(455) HL Decay constant and more detailed equations [IB Physics HL]

Radioactive Decay Overview

• Radioactive decay involves changes to atomic nuclei, resulting in the emission of particles and energy.

• Decay types include:

  • Alpha Decay: Emission of an alpha particle (helium-4, 2 protons, 2 neutrons).

  • Beta Decay:

    • Beta Minus (β-): Emits an electron and an anti-electron neutrino.

    • Beta Plus (β+): Emits a positron and an electron neutrino (antimatter).

  • Gamma Decay: Release of energy in the form of photons (no change in atomic number or mass).

Characteristics of Decay

• Exponential Decay: The amount of radioactive substance decreases exponentially over time.

• Half-Life (T1/2): The time required for half of the radioactive atoms to decay.

  • Example: If N0 is the initial mass, then after one half-life, the remaining mass is N0/2.

Key Concepts

• Decay Constant (λ): Represents the probability of decay per unit time, measured in units like seconds⁻¹.

  • Important note: Decay constant (λ) is not equal to wavelength.

Relevant Equations

• Exponential Decay Equation: N = N0 e^(−λT)

  • N is the remaining quantity after time T.

  • N0 is the initial quantity.

• Activity (A): A = λN

  • Represents the decay rate (number of decays per second).

  • Unit: Becquerel (Bq).

• Half-Life Equation: T1/2 = ln(2)/λ

Half-Life Derivation

• Starting with the exponential decay equation:

  • At half-life: N = N0/2. Substituting gives:

    • N0/2 = N0 e^(−λT1/2)

  • Cancelling N0 yields:

    • 1/2 = e^(−λT1/2)

  • Taking natural logarithm:

    • ln(1/2) = −λT1/2

  • Solve for T1/2:

    • T1/2 = ln(2)/λ

Practical Example

• Given cesium-134 with initial mass N0 and its half-life:

  • To find λ, use T1/2 = ln(2)/λ.

  • Rearranging gives λ = ln(2)/T1/2.

• To calculate the time for mass to decrease from N0 to a lower mass:

  • Use T = ln(N/N0)/−λ.