the amount left after time \(t\) when a quantity decays continuously. Here, \(N_0\) is the starting amount, \(\lambda\) is the decay constant, and \(e\) gives the continuous decay pattern.

• N(t) = the amount remaining after time \(t\)
• \(N_0\) = the initial amount at time 0
• \(\lambda\) = the decay constant, a positive number that controls how fast the quantity decreases
• t = the elapsed time
• e = the base of natural logarithms, about 2.71828

Step by step method

1. Start with the formula \(N(t) = N_0 e^{-\lambda t}\).
2. Substitute the given values for \(N_0\), \(\lambda\), and \(t\).
3. Compute the exponent, then evaluate the exponential.
4. Multiply by the starting amount to get the remaining quantity.

Worked example

Problem. A substance starts with 500 grams and decays at a rate of \(\lambda = 0.2\) per hour. How much remains after 6 hours?

1. Use the formula \(N(t) = N_0 e^{-\lambda t}\).
2. Substitute the values: \(N(6) = 500e^{-0.2\cdot 6} = 500e^{-1.2}\).
3. Evaluate: \(e^{-1.2} \approx 0.3010\), so \(N(6) \approx 500 \cdot 0.3010 = 150.5\).

Tips and common mistakes

• Make sure the time unit matches the decay rate unit, for example hours with per hour or years with per year.
• A larger decay constant means faster decay, so the remaining amount drops more quickly.

Frequently asked questions