كيف يحسب الرقم الهيدروجيني pH للمحاليل؟ How to calculate pH and pOH?

يحسب الرقم الهيدروجيني بالعلاقة: pH = -log[H⁺]، وللمحاليل المائية عند 25°C: pH + pOH = 14. إذا كان pH 7 فالمحلول قاعدي. [الصيغة الرمزية الموزونة: pH = -log[H⁺] | pOH = -log[OH⁻] | pH + pOH = 14 | [H⁺][OH⁻] = 10⁻¹⁴]

Theme E — Nuclear and Quantum Physics

E.3.7 The Law of Radioactive Decay — IB Physics HL Revision Notes & Topical Questions

5 min read 10 parts Lesson 24 of 45 1 quiz

📚 IB Physics HL Hub / Topical Revision

Curriculum Equation Key قوانين الرقم الهيدروجيني والاتزان الأيوني - pH & Acid-Base Equilibrium FormulasFast Exam Solution

pH = -log[H⁺]  |  pOH = -log[OH⁻]  |  pH + pOH = 14  |  [H⁺][OH⁻] = 10⁻¹⁴

Word Equation & Conditions: الرقم الهيدروجيني = -لوغاريتم تركيز أيون الهيدروجين | pH + pOH = 14

Location & Specifications: الاتزان الأيوني، كيمياء 3 ثانوي، التحصيلي، و AP Chemistry.

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Lesson summary

E.3.7 The Law of Radioactive Decay — IB PHYSICS HL notes: Use N = N0 e^−λt, A = A0 e^−λt and C = C0 e^−λt and ln graphs to analyse exponential decay.Pattern — the number of undecayed nuclei falls very rapidly without ever reaching zeroRate — dN/dt = −λN so the decay rate is…

What the symbols mean
Count rate
Corrected rate
Parent nuclei
Time constant
Substitution

Part 1 of 8Exponential decay

Pattern

Pattern the number of undecayed nuclei falls very rapidly without ever reaching zero

Rate

Rate dN/dt = −λN so the decay rate is proportional to the number present

Equation

N = N₀e^−λt
What the symbols mean: N = number of nuclei left after time t · N₀ = number at the start · λ = decay constant (s⁻¹) · t = time (s)

Symbols

N₀ initial number, N number at time t, λ decay constant in s⁻¹ and t in the same time unit

Part 2 of 8Other forms

Activity

A = A₀e^−λt since A ∝ N

Count rate

C = C₀e^−λt since the received count rate is proportional to activity

Corrected rate

Corrected rate use the count rate after subtracting background

Parent nuclei

Parent nuclei the law applies to the undecayed parent nuclei, while a stable daughter grows as N₀(1 − e^−λt)

Part 3 of 8The graph

Start

Start the curve starts on the y-axis at N₀

Steepness

Steepness a steeper curve means a larger decay constant

Gradient

Gradient it equals −λN, so it also decays exponentially

Axis

Axis the curve approaches the time axis but never reaches it

Time constant

1/λ is the time for N to fall to 1/e of its value

Part 4 of 8Half-life form

Substitution

λ = ln 2/T_½ gives N = N₀e^{−(ln 2)t/T_½} = N₀(½)^t/T_½

Validity

Validity it works for any time and not only whole numbers of half-lives

When to use

When to use use the e form with λ and the half-life form with T_½

Check

Check after t = 2T_½ the exponent λt = 1.386 and e^−1.386 = 0.25

Part 5 of 8Solving for time

Rearrangement

Rearrangement ln(A/A₀) = −λt

Time

Time t = (1/λ) ln(A₀/A)

Dating

Dating this is the basis of radiometric dating

Dimensionless exponent

λt must have no units, so use matching time units

Part 6 of 8Linearising and finding λ

Natural log

Natural log ln N = −λt + ln N₀

Graph

Graph ln A or ln C against t is a straight line with gradient −λ and intercept ln A₀

Method

Method measure the corrected count rate at regular times, plot ln C against t and take the gradient of the best-fit line

Alternative

Alternative read half-lives from the curve and use λ = ln 2/T_½

Background

Background correct the raw count rate before taking logs

Part 7 of 8Exponential patterns elsewhere

Capacitor discharge, absorption of radiation and simple cooling all have the same mathematics

Sketching

Sketching label the intercept N₀, show the curve flattening towards the axis and mark T_½

Part 8 of 8More from the specification

Radiometric dating across millions of years

Radiometric dating across millions of years exponential decay of uranium isotopes over millions of years provides geological age determination independent of force.

Radioactive Alpha Decay

An unstable heavy nucleus undergoes alpha decay by ejecting an alpha particle,
- (2 protons and 2 neutrons)
reducing atomic number Z by 2 and nucleon number A by 4.

Beta and Gamma Nuclear Emission

Beta-minus decay converts a neutron into a proton with emission of an electron and antineutrino;
- excess excitation energy is subsequently released as gamma photons.

Binding Energy and Stability

Nuclear stability depends on binding energy per nucleon;
- radioactive decay pathways evolve unstable nuclei towards configurations with higher binding energy.

Exponential Activity Decay Law

The activity of a radioactive sample decays exponentially according to A = A₀ e^(-λt),
where decay constant λ reflects the decay probability per unit time.

Ionising radiation

Radiation with enough energy to remove electrons from atoms

Penetrating power

How far radiation travels through a material.

Proton

Positive particle in the nucleus, relative charge +1 and relative mass 1.

Neutron

Neutral particle in the nucleus, relative mass 1.

Isotope

Atom of the same element with a different number of neutrons.

Isotopes

Nuclei with the same proton number but different neutron numbers.

Source

Reservoir at T_H providing heat Q_H

Electron

Negatively charged particle in an atom.

Binding energy

The minimum energy required to separate the nucleons of a nucleus to infinity.

Alpha particle

Helium nucleus, charge +2.

Key terms

Term	Meaning
Exponential decay	A decrease in which the rate is proportional to the amount present.
N = N₀e^−λt	The number of undecayed nuclei at time t.
Time constant	1/λ, the time for N to fall to 1/e of its initial value.
Corrected count rate	The measured count rate minus the background count rate.
Linearisation	Plotting ln N against t to give a straight line of gradient −λ.

Avoid these mistakes

Mixing units — using λ in s⁻¹ with t in minutes.

Using log₁₀ where the law needs natural logs.

Forgetting the minus sign in the exponent, which would give growth.

Thinking N reaches zero at a finite time — it only approaches zero.

Plotting ln of the raw count rate without correcting for background.

Last updated: 18 October 2026