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.5.8 Determination of Stellar Radii — IB Physics HL Revision Notes & Topical Questions

5 min read 9 parts Lesson 45 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.5.8 Determination of Stellar Radii — IB PHYSICS SL notes: Calculate stellar radii by combining Wien's law, the inverse square law for flux and the Stefan–Boltzmann law.A star's radius is deduced — stars are too far away to measure directly.Treat the star as a black body — L = 4…

Wien's displacement law
A peak of 500 nm gives T
At distance d the power spreads over a sphere of area 4πd2
Rearranged, L
Distance comes from stellar parallax or other methods
Check with the Sun

Part 1 of 7Strategy

1A star's radius is deduced

A star's radius is deduced stars are too far away to measure directly.

2Treat the star as a black body

L = 4πR²σT⁴.

3Then R

√( L / (4πσT⁴) ) — needing luminosity L and surface temperature T.

Procedure combines three laws.

Wien's displacement law gives the surface temperature.
The inverse square law of flux gives the luminosity from radiant flux and distance.
The Stefan–Boltzmann law gives the radius.

Part 2 of 7Surface temperature

Wien's displacement law

λ_maxT = 2.90 × 10^-3 m K, so T = 2.90 × 10^-3 / λ_max.

Peak wavelength is the wavelength of greatest intensity in the spectrum.

A peak of 500 nm gives T

5800 K like the Sun — Betelgeuse at about 850 nm gives about 3400 K.

Temperature can also come from the spectral class found from absorption lines.

Part 3 of 7Luminosity from brightness and distance

Luminosity from brightness and distance

The star radiates power L equally in all directions.

At distance d the power spreads over a sphere of area 4πd2

At distance d the power spreads over a sphere of area 4πd2 so b = L / (4πd²).
b is apparent brightness (radiant flux) in W m^-2, L is in W and d in m.

Rearranged, L

4πd²b.

Distance comes from stellar parallax or other methods

Distance comes from stellar parallax or other methods and b from a photometer.

Check with the Sun

Check with the Sun b = 1.36 × 10³ W m^-2 at 1.50 × 10¹¹ m gives L = 3.8 × 10²⁶ W.

Part 4 of 7Combined procedure

Measure the parallax angle p and find d = 1/p in parsecs, then convert to metres.

Measure apparent brightness b and find L = 4πd²b.

Measure the peak wavelength and find T from Wien's law.

Find R = √( L / (4πσT⁴) ).

If L is given directly, start from step 3.

Solar ratios

R/R☉ = √(L/L☉) × (T☉/T)².
The Sun has L☉ = 3.83 × 10²⁶ W, T☉ = 5800 K and R☉ = 6.96 × 10⁸ m.

For Betelgeuse (L

4.49 × 10³¹ W, peak 850 nm) the radius is about 1000 times the Sun's.

Part 5 of 7Meaning of the results

Star type	Typical radius
White dwarf	About 0.01 solar radii, comparable to Earth
Main-sequence star	About 0.1 to 10 solar radii
Red giant	Tens to hundreds of solar radii
Red supergiant	Up to about a thousand solar radii
Neutron star	About 10 km

These sizes explain the regions of the HR diagram.

2The black-body assumption is an approximation

The black-body assumption is an approximation real spectra have absorption lines.

3Interstellar dust dims starlight

Interstellar dust dims starlight making the calculated luminosity too low unless corrected.

Close eclipsing binaries give radii more directly from eclipse duration and depth.

Part 6 of 7Ratios and checks

Ratios and checks

For two stars use L ∝ R2T4

For two stars use L ∝ R2T4 constants such as 4πσ cancel.
If star A has three times the radius and half the temperature of star B, L_A/L_B = 3² × (1/2)⁴ = 9/16.

Convert temperature to kelvin and distance to metres

Convert temperature to kelvin and distance to metres because T⁴ and d² amplify errors.

Sanity checks

Sanity checks a main-sequence radius lies within a factor of ten of the Sun's, a giant is much larger, a white dwarf is near Earth-sized.

Part 7 of 7More from the specification

Stellar blackbody energy levels

Stefan-Boltzmann calculations combine radiant flux and temperature, governed by underlying quantum photonic energy levels.

Stefan-Boltzmann Law and Stellar Radii

The total power (luminosity L) emitted by a star is related to its surface radius R and absolute surface temperature T by L = 4π R² σ T⁴,
allowing stellar radii to be calculated from observational data.

Hydrostatic Equilibrium in Stars

A stable star maintains hydrostatic equilibrium,
where inward gravitational pressure is precisely balanced by outward gas and radiation pressure generated by core nuclear fusion reactions.

Hydrogen Fusion in Stellar Cores

Main sequence stars fuse hydrogen nuclei into helium nuclei via the proton-proton chain or CNO cycle, with mass converted into released radiant energy via E = m c².

Core Collapse and Post-Main Sequence Evolution

When core hydrogen is exhausted, radiation pressure drops and gravity causes core collapse until temperature and pressure rise sufficiently to trigger helium fusion.

Stefan-Boltzmann Law and Stellar Radii

The total power (luminosity L) emitted by a star is related to its surface radius R and absolute surface temperature T by L = 4π R² σ T⁴,
allowing stellar radii to be calculated from observational data.

Hydrostatic Equilibrium in Stars

A stable star maintains hydrostatic equilibrium,
where inward gravitational pressure is precisely balanced by outward gas and radiation pressure generated by core nuclear fusion reactions.

Hydrogen Fusion in Stellar Cores

Main sequence stars fuse hydrogen nuclei into helium nuclei via the proton-proton chain or CNO cycle, with mass converted into released radiant energy via E = m c².

Core Collapse and Post-Main Sequence Evolution

When core hydrogen is exhausted, radiation pressure drops and gravity causes core collapse until temperature and pressure rise sufficiently to trigger helium fusion.

Energy level

A discrete energy an electron in an atom can have

Kinetic energy

K = ½mv², energy of motion.

Continuous spectrum

Spectrum with no gaps between the regions.

Fusion

Joining of light nuclei into a heavier one.

Pressure

Force per unit area, P = F/A.
What the symbols mean: p = pressure (Pa = N m⁻²) · F = force acting at right angles to the surface (N) · A = area (m²)

Helium

Product of the fusion of hydrogen in stars.

Nucleus

Small, dense, positively charged centre of the atom.

Reaction

The partner force acting on the other object.

Radiation pressure

The pressure exerted on a surface by absorbed or reflected radiation because photons transfer momentum.

Binding energy

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

Emission spectrum

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

Key terms

Term	Meaning
Apparent brightness	Intensity received from a star, b = L/(4πd².
Luminosity	Total power output of a star.
Black body	Ideal emitter whose spectrum depends only on temperature.
Wien's displacement law	λ_max T = 2.90 × 10^-3 m K.
Stefan–Boltzmann law	L = σAT⁴ for a black body.
Photosphere	Visible surface layer of a star.
Solar radius	6.96 × 10⁸ m.

Avoid these mistakes

Using the Celsius scale for T — use kelvin.

Forgetting the factor 4π in L = 4πd² b or in the Stefan–Boltzmann law.

Leaving the distance in parsecs or light-years in b = L/(4πd² — use metres.

Forgetting to take the square root when finding R from R².

Thinking luminosity and apparent brightness are the same — brightness depends on distance.

Last updated: 18 October 2026