A radioactive substance decays exponentially with a half-life of 10 years. If the initial mass is 50 grams, calculate the remaining mass after 25 years.
["Title: Exponential Decay of Radioactive Substances: Decay of a 50-Gram Sample with a 10-Year Half-Life", "---", "Radioactive decay is a fundamental concept in nuclear physics and plays a critical role in fields ranging from medicine to archaeology. One of the key characteristics of radioactive materials is their exponential decay, which means the amount of substance decreases at a rate proportional to its current mass. A classic example is a radioactive isotope with a half-life of 10 years — a well-known and crucial property for predicting how long a substance remains active. In this article, we explore the exponential decay model and calculate how much of an initial 50-gram sample remains after 25 years.", "### Understanding Exponential Radioactive Decay", "Radioactive decay follows an exponential model described mathematically by:", "[ m(t) = m_0 \cdot \left( \frac{1}{2} \right)^{t/T} ]", "Where:\n- ( m(t) ) = remaining mass at time ( t ) (in years)\n- ( m_0 ) = initial mass = 50 grams\n- ( T ) = half-life = 10 years\n- ( t ) = elapsed time = 25 years", "This formula reflects that every 10 years, the mass halves. Instead of calculating fractional decay each year, the exponential formula efficiently computes remaining quantity.", "---", "### Step-by-Step Calculation", "Given:\n- ( m_0 = 50 ) grams\n- Half-life ( T = 10 ) years\n- Time elapsed ( t = 25 ) years", "The decay factor is based on how many half-lives have passed:", "[ \ ext{Number of half-lives} = \frac{t}{T} = \frac{25}{10} = 2.5 ]", "So, the remaining mass is:", "[\nm(25) = 50 \cdot \left( \frac{1}{2} \right)^{2.5} = 50 \cdot 2^{-2.5}\n]", "We calculate ( 2^{-2.5} ):", "[\n2^{-2.5} = \frac{1}{2^{2.5}} = \frac{1}{2^2 \cdot 2^{0.5}} = \frac{1}{4 \cdot \sqrt{2}} \approx \frac{1}{4 \cdot 1.4142} \approx \frac{1}{5.6568} \approx 0.1768\n]", "Thus:", "[\nm(25) \approx 50 \cdot 0.1768 = 8.84 \ ext{ grams}\n]", "Alternatively, using logarithms or direct exponentiation:", "[\n2^{-2.5} = e^{-2.5 \ln 2} \approx e^{-1.732} \approx 0.177\n]", "So,", "[\nm(25) \approx 50 \cdot 0.177 = 8.85 \ ext{ grams (rounded)}\n]", "For higher precision, using a calculator on ( 2^{-2.5} \approx 0.1767767 ):", "[\nm(25) = 50 \ imes 0.1767767 \approx 8.8388 \ ext{ grams}\n]", "---", "### Conclusion: The Remaining Mass After 25 Years", "After 25 years, a 50-gram sample of this radioactive substance—with a half-life of 10 years—decays to approximately 8.84 grams. This demonstrates the rapid reduction over time, highlighting the effectiveness of radioactive decay processes in applications like radiometric dating, medical tracers, and nuclear waste management.", "Understanding half-life and exponential decay enables scientists and engineers to predict material behavior, plan safety protocols, and utilize radioactive isotopes responsibly.", "---", "Keywords for SEO:\nradioactive decay, half-life formula, exponential decay, half-life calculation, remaining mass after 25 years, 50 gram decay, nuclear physics, radioactive halves, decay calculation, time and mass decay, half-life explanation, decay percentage workshop.", "---", "This simple yet powerful model remains indispensable in science and industry, offering precise predictions of how long and how much of a radioactive substance remains after any given time."]
