Physical quantities and Units
Measurement uncertainties
Prof. H. ASCHAWA
Biophysics
Semester II
School year: 2022-2023
www.um6ss.ma
BIOPHYSICS OF THE INTERNAL
ENVIRONMENT AND MAJOR FUNCTIONS
Quantities and Units / Measurement Uncertainties
BIOPHYSICS OF THE INTERNAL ENVIRONMENT AND MAJOR FUNCTIONS
GENERAL
• Quantities and Units - Dimensional Equation - Uncertainties of measurements
• Extensive and intensive elementary quantities
THE DIFFERENT STATES OF MATTER - ENERGY
• Biophysics of water and solutions
• Biophysics of the gaseous state
• Work - Potential energies and potentials
FREE DISPLACEMENTS OF MATTER
• Notion of Viscosity applied to biological fluids
• Convection movements of fluids
• Migration movements within fluids
PHYSICAL PHENOMENA OF MEMBRANES :
• Characteristics of membranes - Different membrane potentials
• Fluid transfers in the organism
BIOPHYSICS OF MAJOR FUNCTIONS :
• Biophysics of the Blood Circulation
• Electrocardiography (E.C.G)
• Biophysics of the Respiration -Transport of Gases in the Blood
• Acid-Basic Balance
Lesson Plan
Chapter 1: Physical quantities and Units
Chapter 2: Dimensional Equations
Chapter 3: Errors and Calculation of Uncertainties
• Sub-Chapter 3.1: Systematic errors
• Sub-Chapter 3.2: Random errors
• Sub-Chapter 33: -Calculation of uncertainties
Chapter 4: Extensive and Intensive Quantities
Educational objectives:
• To acquire the basic notions of quantification for reasoning in biophysics of
the living organism.
• Integrate the notion of uncertainty and variability of a measurement in life
science.
• Calculate and estimate the error of biological parameters.
Quantities and Units / Measurement Uncertainties
Quantities and Units
• A physical quantity :
It is a quantifiable property of matter, space or a phenomenon.
It is distinguished qualitatively and determined quantitatively.
• The International System (SI) comprises 7 basic quantities:
Meter - Kilogram - Second - Ampere - Kelvin - Mole - Candela
• Units derived from the international system :
Formed by combining base units according to the algebraic relationships between quantities
corresponding: distance, mass, time, quantity of matter, electricity, mechanics, heat, IR.
• Additional SI quantities
Plane angle and solid angle
Magnitude
Name
Symbol
Plane angle
radian
rad
Solid angle
steradian
sr
Dimension
(𝛀)
Quantities and Units / Measurement Uncertainties
SI: base quantities and units
Size
Dimension
Unit
Symbol
Length
L
meter
m
Mass
M
kilogram
kg
Time
T
second
s
Electric current intensity
I
ampere
A
Temperature
Θ
kelvin
K
Amount of substance
N
mole
mol
Light intensity
J
candela
cd
Quantities and Units / Measurement Uncertainties
SI derived units or quantities
Derived units: Formed by combining basic units according to algebraic relationships:
•
Space: volume, area, wavelength...
•
Mass: density ....
•
Time: frequency....
•
Quantity of material: concentration...
•
Mechanics: speed, acceleration, force, surface tension, work, energy, pressure, viscosity, etc.
•
Electricity: electric current, potential difference, quantity of electricity, resistance...
•
Heat: capacity, conductivity, convection, ...
•
Ionising radiation: activity of a source, absorbed dose, equivalent dose, etc.
Quantities and Units / Measurement Uncertainties
SI derived units or quantities
Size
Name
Symbol
Size
Name
Surface
Square meter
m2
Power
Watt
Volume
Cubic meter
m3
Electric charge
Coulomb
Speed
Meter per
second
m.s-1
Electrical
potential
Volt
Acceleration
Meter per
square second
m.s-2
Electric fields
Volt per
meter
Force
Newton
N = kg.m.s-2
Frequency
Hertz
Pressure
Pascal
Pa = N.m-2
Density
Kilogram per
cubic meter
Energy
Joule
J = N.m
Activity
Becquerel =
1 d.s-1
Bq
Molarity
Mole per
cubic meter
.....
........................
Symbol
W = J.s-1
C = A.s
V = J.C-1
= W.A-1
E = V.m-1
Hz = s-1
𝞺 = Kg.m-3
mol.m-3
.........
Quantities and Units / Measurement Uncertainties
Table of Prefixes
Quantities and Units / Measurement Uncertainties
Some Physical Constants
§ Avogadro number: Ɲ: 6.02.1023 atoms (Unit: mol )-1
§ Velocity: Speed of light in a vacuum: c = 3.108 m.s-1
§ Charge of an electron: e = 1.6.10-19 C (Coulomb)
§ Electron-volt (eV): energy acquired by an elementary charge q subjected to a potential difference
of 1 volt => 1eV = 1.602 . 10 J -19
§ Planck's constant: h = 6.626 . 10- 34 J.s
Reminder
§ Gradient: Measures the variation of a parameter in space/unit length
Velocity gradient dv/dx; Concentration gradient dC/dx; Volume gradient dV/dx
Force gradient dF/dx; Pressure gradient dP/dx
§ Einstein's relativity: E0 = m0 c02 ;
E=
mc2
;
E0 : E at rest (J); m0 : mass (kg); c: velocity (m.s )-1
m : mass (Kg)
m = m0 / 𝟏 −
𝒗𝟐
𝒄𝟐
Quantities and Units / Measurement Uncertainties
Dimensional equations
Dimension: Characterises the proper nature of a physical quantity.
•
All dimensions are expressed in terms of 7 fundamental quantities:
- Length [L] - Mass [M] - Time [T] - Intensity [I]
- Temperature [𝛉]; - Amount of material [N] - Light intensity [J].
•
Example: velocity [V] = L.T-1
;
[𝛄]: L.T-2
•
The equation with dimensions : Allows in a relationship between quantities to:
•
Replace each term with the corresponding fundamental quantity.
•
Determine the composite unit of a quantity according to the fundamental quantities, make
changes of units
•
Check if a formula is homogeneous => Any inhomogeneous formula is necessarily false
Quantities and Units / Measurement Uncertainties
Dimensional equations
• The equation in dimensions of a quantity G is written in the
general form :
[G] = Ma x Lb x Tc x Id x 𝛉e x Jf x Ng
Derived quantities
• Examples:
• If [G] = L , G is said to have the dimension of a length,
or that G is homogeneous to a length
• If [G] = L2 , G is said to have the dimension of a
surface.....
• If [G] = 1, G is said to be dimensionless
Size
Dimension
Units S
Speed
L.T-1
m.s-1
Density
M . L-3
kg . m -3
Force
M.L.T-2
N = kg. M.s-2
Molality
N/M = nM -
mol / kg
1
Molarity
N L-3
mol/m3
Quantities and Units / Measurement Uncertainties
Measurement uncertainties
Measuring a quantity:
•
Assigning a measured value to a quantity
•
Compare this value to another chosen conventional value: unit
Any measurement always contains errors, hence: uncertainty
•
Causes: Limited accuracy of the equipment, human error...
•
It is a parameter associated with the result of a measurement, which characterises the
dispersion of values attributed to a quantity.
•
This parameter: reference value: true (A0 ) and measured value (A)
A0 - A = error
Mearured
value
A
True
0
value
A
Measurement error
Measurement
Error
Quantities and Units / Measurement Uncertainties
Systematic Errors
§ Causes (avoidable!): The measuring device: Poorly calibrated devices++.
The way the instrument is used: Poorly controlled handling...
§ Two types of systematic errors: A: shifted/A0
•
Zero or translation errors: A = A0 + K (A - A0 ) = K cte)
•
Proportional or amplification errors: A = KA0
A0
True
value
§ Features
A1 A2 A3. A4
Measured
values
•
Always contributes to over (or under) valuing the measured value.
•
Not diminished by a series of measurements (unlike the random error)
•
But must be spotted by the experimenter and eliminated
•
Example: if an empty scale already shows a few grams => all measurements will be
overestimated.
Quantities and Units / Measurement Uncertainties
Random errors
Features:
• Always present (unavoidable), not predictable. Varies randomly (chance)
• The value obtained: over- or undervalued / the real value.
• Repeat measurements reduce random error.
Absolute uncertainty: ΔA
• This is the absolute value of the maximum error: ΔA = IA - A0 I max;
• A0 : (not known exactly from where measurements are repeated)
A - ΔA < A0 < A + ΔA or A0 = A ± ΔA
Relative uncertainty: ΔA/A:
• Value per unit: assessed by : ΔA/A expressed as a percentage
A0
A1 A2 A3.
A4 A5
Measured
values
Quantities and Units / Measurement Uncertainties
Calculation of uncertainties
Direct measurements (length: metre, mass: balance ...), the uncertainty is made:
• From the smallest scale ΔA of the measuring device (mm: 0.5mm)
• Or by repeating the same measurement several times: average value
Art max pressure (in mm Hg): 125 ; 130 ; 135 hence A ±ΔA = 130 ± 5 mm Hg
Indirect measurements (surface, volume, equivalent dose, effective dose, etc.):
• Obtained using a formula
• Know the uncertainties of the measured values to estimate the uncertainty of the final
calculated result
• ΔA ≈ equated to the differential of A
Quantities and Units / Measurement Uncertainties
Calculation of uncertainties
Operation
A
IA: ΔA
IR: ΔA/A
Sum
A+B
ΔA + ΔB
ΔA + ΔB /(A + B)
Difference
A-B
ΔA + ΔB
ΔA + ΔB /(A - B)
Product
AxB
AΔB +BΔA
ΔA/A + ΔB/B
Quotient
A/B
(AΔB +BΔA)/B2
ΔA/A + ΔB/B
Power
An
nAn-1 ΔA
nΔA/A
∆X is the sum of the absolute ∆ for sums and differences.
∆X/X is the sum of relative ∆ for products and quotients.
Quantities and Units / Measurement Uncertainties
Calculation of uncertainties
Do not confuse
•
Measurement uncertainty and...
•
Biological variability (true A0 ): linked to the individual
Punctual uncertainty in case of single measurement (Type B)
•
Absolute uncertainty: ΔA
•
Relative uncertainty: ΔA/A
Statistical uncertainties in case of repetitive measurements (Type A)
•
Arithmetic mean (measure of central tendency)
•
Standard deviation (measure of dispersion around the mean: SD)
Quantities and Units / Measurement Uncertainties
Writing the results
A0 = A ± ΔA with a confidence level.
Expression of the result:
• The result of a measurement will be characterised by : ΔA/A in %.
• The smaller the result, the more accurate the result.
• Determine most often a significant number of A and ΔA
Example
• We measure r = 100.251389 Ω with an uncertainty ΔA= 0.812349 Ω.
• The result (R) is then written as R = (100.3 ± 0.8) Ω.
Quantities and Units / Measurement Uncertainties
Presentation of numerical results: examples
Scientific notation: the second significant figures of a number.
Examples:
•
m = 11.597 kg means that 11.5975 kg > m > 11.5965 kg
•
m = 11.60 kg means that 11.605 kg > m > 11.595 kg
•
m = 11.6 kg means that 11.65 kg > m > 11.55 kg
Rounded to the nearest whole number :
1.349 becomes 1.3 (because the number after 3 is strictly <5)
1.350 becomes 1.4 (because the number after 3 is at least 5)
For consistency: The first five: 0, 1, 2, 3 and 4, go to the lower value.
The next five: 5, 6, 7, 8 and 9, we go to the next value.
Quantities and Units / Measurement Uncertainties
Extensive and intensive quantities Elementary
• The state of a system can be described by two types of quantities: intensive quantities
and extensive quantities.
• An intensive quantity is independent of the quantity of matter in the system.
• Examples of intensive variables:
•
Temperature
•
Pressure
•
Density (mass per volume)
Quantities and Units / Measurement Uncertainties
Elemental Extents
• An Extensive Quantity is proportional to the size and amount of matter in the system.
• Extensive grades are additive grades.
• Examples of extensive variables :
- Volume - Mass
- Electrical charge - Energy...
Quantities and Units / Measurement Uncertainties
Quantity of electricity: Electrical charge
Definition:
•
The quantity of electricity dq that flows through the cross-section of a conductor during
a time dt is the intensity of an electric current:
Dq
I = ----- ⇒ dq = Idt
dt
- Dimension dq: SI: Q
- Unit dq: S: Coulomb (Cb); 1Cb 1.1s
- Electric charges: positive or negative - dq: algebraic values
Point load :
•
If dq: in infinitely small volume dV ≈ a point
Quantities and Units / Measurement Uncertainties
Electrical dipole
•
Consists of two point charges: −𝒒 = +𝒒 , distant by l
•
Its electric charge: zero
•
Characterised by its moment: vector 𝑴
-
Origin of vector : O = medium of the dipole
-
Directed line: (- q ,+ q )
-
Direction : from - q to + q
-
Magnitude: ½ 𝑴 ½= q . l
-q
O
𝐌
+q
Quantities and Units / Measurement Uncertainties
Quantity of matter: the Mass
•
The same force 𝑭 applied to different systems generates different accelerations :
𝜸 = dv/dt (L T-2 ; m /s²)
•
𝑭 / 𝜸 : Inertia of the system is the mass of the system: a quantity that is conserved
(except for ionizing radiations : ­ E et v ­­ )
•
In an isolated system with zero or low velocity / light velocity (c = 3.108 m.s-1 ), mass is
conserved.
Quantities and Units / Measurement Uncertainties
Quantity of matter: the Mass
• At very high speeds, the mass varies: m0 and mv (E = m c )02
• Density : non-extensive quantity: r = dm/dV :
m0
mv = -----------v2
1 - ----c2
Size
Dimension
S
cgs
Mass
M
kg
g
Density
M . L-3
kg . m -3
g . cm -3
Quantities and Units / Measurement Uncertainties
Quantity of matter: the Mass
§ The relative density of a liquid or solid (d) :
• if m is the mass of a volume V of the liquid or a solid
• m0 : mass of the same volume V of water at the same temperature:
• The relative density: d = m / m 0
§ Relative Density: dimensionless and unitless quantity
• r : liquid
•
r0 : water (reference)
Þ d = rV / r0V = r / r0
Quantities and Units / Measurement Uncertainties
Amount of material: the Mole
•
Physico-chemical phenomena depend on the number of particles present in the system:
molecules, atoms, ions, electrons, etc.
•
AVOGADRO number = 6.02.1023 particles: molecules; ions; atoms...
•
Mole: dimensionless but the Mass of a mole: Molar mass ( SI: kg)
Monovalent (valence 1) or z-valent ions:
•
Cation has lost an electron (or ze), its charge is: + 1,6.10-19 Cb (or +z .1,6 .10-19 Cb)
•
Cl anion- has gained an electron (or ze), its charge is: - 1.6.10-19 Cb (or - z .1.6 .10-19 Cb)
For a mole :
•
One mole of ions of valence z, carries = z .1,6 .10–19 .6.1023 = z . 96500 Cb
= z . Q0
Quantities and Units / Measurement Uncertainties
Amount of material: Expression of biological results
According to the International Rules: the SI unit being mole
• a body with a well-defined chemical composition such as glucose can be
expressed in mmol/l: glucose: 4.5 - 6 mmol/l
• If the body is of poorly defined chemical structure (because it is complex or
mixed), the biological result is expressed in g/l.
• e.g. Protein: 70 g/l of which well-defined albumin: 45 g/l or 0.7 mmol/l
Quantities and Units / Measurement Uncertainties
Forces and Derivatives
•
A force is a physical quantity having :
- One direction
•
- An intensity or module
- An application point
Force: vector quantity: Any mass m exposed to external forces whose resultant is F,
moves with an acceleration force g :
𝑭 =m𝜸
•
Dimension:
F : MLT-2
•
Units:
SI: Kg.m.s-2 or Newton (N)
•
Weight: if. 𝜸 = g (pesanteur) : 𝑭 = m g is the weight of the mass m.
Quantities and Units / Measurement Uncertainties
FIELD OF FORCE
•
Field: Region of space where a force is exerted at a distance without a material connection
•
Field strength:
The field A exerts a force 𝑭 on B (quantity X) located at a point P in the field.
-
𝑭 ∶depends on the field A and B
-
B: intervening by its quantity X: 𝑭 = X 𝒁
-
𝒁 : the force exerted on the unit of quantity: it’s the intensity of field A at point P
(vector quantity depending on the characteristics of A and the position P)
•
Uniform field:
The field strength has at any point in the field the same direction and intensity
Quantities and Units / Measurement Uncertainties
Particular force fields
§ Gravity field:
• The quantity involved is the mass: X = m; or 𝑭 = m 𝜸
• 𝒁 = g : Intensity of the earth's gravity field is the acceleration of gravity
§ Electric field E:
• The quantity: electric charge q (positive or negative)
• Field strength is E, at a point P: this is the force exerted on the unit of
positive electric charge (+1 Cb in SI)
• Any electric charge q placed in the field E is subject to a force: 𝑭 = q 𝑬
Quantities and Units / Measurement Uncertainties
Field produced by a point charge: Coulomb's law
Two point charges q and q':
• Distances "d" => Subjected to a force F :
𝑲𝒒𝒒′
𝑭=
𝜺𝒅𝟐
- Direction: straight line from q to q'.
- Attraction between q and q': signs ≠ and Repulsion between q and q': signs =
- Intensity: F
- K constant: depends on the system of units
- e : dielectric constant, dimensionless; In vacuum: e = 1; In water: e = 80
Quantities and Units / Measurement Uncertainties
Field produced by a dipole
• Field produced by dipole:
E
E1
E2
-q
P
Au point P :
+ q produit un champ E1
- q produit un champ E2
+q
• The field strength produced by the dipole is: 𝑬 = 𝑬𝟏 + 𝑬𝟐 :
•
an electrically neutral dipole
field