Half Life Remaining Energy Calculator

Formula used

This calculator models exponential half‑life behavior for energy:

E = E₀ × (1/2)^(t / t₁/₂)

• E₀ is initial energy.
• t₁/₂ is half life.
• t is elapsed time.
• Half‑lives elapsed: n = t / t₁/₂.
• Decay constant: λ = ln(2) / t₁/₂ (in 1/s).
• Mean life: τ = 1/λ (in seconds).

How to use this calculator

1. Enter the initial energy value and select its unit.
2. Enter the half life and choose the time unit.
3. Enter the elapsed time and choose the time unit.
4. Select an output energy unit for the results.
5. Press Calculate to view results above.
6. Use the export buttons to download a CSV or PDF report.

Example data table

Sample values to understand typical outputs.

Half life remaining energy guide

1) Why half life fits energy decay

Many physical processes reduce a tracked energy quantity by a constant fraction per equal time interval. A half life model assumes the quantity halves every t₁/₂, producing a smooth exponential curve. It is useful when proportional change dominates across time scales.

2) Core equation and key constants

The calculator applies E = E₀ × (1/2)^(t/t₁/₂). The constant ln(2) ≈ 0.693147 connects half life to the decay constant: λ = ln(2)/t₁/₂. Mean life follows τ = 1/λ.

3) Reading the “half-lives elapsed” value

The ratio n = t/t₁/₂ tells how many half life intervals have passed. Whole numbers are intuitive: n = 1 means 50% remains, n = 2 means 25%, n = 3 means 12.5%. Fractional n values are valid, such as n = 0.5 giving about 70.71% remaining.

4) Practical unit handling and conversions

Inputs support joules and multiples (kJ, MJ), electron-volts (eV to GeV), and watt-hours (Wh, kWh). Time can be seconds, minutes, hours, days, or years. Internally, the page converts energy to joules and time to seconds, then converts results back to your chosen output unit.

5) Example interpretation with real numbers

If E₀ = 1000 J, t₁/₂ = 10 s, and t = 25 s, then n = 2.5. Remaining fraction is (1/2)^{2.5} ≈ 0.1767767, so remaining energy is about 176.7767 J and percent remaining is 17.6777%.

6) Sensitivity and uncertainty notes

Small uncertainty in half life can noticeably shift long-duration results. A 5% uncertainty in t₁/₂ can accumulate across multiple half lives and change the remaining percentage. Use consistent units, precise timing, and repeat measurements when possible.

7) When the half life model may not apply

If energy loss is dominated by a constant rate (linear decrease) or by changing environmental conditions, the half life assumption can mislead. Examples include fixed power draw, saturating systems, or externally forced behavior. Treat the output as an approximation and compare against direct measurements.

8) Reporting and sharing results

Export tools generate a CSV for analysis and a PDF for documentation. Include initial energy, half life, elapsed time, and output unit so others can reproduce the calculation. Decay constant and mean life provide standardized descriptors for comparisons across processes.

FAQs

1) What does “remaining energy” mean here?

It is the portion of the starting energy that remains after exponential half life decay over the elapsed time, computed as E = E₀ × (1/2)^(t/t₁/₂).

2) Can I use fractional half lives?

Yes. The exponent t/t₁/₂ can be any non‑negative real number. Fractional values represent partial progress through a half life and are handled using the same formula.

3) Why is the decay constant shown in 1/s?

The decay constant λ is computed in seconds to keep a consistent base unit. It is derived using λ = ln(2)/t₁/₂ after converting half life to seconds.

4) How are eV and joules related?

Electron‑volt units are converted using 1 eV = 1.602176634×10⁻¹⁹ J. The calculator converts to joules internally, then converts back to your selected output unit.

5) What is mean life and when is it useful?

Mean life is τ = 1/λ. It provides another decay time scale and is convenient when comparing systems that report decay constants rather than half lives.

6) Why are some results in scientific notation?

Very large or very small energies can be hard to read in standard decimals. Scientific notation keeps results compact while retaining precision, especially for keV–GeV values or long time spans.

7) Does “energy lost” mean energy is destroyed?

Not necessarily. “Energy lost” here is the decrease in the tracked quantity relative to the half life model. In real systems, that energy may transform into other forms, be emitted, or move to another part of the system.