A radioactive substance decays exponentially, halving every 8 days. If the initial mass is 128 grams, how many grams remain after 32 days?
["Understanding Exponential Decay: Radioactive Substances Halving Over Time", "Radioactive decay is a natural process that describes how unstable atomic nuclei lose energy by emitting radiation, transforming into lighter elements over time. One of the most well-known features of radioactive decay is its exponential nature—each substance decays at a consistent, predictable rate, halving its quantity in a fixed period, a phenomenon known as half-life.", "### What Is Half-Life and Why Does It Matter?", "The half-life of a radioactive substance is the time required for half of the original material to decay. In this article, we explore a classic example: a radioactive substance that decays with a half-life of 8 days, starting from an initial mass of 128 grams. We will calculate how much remains after 32 days—a period that spans five half-lives.", "### How Exponential Decay Works Mathematically", "Exponential decay follows the formula:\n[\nm(t) = m_0 \ imes \left(\frac{1}{2}\right)^{\frac{t}{T}}\n]\nwhere:\n- ( m(t) ) = mass remaining at time ( t )\n- ( m_0 ) = initial mass\n- ( T ) = half-life (in days)\n- ( t ) = elapsed time (in days)", "In our case:\n- ( m_0 = 128 ) grams\n- ( T = 8 ) days\n- ( t = 32 ) days", "Since ( 32 \div 8 = 4 ), the substance undergoes 4 half-lives, so the decay is:\n[\nm(32) = 128 \ imes \left(\frac{1}{2}\right)^4\n]", "Calculating step-by-step:\n[\n\left(\frac{1}{2}\right)^4 = \frac{1}{16}\n]\n[\nm(32) = 128 \ imes \frac{1}{16} = 8 \ ext{ grams}\n]", "### The Result: 8 Grams Remain After 32 Days", "After 32 days—spanning four half-lives—the remaining mass of the radioactive substance is 8 grams. This demonstrates how quickly decay progresses when measured in half-life intervals. Despite starting with a substantial 128 grams, the material decays efficiently, halving every 8 days and dropping to just 8 grams after four periods.", "### Real-World Implications", "Understanding exponential decay is essential in nuclear physics, medicine, environmental science, and energy. For example, radioisotopes used in cancer treatment or medical imaging rely on predictable decay rates. Knowing the remaining quantity after a given time helps with dosing, waste management, and safety planning.", "### Summary", "- Half-life: the time for half the substance to decay\n- After 32 days (4 half-lives) and a half-life of 8 days:\n[\n128 \ imes \left(\frac{1}{2}\right)^4 = 8 \ ext{ grams}\n]\nExponential decay transforms large initial masses into small, measurable amounts in known intervals—an elegant and predictable natural law.", "---", "Key Takeaway:\nAfter 32 days, only 8 grams remain from the original 128 grams, illustrating the powerful efficiency of exponential decay with a half-life of 8 days."]
