كيف تحسب فترة عمر النصف والكمية المتبقية من عنصر مشع؟ How to calculate radioactive half-life?

تحسب الكمية المتبقية بالقانون N(t) = N₀(1/2)^n، حيث n = الزمن الكلي (t) ÷ زمن عمر النصف (t½). وثابت التحلل الإشعاعي: λ = 0.693 / t½. [الصيغة الرمزية الموزونة: N(t) = N₀ × (1/2)^(t / t½) | λ = ln(2) / t½ ≈ 0.693 / t½ | E = Δm × c²]

Theme E — Nuclear and Quantum Physics

E.1.2 Nuclear Notation — IB Physics HL Revision Notes & Topical Questions

5 min read 10 parts Lesson 4 of 45 1 quiz

📚 IB Physics HL Hub / Topical Revision

Curriculum Equation Key Radioactive Decay & Half-Life FormulasFast Exam Solution

N(t) = N₀ × (1/2)t / t½  |  λ = ln(2) / t½ ≈ 0.693 / t½  |  E = Δm × c²

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Lesson summary

E.1.2 Nuclear Notation — IB PHYSICS SL notes: Use nuclide notation to give numbers of protons, neutrons and nucleons, and describe isotopes and nuclear equations.Atoms — they are made of protons, neutrons and electronsProtons and neutrons are in the nucleus and electrons orbit it…

Neutral atom
Electron mass
Unified atomic mass unit (u)
Nucleon masses
Charge and mass units
Nucleon number A (mass number)

Part 1 of 8Particles in the atom

Particles in the atom

Atoms

Atoms they are made of protons, neutrons and electrons
Protons and neutrons are in the nucleus and electrons orbit it.

Nucleons

Nucleons protons and neutrons together

Neutral atom

Neutral atom the number of electrons equals the number of protons

Electron mass

Electron mass about 1/1840 of the proton mass, so it is negligible in the nuclear mass

Particle	Charge / C	Charge / e	Mass / kg	Mass / u
Proton	+1.60 × 10⁻¹⁹	+1	1.673 × 10⁻²⁷	1.007276
Neutron	0	0	1.675 × 10⁻²⁷	1.008665
Electron	−1.60 × 10⁻¹⁹	−1	9.109 × 10⁻³¹	0.000549

Part 2 of 8Atomic mass unit

Atomic mass unit

Unified atomic mass unit (u)

Unified atomic mass unit (u) one twelfth of the mass of a carbon-12 atom
1 u = 1.66 × 10⁻²⁷ kg.

Nucleon masses

Nucleon masses a proton or neutron has a mass of almost exactly 1 u

Charge and mass units

Charge and mass units charge can be given in C or in units of e, and mass in kg or u

Part 3 of 8Nuclide notation

Nuclide notation

Nuclide

Nuclide a specific type of nucleus, written ^A_ZX
X is the chemical symbol of the element.

Nucleon number A (mass number)

Nucleon number A (mass number) the total number of protons and neutrons, written at the top

Proton number Z (atomic number)

Proton number Z (atomic number) the number of protons, written at the bottom
It fixes the element and the nuclear charge +Ze.

Neutron number

N = A − Z

Meaning of A

Meaning of A the nuclear mass in u is close to A, but not exactly A because of binding energy

Example

²³⁸₉₂U has 92 protons, 238 nucleons and 146 neutrons

Periodic table

Periodic table it is ordered by atomic number

Part 4 of 8Isotopes

1Isotopes

Isotopes nuclides of the same element with the same Z but different numbers of neutrons, so different A

2Chemistry

Chemistry isotopes have the same chemical properties because the electron configuration depends only on Z

3Nuclear properties and mass

Nuclear properties and mass these differ between isotopes

4Hydrogen

Hydrogen protium ¹₁H, deuterium ²₁H and tritium ³₁H (one proton, two neutrons, radioactive)
- (one proton and one neutron)

5Carbon

Carbon mostly ¹²₆C with some ¹³₆C and a trace of radioactive ¹⁴₆C used in carbon dating

6Separation

Separation isotopes can be separated by mass, for example in a mass spectrometer where r = mv/(qB) depends on the ion's mass

Part 5 of 8Nuclear equations

Conservation

Conservation the total nucleon number A and the total charge (sum of Z) are conserved

Balancing

Balancing check the top and bottom numbers separately

Alpha decay

²³⁸₉₂U → ²³⁴₉₀Th + ⁴₂He
Nucleons238 = 234 + 4.
Charge92 = 90 + 2.

Beta-minus decay

Beta-minus decay a neutron changes into a proton and an electron and antineutrino are emitted, so Z rises by one and A is unchanged
¹⁴₆C → ¹⁴₇N + ⁰₋₁e + antineutrino.

Part 6 of 8Size and density

Nuclear radius

R = R₀A^1/3 with R₀ ≈ 1.2 × 10⁻¹⁵ m

Density

Density volume and mass are both proportional to A, so all nuclei have roughly the same density, about 2 × 10¹⁷ kg m⁻³

Empty atom

Empty atom nearly all the mass lies in a volume about 10⁻¹⁵ times that of the atom

Part 7 of 8Nuclear stability

Repulsion

Repulsion protons repel each other electrostatically

Strong nuclear force

Strong nuclear force an attractive force between nucleons with a range of about 3 fm that overcomes the electrostatic repulsion at those distances

Light stable nuclei

Light stable nuclei they have about equal numbers of protons and neutrons

Heavy stable nuclei

Heavy stable nuclei they need more neutrons than protons, because extra neutrons add attraction without adding repulsion

Unstable nuclides

Unstable nuclides those with too many or too few neutrons decay

Part 8 of 8More from the specification

Discrete Atomic Energy Levels

Bound electrons occupy quantized, discrete energy levels;
- transitions between these levels require absorption or release of specific quanta of energy.

Emission and Absorption Spectra

An emission spectrum displays bright coloured lines where downward electronic transitions release a photon with energy ΔE = hf = hc/λ.

Photon Wavelength Calculation

The discrete wavelengths observed in hydrogen emission experiments correspond precisely to energy differences between initial and final quantum shells.

Continuous vs Line Spectra

An incandescent black-body source emits a continuous spectrum across all wavelengths,
whereas gaseous atoms emit sharp spectral lines characteristic of their element.

Spectral lines

Dark or bright lines in a spectrum at wavelengths characteristic of elements.

Rutherford scattering

Scattering of charged particles by a nucleus through electrostatic repulsion alone.

Subatomic particle

A particle smaller than an atom, e.g. the electron.

Photon

A quantum of electromagnetic energy, E = hf

Wavelength

Distance between a point on one wave and the same point on the next wave

Absorption

The conversion of wave energy to internal energy of the medium.

Scattering

Deflection of particles from their original path.

Absorbed

Taken in by an object, with light energy transferred to thermal energy.

Energy level

A discrete energy an electron in an atom can have

Photon energy

E = hf = hc/λ, the energy carried by one photon.

Emission spectrum

The set of discrete wavelengths emitted by excited atoms of an element.

∆E

∆E hf = E2 − E1

What the symbols mean: E = energy of one photon (J) · h = Planck constant (6.63 × 10⁻³⁴ J s) · f = frequency (Hz)

Key terms

Term	Meaning
Nucleon	A proton or a neutron.
Proton number Z	Number of protons in the nucleus.
Nucleon number A	Number of protons plus neutrons.
Isotopes	Nuclides with the same Z but different A.
Nuclide	A specific type of nucleus with given Z and A.
Unified atomic mass unit	1/12 of the mass of a carbon-12 atom, 1.66 × 10⁻²⁷ kg.

Avoid these mistakes

Mixing up A and Z — Z is the number of protons (bottom) and A is the nucleons (top).

Saying isotopes have different numbers of protons — they have the same Z and different neutron numbers.

Thinking isotopes differ in chemistry — the electrons are the same.

Forgetting to balance both A and Z in nuclear equations.

Using the electron mass as part of the nuclear mass — it is negligible.

Last updated: 15 October 2026