Ch 41: Atomic Physics

Knight Calc - Physics for Scientists and Engineers 5th Edition

Knight Calc5th EditionPhysics for Scientists and EngineersISBN: 9780137344796Not the one you use?Change textbook

Knight Calc 5th Edition

Ch 41: Atomic Physics

Problem 47b

Chapter 41, Problem 47b

During what interval of time will 10% of a sample of 2p hydrogen atoms decay?

Verified step by step guidance

Understand the problem: The question involves radioactive decay, which follows an exponential decay law. The goal is to find the time interval during which 10% of the sample decays. This means 90% of the sample remains undecayed.

Write the exponential decay formula:

N(t) = N_0 e^{-λt}

, where

N(t)

is the number of atoms remaining at time

t

N_0

is the initial number of atoms, and

λ

is the decay constant.

Set up the equation for 90% of the sample remaining:

0.9N_0 = N_0 e^{-λt}

. Simplify by dividing through by

N_0

, giving

0.9 = e^{-λt}

Take the natural logarithm of both sides to solve for

t

ln(0.9) = -λt

. Rearrange to isolate

t

t = -ln(0.9)/λ

Determine the decay constant

λ

if not provided: The decay constant is related to the half-life

T_{1/2}

by the formula

λ = ln(2)/T_{1/2}

. Substitute this into the equation for

t

if the half-life is known.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Radioactive Decay

Radioactive decay is a stochastic process by which an unstable atomic nucleus loses energy by emitting radiation. This process occurs at a characteristic rate for each isotope, defined by its half-life, which is the time required for half of the radioactive atoms in a sample to decay. Understanding this concept is crucial for calculating the decay of a specific percentage of a sample.

Half-Life

Half-life is the time it takes for half of a given quantity of a radioactive substance to decay. It is a key parameter in nuclear physics and helps predict how long it will take for a certain fraction of a sample to decay. For example, if the half-life of a substance is known, one can determine the time required for 10% of the sample to decay using exponential decay formulas.

Exponential Decay

Exponential decay describes the process where the quantity of a substance decreases at a rate proportional to its current value. This means that as time progresses, the amount of substance decreases rapidly at first and then slows down. The mathematical model for exponential decay can be expressed as N(t) = N0 * e^(-λt), where N0 is the initial quantity, λ is the decay constant, and t is time, allowing for precise calculations of decay over time.

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Related Practice

Textbook Question

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