Understanding Radioactive Decay: What Fraction of a 150 g Sample Decays in 10 minutes?
Radioactive decay is a stochastic process that governs how unstable nuclei transform into more stable configurations, releasing energy in the form of particles or electromagnetic radiation. Day to day, when you ask “what fraction of a 150 g sample decays in 10 minutes? ”, the answer hinges on the isotope’s half‑life, the mathematical description of decay, and the way we calculate fractions from exponential laws. This article walks you through the concepts, the step‑by‑step calculation, and the scientific reasoning behind the numbers, so you can confidently determine the decayed fraction for any radioactive material Small thing, real impact..
1. Introduction to Radioactive Decay
1.1 What Is Decay?
Radioactive decay occurs when an unstable atomic nucleus loses energy by emitting radiation. The process is random for any single nucleus, but when billions of atoms are considered, the overall behavior follows a precise mathematical pattern And that's really what it comes down to..
1.2 The Role of Half‑Life
The half‑life (t½) is the time required for half of the radioactive atoms in a sample to transform. It is a characteristic property of each isotope and is independent of the sample’s size, chemical form, temperature, or pressure Easy to understand, harder to ignore..
1.3 Why Focus on Fractions?
Instead of dealing with absolute masses, scientists often express decay in terms of fractions or percentages because the underlying exponential law is easier to manipulate mathematically. Knowing the fraction that decays in a given period allows you to predict radiation levels, plan safety protocols, and estimate the remaining activity of a source.
2. The Exponential Decay Law
The number of undecayed nuclei (N(t)) at time (t) is described by the first‑order kinetic equation:
[ N(t) = N_0 , e^{-\lambda t} ]
- (N_0) – initial number of radioactive nuclei (or mass, if proportional)
- (\lambda) – decay constant (s(^{-1}))
- (t) – elapsed time
The decay constant is related to the half‑life by:
[ \lambda = \frac{\ln 2}{t_{½}} ]
Using these relationships, the fraction remaining after time (t) is:
[ \frac{N(t)}{N_0} = e^{-\lambda t} ]
As a result, the fraction decayed is:
[ \text{Fraction decayed} = 1 - e^{-\lambda t} ]
3. Step‑by‑Step Calculation for a 150 g Sample
3.1 Identify the Isotope and Its Half‑Life
The problem statement does not specify the isotope, so we will illustrate the calculation with a generic half‑life and then show how to plug in a real example (e.g., Cobalt‑60, (t_{½}=5.27) years). For a 10‑minute interval, the half‑life will dominate the result:
If the half‑life is much longer than 10 minutes, only a tiny fraction decays; if it is shorter, a large fraction will vanish.
3.2 Convert Units Consistently
All time units must match. Convert the half‑life to minutes (or seconds) and the elapsed time (t = 10) minutes accordingly.
Example with Cobalt‑60: [ t_{½}=5.27 \times 365.27\ \text{years}=5.25 \times 24 \times 60 \approx 2 Worth keeping that in mind..
3.3 Compute the Decay Constant
[ \lambda = \frac{\ln 2}{t_{½}} = \frac{0.6931}{2.77 \times 10^{6}\ \text{min}} \approx 2.50 \times 10^{-7}\ \text{min}^{-1} ]
3.4 Apply the Decay Formula
[ \text{Fraction decayed} = 1 - e^{-\lambda t} = 1 - e^{-(2.50 \times 10^{-7})(10)} ]
Because (\lambda t) is very small, we can use the first‑order Taylor approximation (e^{-x}\approx 1-x):
[ \text{Fraction decayed} \approx 1 - (1 - 2.5 \times 10^{-6}) = 2.5 \times 10^{-6} ]
Thus, only about 0.00025 % of the 150 g Cobalt‑60 sample decays in 10 minutes.
3.5 Convert Fraction to Mass (Optional)
If you need the actual mass that decayed:
[ \Delta m = \text{Fraction decayed} \times 150\ \text{g} = 2.5 \times 10^{-6} \times 150\ \text{g} \approx 3.8 \times 10^{-4}\ \text{g} = 0.
So, roughly 0.38 mg of the original 150 g sample transforms in ten minutes That's the part that actually makes a difference..
4. General Formula for Any Isotope
To avoid recalculating each time, keep a template:
- Gather data – half‑life (t_{½}) (in minutes) and initial mass (m_0).
- Compute decay constant: (\displaystyle \lambda = \frac{\ln 2}{t_{½}}).
- Insert elapsed time (t) (10 min) into the fraction equation: [ f_{\text{decayed}} = 1 - e^{-\lambda t} ]
- Multiply by (m_0) if the actual mass loss is needed.
This workflow works for any isotope, from short‑lived Iodine‑131 (half‑life ≈ 8 days) to long‑lived Uranium‑238 (half‑life ≈ 4.5 × 10⁹ years) Simple, but easy to overlook..
5. Scientific Explanation: Why the Exponential?
Radioactive decay follows a first‑order kinetic process because each nucleus has the same probability per unit time of decaying, regardless of how many neighbors have already decayed. This constant probability is the decay constant (\lambda). Mathematically, the rate of change of undecayed nuclei is:
[ \frac{dN}{dt} = -\lambda N ]
Integrating this differential equation yields the exponential law shown earlier. The memoryless nature of the process (the probability of decay in the next instant does not depend on the past) is a hallmark of Poisson processes, which underlie many natural phenomena such as radioactive decay, photon emission, and even certain biological events.
6. Frequently Asked Questions (FAQ)
6.1 Does the sample’s mass affect the half‑life?
No. Half‑life is an intrinsic property of the isotope; a 1 g or 150 g sample of the same nuclide will have the same half‑life. Only the number of nuclei changes, which scales linearly with mass No workaround needed..
6.2 What if the half‑life is comparable to 10 minutes?
When (t_{½}) ≈ 10 min, the fraction decayed becomes significant. Here's one way to look at it: with (t_{½}=10) min: [ \lambda = \frac{0.693}{10}=0.0693\ \text{min}^{-1} ] [ f_{\text{decayed}} = 1 - e^{-0.0693 \times 10}=1 - e^{-0.693}=1 - 0.5 = 0.5 ] Exactly 50 % of the sample would decay in 10 minutes Easy to understand, harder to ignore..
6.3 Can I use a linear approximation for very short times?
Yes, for (t \ll t_{½}) the fraction decayed ≈ (\lambda t). This is the same as the Taylor series first‑order term used earlier and is handy for quick mental estimates.
6.4 How do I handle mixtures of isotopes?
Treat each isotope separately, compute its own decayed fraction, then sum the contributions weighted by the initial mass fraction of each nuclide.
6.5 Is temperature a factor?
In most cases, temperature has negligible impact on nuclear decay rates. Exceptions exist for electron‑capture decays under extreme pressure, but they are rare and not relevant for typical laboratory or medical scenarios Easy to understand, harder to ignore..
7. Practical Applications
| Field | Why Knowing the Decayed Fraction Matters |
|---|---|
| Medical Imaging | Dosimetry calculations for radiopharmaceuticals (e.g.Even so, |
| Archaeology | Radiocarbon dating uses the known half‑life of (^{14})C (≈ 5,730 years) to calculate the fraction of carbon‑14 remaining in organic artifacts. |
| Industrial Radiography | Planning exposure times for gamma‑ray sources (e.Worth adding: , Technetium‑99m, half‑life ≈ 6 h) require precise knowledge of activity loss during preparation and administration. So |
| Nuclear Power | Fuel management relies on predicting how much uranium‑235 remains after a given operating period to schedule refueling. g. |
| Environmental Monitoring | Estimating how quickly fallout from a nuclear accident diminishes helps set evacuation timelines and food‑safety limits. , Iridium‑192) depends on the decay fraction to ensure sufficient penetration without over‑exposure. |
Worth pausing on this one.
Understanding the fraction that decays in a short interval like 10 minutes is especially critical when timing matters—such as synchronizing a PET scan with the peak activity of a short‑lived tracer.
8. Common Pitfalls and How to Avoid Them
- Mixing Units – Always convert half‑life and elapsed time to the same unit (seconds, minutes, or years).
- Neglecting the Exponential Nature – Using a linear decay model for long periods yields wildly inaccurate results.
- Assuming 100 % Decay – Even after many half‑lives, a tiny fraction of nuclei remains; the exponential tail never truly reaches zero.
- Forgetting Mass‑Number to Nuclei Conversion – If you need the exact number of atoms, use Avogadro’s number: (N = \frac{m}{M} N_A), where (M) is molar mass.
By keeping these points in mind, you’ll produce reliable decay calculations every time.
9. Conclusion
The fraction of a 150 g radioactive sample that decays in 10 minutes is dictated solely by the isotope’s half‑life and the exponential decay law. By converting the half‑life to a decay constant, inserting the 10‑minute interval, and applying the simple formula
[ \text{Fraction decayed}=1-e^{-\lambda t}, ]
you can quickly determine the decayed portion for any nuclide. Day to day, for a long‑lived isotope like Cobalt‑60, the fraction is minuscule (≈ 2. 00025 %). 5 × 10⁻⁶, or 0.Conversely, for a short‑lived isotope with a half‑life near 10 minutes, the fraction can be as high as 50 % or more.
Mastering this calculation empowers you to assess radiation safety, plan medical procedures, interpret archaeological dates, and manage nuclear materials with confidence. Remember: the key steps are identify the half‑life, compute the decay constant, apply the exponential formula, and, if needed, translate the fraction into mass or activity. With these tools at hand, you’ll figure out the world of radioactive decay with both precision and insight.
