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 C — Wave Behaviour

C.4.2 Nodes and Antinodes — IB Physics HL Revision Notes & Topical Questions

5 min read 11 parts Lesson 28 of 37 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

C.4.2 Nodes and Antinodes — IB PHYSICS SL notes: Locate and compare nodes and antinodes, relate their spacing to wavelength and describe phase relationships between points.Node — a point in a standing wave where the amplitude is zero, so the displacement is permanently zeroAntino…

Arrangement
Fixed positions
Node to next node
Antinode to next antinode
Node to adjacent antinode
Counting in quarter-wavelengths

Part 1 of 9Definitions

Node

Node a point in a standing wave where the amplitude is zero, so the displacement is permanently zero

Antinode

Antinode a point where the amplitude of oscillation is a maximum

Arrangement

Arrangement nodes and antinodes alternate, with every node midway between two antinodes

Fixed positions

Fixed positions nodes and antinodes do not move along the wave
Nodes are fixed and at rest.
Antinodes oscillate to and fro, so a snapshot finds them at any height between the positive and negative maximum.

Part 2 of 9Spacing in a standing wave

1Node to next node

λ/2

2Antinode to next antinode

λ/2

3Node to adjacent antinode

λ/4

4Counting in quarter-wavelengths

Counting in quarter-wavelengths node to node is two quarters and node to antinode is one quarter
A pipe open at both ends needs an even number of quarters.
A pipe closed at one end needs an odd number.

Part 3 of 9Why nodes and antinodes form

At nodes

At nodes the two waves are in antiphase, so a crest meets a trough and destructive interference cancels them

At antinodes

At antinodes the waves are in phase, so crest meets crest and constructive interference adds them

Origin

Origin nodes and antinodes result from destructive and constructive interference respectively

Part 4 of 9Amplitude varies with position

1Amplitude at a point

Amplitude at a point it depends on the distance from the nearest node
With x measured from a node it is 2A sin(2πx/λ).
It is zero at x = 0, 2A at x = λ/4 and zero again at x = λ/2.

2Graph

Graph amplitude against position is a series of arches, each half a wavelength wide

3Difference from a travelling wave

Difference from a travelling wave a travelling wave has the same amplitude everywhere

4Example point

Example point a point λ/8 from a node has amplitude 2A sin(π/4), about 1.41A

Part 5 of 9Phase relationships

1Only two cases

Only two cases two points on a standing wave are either in phase or in antiphase

2Same loop

Same loop all points within one loop between adjacent nodes oscillate in phase
They reach maximum displacement together and pass through equilibrium together, with different amplitudes.

3Adjacent loops

Adjacent loops particles in neighbouring loops are in antiphase

4Counting nodes

Counting nodes an even number of nodes between two points means in phase, and an odd number means antiphase

5Contrast

Contrast on a travelling wave the phase difference can take any value from 0 to 2π

Part 6 of 9Pressure and displacement in sound

Exchange

Exchange in a standing sound wave the displacement and pressure nodes and antinodes are interchanged

Displacement node

Displacement node the air cannot move there but is squeezed and stretched from both sides, so it is a pressure antinode

Displacement antinode

Displacement antinode the air moves freely and the pressure stays near atmospheric, so it is a pressure node

Kundt's tube

Kundt's tube powder in a horizontal glass tube gathers at displacement nodes, where the air is still, and its spacing gives λ/2

Part 7 of 9Measuring with nodes

Principle

Principle adjacent nodes are exactly λ/2 apart, so locating them measures the wavelength precisely

Microwaves

Microwaves move a detector between the transmitter and a reflector and note the successive minima

String

String measure the length of several loops and divide

Precision

Precision measure across several nodes and divide by the number of intervals, not by the number of nodes

Part 8 of 9Example: third harmonic on a string

Example: third harmonic on a string

Features

Features a string fixed at both ends in its third harmonic has four nodes and three antinodes

Neighbouring antinodes

Neighbouring antinodes they have equal amplitude and move in opposite directions at any instant

Test

Test a small paper rider stays put at a node but is thrown off at an antinode

Part 9 of 9More from the specification

1Nodal displacement scale

Nodal displacement scale antinodal points achieve maximal displacement amplitudes while true nodal boundaries show zero net oscillation on any scale.

2Standing Wave Nodes and Antinodes

In a standing wave system formed by superposition, nodes are points of zero displacement due to destructive interference,
while antinodes are points of maximum oscillation amplitude at boundary conditions.

3Resonance and Natural Frequency

When a periodic external driving force matches the natural frequency of an oscillating system, resonance occurs, transferring maximum energy and generating maximum amplitude of vibration.

4Damping and Energy Dissipation

Damping forces oppose motion and dissipate mechanical energy from an oscillating system as thermal energy,
causing oscillation amplitude to decay over time.

5Critical Damping vs. Heavy Damping

Critical damping returns a displaced system to its equilibrium position in the minimum possible time without overshooting,
whereas overdamped (heavy) systems return sluggishly without oscillating.

6Standing Wave Nodes and Antinodes

In a standing wave system formed by superposition, nodes are points of zero displacement due to destructive interference,
while antinodes are points of maximum oscillation amplitude at boundary conditions.

7Resonance and Natural Frequency

When a periodic external driving force matches the natural frequency of an oscillating system, resonance occurs, transferring maximum energy and generating maximum amplitude of vibration.

8Damping and Energy Dissipation

Damping forces oppose motion and dissipate mechanical energy from an oscillating system as thermal energy,
causing oscillation amplitude to decay over time.

9Critical Damping vs. Heavy Damping

Critical damping returns a displaced system to its equilibrium position in the minimum possible time without overshooting,
whereas overdamped (heavy) systems return sluggishly without oscillating.

10Boundary condition

Constraint at an edge fixing whether there is a node or antinode.

11Forced oscillation

Oscillation caused by a periodic driving force

12Progressive wave

Wave that transfers energy without transferring matter.

13Heavy damping

Damping greater than critical, giving a slow return without oscillation

14Resistive force

Friction and drag opposing motion.

15Frequency

Revolutions per second, f = 1/T.
What the symbols mean: T = period, the time for one complete wave (s) · f = frequency (Hz)

16Damping

Energy loss to friction that reduces amplitude.

17Resonance

Large amplitude when driving frequency equals natural frequency.

18Boundary

Surface between two media at which waves are partially reflected

19Stationary

Not moving.

20Force

A push or pull measured in newtons.

21Natural frequency

The frequency at which a system oscillates freely

22Driving frequency

The frequency of the periodic external force

Key terms

Term	Meaning
Node	Point of zero amplitude.
Antinode	Point of maximum amplitude.
Loop	Region between adjacent nodes.
Pressure antinode	Point of maximum pressure variation, at a displacement node.

Avoid these mistakes

Treating antinodes as stationary maximum-height points instead of points of maximum oscillation.

Confusing node-to-node spacing with the wavelength.

Thinking points on either side of a node oscillate in phase.

Forgetting that displacement nodes in sound are pressure antinodes.

Counting nodes instead of intervals when measuring.

Last updated: 15 October 2026