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]

Tools and Inquiry — IB Physics skills

T.3.6 Using Dimensional Analysis — IB Physics HL Revision Notes & Topical Questions

5 min read 9 parts Lesson 11 of 22 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

T.3.6 Using Dimensional Analysis — IB PHYSICS SL notes: Check the homogeneity of equations using SI base units, deduce units of constants and predict the form of a relationship.Dimensional analysis = checking the homogeneity of a physical equation using SI base units.The units on…

The units on both sides of a valid equation must be the same
A mismatch shows an algebra error
Dimensions show how a quantity is built from base quantities
Speed is [L T-1]
Pressure in base units
What the symbols mean

Part 1 of 7Homogeneity of equations

1Dimensional analysis

Dimensional analysis checking the homogeneity of a physical equation using SI base units.

2The units on both sides of a valid equation must be the same

The units on both sides of a valid equation must be the same and every term added or subtracted must match.

Procedure.

Write the units of each quantity in base units.
Work out the units of each side of the equation.
Compare them — if they differ, the equation needs adjusting.

4A mismatch shows an algebra error

A mismatch shows an algebra error a match does not prove the equation is correct.

Part 2 of 7Dimensions of quantities

Dimensions of quantities

Dimensions show how a quantity is built from base quantities

Dimensions show how a quantity is built from base quantities mass [M], length [L], time [T], current [I], temperature [Θ] and amount [N].

Speed is [L T-1]

Speed is [L T-1] acceleration [L T^-2], force [M L T^-2] and energy [M L² T^-2].

Pure numbers, radians, ratios and the exponent of e are dimensionless.

Pressure in base units

Pressure in base units p = F/A,
so kg m s^-2 divided by m² gives kg m^-1 s^-2.
What the symbols mean: p = pressure (Pa = N m⁻²) · F = force acting at right angles to the surface (N) · A = area (m²)

Quantities with the same dimensions need not be the same kind

Quantities with the same dimensions need not be the same kind torque and energy both have [M L² T^-2].

Part 3 of 7Checking an equation

Check s

Check s ut + ½at².

[s] = L, [ut] = (L T^-1)(T) = L and [½at²] = (L T^-2)(T²) = L.
All terms are lengths, so the equation is consistent.

Check v

√(γp/ρ), with γ a constant.

p has units kg m^-1 s^-2 and ρ has units kg m^-3.
p/ρ has units m² s^-2, so the square root gives m s^-1 and γ has no units.

Dimensional analysis cannot detect numerical factors (½, 2π) or dimensionless functions.

Part 4 of 7Finding units of constants

Rearrange the equation to isolate the constant and substitute units.

In F = kx, k has dimensions [M L T^-2] / [L] = [M T^-2].
For the Stefan–Boltzmann law P = σAT⁴, σ has units W m^-2 K^-4.

For drag F

½ρAv²C_d — ρAv² has the dimensions of force, so C_d is dimensionless.

Arguments of trigonometric, exponential or log functions must be dimensionless.

In e^-λt, λt has no dimension, so λ is [T^-1].

Part 5 of 7Deducing a relationship

If a quantity depends on others by a power law, dimensions can fix the exponents.

Write T = k L^a m^b g^c for a pendulum.
Replace each quantity by its dimensions and equate powers of M, L and T.
M gives b = 0, T gives c = −½ and L gives a = ½.

Result

T ∝ √(L/g), independent of mass.

The constant (2π) must be found by experiment or full theory.

Cautions.

If an important variable is left out, the deduced law is incomplete.
If too many variables are included, the exponents are not uniquely fixed.

Part 6 of 7Practical uses

Practical uses

Check the algebra at the end of a derivation

Check the algebra at the end of a derivation before substituting numbers.

Check a remembered formula

Check a remembered formula only mv²/r has the dimensions of force, not mv²r.

Work out the base units of a derived quantity

Work out the base units of a derived quantity see T.3.5.

Part 7 of 7More from the specification

1Experimental Uncertainty and Significant Figures

Physical measurements carry experimental uncertainty;
- calculating absolute and percentage uncertainty ensures final calculated values are appropriately quoted to the correct number of significant figures.

2Scalar vs Vector Quantities

A scalar has magnitude only,
whereas a vector quantity possesses both magnitude and spatial direction; vectors add using vector triangles or component resolution.
- (such as forces and velocities)

3Data Analysis Using Linear Graphs

Plotting experimental data on linear graphs allows researchers to determine physical constants from the line of best fit's gradient and y-intercept,
while error bars indicate the bounds of measurement uncertainty.

4Percentage uncertainty

The absolute uncertainty divided by the value, times 100%.

5Absolute uncertainty

The range ±Δx in the same unit as the measurement.

6Percentage error

Difference from the accepted value as a percentage of it.

7Percentage difference

Difference divided by accepted value × 100

8Scale drawing

Accurate drawing with a chosen scale.

9Parallelogram method

Vector addition using a parallelogram of vectors.

10Uncertainty

An estimate of the range in which the true value is expected to lie.

11Vector

Quantity with magnitude and direction.

12Gradient

Slope of a graph line.

13Distance

Total length of path travelled, a scalar.

14Displacement

Change in position.

15Proportional

Changing by the same factor as another quantity.

16Scalar

Quantity with magnitude only.

17Significant figures

Digits carrying meaningful information in a number.

18a

− ω2x

What the symbols mean: a = acceleration (m s⁻²) · ω = angular frequency (rad s⁻¹) · x = displacement from equilibrium (m); the minus sign means a always points back to the centre

Key terms

Term	Meaning
Dimensional analysis	Checking equations through their base units or dimensions.
Homogeneity	All terms of an equation have the same dimensions.
Dimensionless	Quantity with no units, such as a ratio.
Base quantity	Fundamental quantity such as length, mass or time.
Power law	Relationship of the form y = kx^n.
Derived unit	Unit built from SI base units.

Avoid these mistakes

Assuming matching units prove an equation correct — numerical factors are not checked.

Forgetting to expand derived units such as pascal into base units.

Adding terms with different dimensions.

Treating angles and ratios as having dimensions.

Using the exponent of e or a trigonometric argument with dimensions.

Last updated: 18 October 2026