A sample containing 88^224 Ra, which decays by α-particle emission, disintegrates at the followi

step 1 Okay, so for this question, we're trying to find the half -life of radon. So, and then we're giv

A sample containing 88^224 Ra, which decays by α-particle...

A sample containing $_{88}^{224} \mathrm{Ra},$ which decays by $\alpha$ -particle emission, disintegrates at the following rate, expressed as disintegrations per minute or counts per minute $(\mathrm{cpm}): t=0,1000 \mathrm{cpm} ; t=1 \mathrm{h}$ $992 \mathrm{cpm} ; t=10 \mathrm{h}, 924 \mathrm{cpm} ; t=100 \mathrm{h}, 452 \mathrm{cpm}$ $t=250 \mathrm{h}, 138 \mathrm{cpm} .$ What is the half-life of this nuclide?

A sample containing $_{88}^{224} \mathrm{Ra},$ which decays by $\alpha$ -particle emission, disintegrates at the following rate, expressed as disintegrations per minute or counts per minute $(\mathrm{cpm}): t=0,1000 \mathrm{cpm} ; t=1 \mathrm{h}$
$992 \mathrm{cpm} ; t=10 \mathrm{h}, 924 \mathrm{cpm} ; t=100 \mathrm{h}, 452 \mathrm{cpm}$ $t=250 \mathrm{h}, 138 \mathrm{cpm} .$ What is the half-life of this nuclide?

Key Concepts

Radioactive Decay

Radioactive decay is the process by which unstable atomic nuclei lose energy by emitting particles or electromagnetic radiation. This decay follows a spontaneous and random process that, when observed in large numbers, exhibits a predictable exponential decrease in activity.

Exponential Decay

Exponential decay describes a decrease where the rate of change is proportional to the current amount. In the context of radioactive materials, the number of undecayed nuclei (and the associated disintegration rate) diminishes exponentially over time, forming the basis of decay equations.

Half-life

Half-life is the time required for half of the initial number of radioactive nuclei to decay. It is a fundamental concept in nuclear physics that characterizes the rate at which a radioactive material loses its activity through exponential decay.

Decay Constant

The decay constant is a parameter that quantifies the probability per unit time that a given nucleus will decay. It is mathematically related to the half-life through the equation ? = ln(2)/T?/?, making it essential for relating the exponential decay law to the observed disintegration rate.

Data Analysis in Radioactivity

Data analysis in radioactivity involves measuring the activity of a sample over time and using the exponential decay model to extract key parameters like the decay constant and half-life. Techniques such as using logarithms to linearize the decay data are commonly employed for this purpose.

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