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(2005-08-20)  
In "force n" weather, the wind speed is proportional to   n^3/2 =  nn

The widely-used Beaufort scale  was devised in 1806,by Sir Francis Beaufort (1774-1857), rear admiral, hydrographer to the Royal Navy. It was adopted by the British Admiralty in 1838,and has been in international use since 1874. Originally, the Beaufort Wind Scale did not refer to specific wind speeds,but to the effect of the wind on a full-rigged ship, and the amountof sail which should be carried. Since "force 12" meant a wind that 'no canvas can withstand',the original scale did not extend beyond that point.

Each Beaufort number still corresponds to a variety of common observationswhich can be made at sea or inland. For example, in a "force 0" condition: 'Smoke rises vertically. Sea is like a mirror.'

Since 1946, the Beaufort scale has been defined in terms of the speed of the wind,measured by an anemometer placed 10 meters above the ground.

"Force n" means a wind speed around  V.n^3/2, where V is a speed of about 1.871 mph (we're told that the 1946 scale was officially based on a speed of 0.836 m/s, or about1.87008 mph, which is slightly too low to be consistent with modern tables). Any speed  V, in mph, between 6226/169 and14646/529 yields agreement withtherounded "mph" scale below (and also with the "km/h" scale, which issomewhat less restrictive).

To find the Beaufort number corresponding to agiven speed, one divides that speed by V, and finds the whole number closestto the cubic root of the square of that ratio. As a result of this modern definition,the Beaufort scale can be extended beyond the traditional limitof "force 12" for extremely violent winds.

We have not traced the existence of a "standard" value of V; we shall simplynote that a value V = 0.8365 m/s (or any value between 0.83626 m/s and0.8368 m/s) will agree with the above tables in mph or km/h, butthat (inexplicably) tables published in knots imply a value of V fallingin the incompatible range of 0.8401 m/s to 0.8433 m/s (once the inconsistent value of 16knots published for the upper limit of a moderate breeze is lowered to 15 knots).

Wheather reports for sailors commonly use the Beaufort scale or quote wind speedsin knots. Otherwise,the media may prefer different units for wind speeds in different parts of theWorld:  m/s (Sweden, Denmark), km/h (France, Germany, Canada), mph (United States).

(2005-08-20)  
The customary scales for hurricanes  (Beaufort force 12  and "above").

In August 1969, Hurricane "Camille" hit the Mississipi-Alabama coastwith what would be "force 23" winds in anextended Beaufort scale: 200 mph to 213 mph. However, the Beaufort scale is rarely extended (if ever)  beyond force 12. Instead, the strength of hurricanes is described with the following scale,which was originally devised in 1969 byHerbertSaffir (1917-2007) a consulting structural engineer,  and   Dr.Robert H. Simpson (b. 1912)director of the National Hurricane Center  (NHC)  from 1967 to 1974.

The  NHC  considers anything below category 1to be either a tropical depression  (D)  or a tropical storm  (S). Categories 1 and 2 are Hurricanes  (H)  above the Beaufort force 12  threshold. Categories 3, 4 and 5 are major hurricanes  (M). There's no need for a category 6.

In the Atlantic, the record-breaking hurricane season of 2005 included threecategory-5 hurricanes, named Katrina, Rita and Wilma (in chronological order). At this writing (Oct. 2005) Wilma is the most intense hurricane everobserved in the Atlantic basin, featuring the lowest sea-level atmospheric pressureever recorded in the Western Hemisphere outside oftornadoes  (882 hPa). In the Northwest Pacific Ocean, only 9 typhoons have surpassed the intensity of Wilma. (The terms typhoon  and hurricane  describe thesame phenomenon, but are used in different parts of the Globe.)

The costliest hurricane ever was hurricane Katrina (August 23 to 31, 2005) which caused an estimated $200 billion in damages and atleast 1281 fatalities  (official count at this writing). After hitting land as a mere category-1 hurricane north of Miami on August 25,the eye of Katrina made landfall again in Louisianaat 6:10am (CDT) on Monday, August 29, 2005. as a category-4 hurricane... By 11 am, the storm surge had breached the leveesystem protecting New Orleans  from Lake Pontchartrain. Most of the city was subsequently flooded.

Hurricane Names

The names of Hurricanes comes from a preapproved yearly listof 21 names with initials A through W (skipping Q and U) which is reusedevery 6 years, except that names of violent hurricanes areretiredand replaced... The 2005 season had so many major storms that the last oneshad to be named after letters from the Greek alphabet (Alpha, Beta, Gamma, Delta, Epsilon, Zeta).

The list of retired names is typically decided in March of the following year. At this writing (2012-10-28, 11am EDT) there are great fears that Hurricane Sandy will make the list, as it threatens New-York City and other areas to the Northof the region commonly affected by Atlantic hurricanes. It made landfall in New-Jersey on Monday night (2012-10-29) after falling just below hurricane strength.

The following names have been retired, before the 2019 season:  Agnes (1972),Alicia (1983),Allen (1980),Allison (2001),Andrew (1992),Anita (1977),Audrey (1957),Betsy (1965),Beulah (1967),Bob (1991),Camille (1969),Carla (1961),Carmen (1974),Carol (1954),Celia (1970),Cesar (1996),Charley (2004),Cleo (1964),Connie (1955),David (1979),Dean (2007),Dennis (2005),Diana (1990),Diane (1955),Donna (1960),Dora (1964),Dorian (2019),Edna (1968),Elena (1985),Eloise (1975),Erika (2015),Fabian (2003),Felix (2007),Fifi (1974),Flora (1963),Floyd (1999),Fran (1996),Frances (2004),Frederic (1979),Georges (1998),Gilbert (1988),Gloria (1985),Gracie (1959),Gustav (2008),Harvey (2017),Hattie (1961),Hazel (1954),Hilda (1964),Hortense (1996),Hugo (1989),Igor (2010),Ike (2008),Inez (1966),Ingrid (2013),Ione (1955),Irene (2011),Iris (2001),Irma (2017),Isabel (2003),Isidore (2002),Ivan (2004),Janet (1955),Jeanne (2004),Joan (1988),Joaquin (2015),Juan (2003),Katrina (2005),Keith (2000),Klaus (1990),Lenny (1999),Lili (2002),Luis (1995),Maria (2017),Marilyn (1995),Matthew (2016),Michelle (2001),Mitch (1998),Nate (2017),Noel (2007),Opal (1995),Otto (2016),Paloma (2008),Rita (2005),Roxanne (1995),Sandy (2012),Stan (2005),Tomas (2010) andWilma (2005).

Because of the COVID-19 pandemic, the week-long WMO Committe couldn't be held normally in the Spring of 2020 to officiallyretireDorian.  Other possible retirees from the 2019 season include Imelda and Lorenzo.

(2005-08-20)  
Local twisters are primarily measured against a 6-rung scale  (F0 to F5).

Within tornadoes, the wind can reach speedsin excess of 280 mph  (450 km/h). If the Beaufort scale was applicable, this would mean force 28 or 29. Instead, all tornadoes are ranked using the following scale, from weakest to strongest,which was devised in 1971 byTed Fujita (1920-1998) at the University of Chicago.

The Original Fujita Tornado Scale  (1971-2007)

Fn	Effects	Wind speed  (km/h)
F0	Twisted antennas, broken branches	60 to 110
F1	Uprooted trees, vehicles turned over	120 to 170
F2	Lifted roofs, small projectiles	180 to 250
F3	Walls tipped over, large projectiles	260 to 330
F4	Houses destroyed, some trees lifted	340 to 410
F5	Large structures lifted, incredible damages	420 to 510

Since February 1, 2007, a revised scale has been used which is knownas the Enhanced Fujita  scale (EF).

The Enhanced Fujita Tornado Scale

EFn	Effects	Wind
EF0	Minor damage.  Some roof damage, shallow trees knocked over, etc.	65 to 85 mph
EF1	Moderate damage.  Roofs stripped, mobile homes turned over, windows broken.	86 to 110 mph
EF2	Considerable damage.  Cars lifted off the ground, roofs torn off houses, large trees uprooted.	111 to 135 mph
EF3	Reports of well-constructed houses totaled, trains and big rigs overturned, heavy cars thrown off ground.	136 to 165 mph
EF4	Houses and whole frame houses completely leveled, cars thrown and small missiles generated.	166 to 200 mph
EF5	Houses and frames swept away, steel enforced concrete structures badly damaged. Cars thrown like toys.	over 200 mph

(2006-12-02)  
A general-purpose logarithmic scale for physical power.

In a given medium, a signal carries a certain power (or a power flux)  proportional to thesquare of an associated"amplitude"  (which may be variously defined).

)  the "amplitude"of theelectromagnetic fieldshould be expressed in V/m, which identifies the electric field (E).

The relative magnitude  of two signals may be expressed equivalentlyas a logarithmic function of the ratios of their powers (P) or as the same logarithmic functionof the squares of their amplitudes (A). If decibels (dB) are used, the relative magnitude of the signal  (compared to some other signal of reference) is defined by either of the following expressions,which involve decimal logarithms.

Relative magnitude (or level)  in dB   =  10  log( P/P₀ )   =   20  log( A/A₀ )

When the amplitude doubles, the power becomes 4 times  as high andthe level is raised by roughly  6 dB. If the amplitude is multiplied by 10, the power is 100 times higherand the level is raisedexactly 20 dB.

From relative ratios to absolute measurements :

Decibels are most useful to express ratios of related signals  (for example thesignals at the input and the output of an electronic amplifier). However, specifying a conventional "reference" signal readily establishesan "absolute" decibel scale. Each choice of a particular reference establishes a different "absolute" scale. 

The most popular such scale (especially among electrical engineers)  is the decibel-milliwatt (dBm) for which the zero level (0 dBm)is a signal whose total (harmonic) power is onemilliwatt  (1 mW).

L   =   10  log ( P / 1 mW )  dBm

Power (P)	0.1 mW	1 mW	10 mW	100 mW	1 W	10 W
Level (L)	-10 dBm	0 dBm	10 dBm	20 dBm	30 dBm	40 dBm
	-40 dBW	-30 dBW	-20 dBW	-10 dBW	0 dBW	10 dBW

(2010-01-03) 
Sound Intensity Level  (SIL)  and Sound Pressure Level  (SPL)

As sound propagates, it carries a certain power per unit area of a small surfaceperpendicular to the direction of propagation. This physical quantity, called sound intensity, is measured in watt per square meter (W/m).

When expressed indecibels, that acoustic power (per unit of receiving area) is called sound intensity level. The reference level  (0 dB)  is, by convention,a sound whose intensity is W/m. The level  (L) of a sound whose intensity is I  (expressed in  W/m) is, therefore:

L   =   [ 10  log ( I ) + 120 ]  dB (SIL)

According to the general scheme outlinedabove, the amplitude  of a soundwave is most commonly defined as its acoustic pressure  p (which is equal to thetheRMS of the rapid local variations in air pressure).

A sound intensity of  10^-12 W/m² corresponds to an acoustic pressure  p_o  which dependson temperature and pressure. The above is rigorously  equivalent to:

L   =   [ 20  log ( p / p_o )]  dB (SIL)

However, in daily practice, a different  sound reference is often usedwhich is defined by an acoustic pressure of exactly 20 Pa, regardless of ambient conditions This gives rise to a slightlydifferent scale, called  "sound pressure level" and identified by the acronym SPL  (which is, unfortunately, often omitted).

L	=	20  log ( p / 20 Pa )	dB (SPL)
		[ 20  log ( p )  +  94 ]	dB (SPL)

The SPL approximation is commonly usedby practitioners who are satisfied with the mere measurement of acoustic pressure. The SPL scale is usually assumed  to coincide numericallywith the (correct) SIL scalefor dry air at room temperature under normal pressure... Let's check  that:

Thecharacteristic acoustic impedance corresponding to a sound having an intensity  I = 10^-12 W/m²  and an acoustic pressure  p = 20 Pa   is equal to:

Z   =   p² / I   =   400 Pa.s / m

For dry air under normal pressure, this would correspond to a toasty temperatureof about  40°C.  Conversely, at room temperature (20°C)  Z  would be around  413.2 Pa.s/m which yields  p_o = 20.33 Pa. This gives:

L   =   [ 20  log ( p )  +  93.84 ]  dB (SIL)      (air,  1 atm,  20°C)

So, the two formulas would match perfectly around  40°C and would be less than  0.2 dB  off at room temperature. Good enough.

The loudest possible sound is 191 dB. Isn't it?

This popular piece of trivia is to be takenwith a grainof salt, since some of the natural assumptionsnormally describing sound make little or no physical sense when thesaturation limit is approached.  Never mind,here goes nothing...

If a sound is a perfect sinewave, the acoustic pressure which appears in theSPL formula is about  70% (i.e., 1/2)  of the maximumdeviation from ambient pressure. Disallowing negative pressures, the latter quantitycannot exceed the ambient pressure  (which we assume to be the normalatmospheric pressure of  101325 Pa). So, the acoustic pressure (RMS)cannot exceed 71647.6 Pa.

The saturation level  for a sinewavewould thus be about  191 dB. Formally, a  square wave could be  3 dB louder  (194 dB). However, neither answer is satisfactory, because most assumptions about soundcollapse well below such pathological levels. In particular, large pressure disturbances are dissipative  (they heat up the airitself)  and cannot be described as waves in a linear system  (power flux need not be proportional to thesquare of acoustic pressure).

(2020-06-23)  
Logarithmic scale for frequencies.

Our musical perception of tone is esstially a logarithmic one. We perceive tones mostly in relation to each other (less than 0.01% of people have acquired as youngsters the ability, known as perfect pitch,  torecognize accurately a tone played by itself).

The most fundamental unit for differences in pitch is the octave,  which corresponds to frequencies in a two to one ratio. We give musical notes the same name if they are separated by a wholenumber of octaves. A semitone  is 1/12 of an octave. In Western art music, notes are always separated by a whole number of semitones. In musical notation,  a sharp  (#) raises a note by one semitone and a flat  (b) lowers it by one semitone.

To evaluate different tuning systems,  the most common unit is the cent, worth  0.01 semitones.

Because decimal logarithms are more common than binary logaritms, engineers often use the decade  as a convenient substitutefor the octave.  The conversion factor  is  0.30103. An octave is about 30% of a decade.

(2006-12-11)  
The absolute magnitude of a star is its apparent magnitude 10 pc away.

On June 22, 134 BC (proleptic Julian calendar). a new star (nova) appeared in Scorpius which was as bright as Venusand could be spotted in the daytime.  It remained visible to thenaked eye until July 21.

According toPliny,this rare event is what prompted Hipparchus of Nicaea (c.190-126 BC) to compile a new catalog of all visible stars (you can't spot new things without an inventory of old ones). Eratosthenes(276-194 BC) had previously listed only 675 relatively bright stars. The catalog that Hipparchus produced in  129 BC listed 1080 stars that he classified into six magnitudes, from brightest (first magnitude) to faintest (sixth magnitude).

The magnitude system of Hipparchus was popularized byPtolemy's Almagest  and became standard. Quantitatively,it turns out that a star of the first magnitude in that system is about100 times as bright as a star of the sixth magnitude. Thus, in 1854, the British astronomerN.R. Pogson(1829-1891)proposed to refine the Ptolemaic rating system by turning it into astrict logarithmic scale, where a difference of 5 magnitudes would separatetwo stars whose brightnesses are in a ratio of 100 to 1.

So specified, the modern system of stellar magnitudes extends to faint objects  (beyond magnitude 6)  and very bright ones (the brighteststars, theplanets,the Moon, the Sun)  which are assigned a magnitudebelow 1, or even a negative one... The Sun has a magnitude of  -26.7. At a magnitude of -1.6, Sirius is the brightest object outside thesolar system. The faintest stars detected so far by the largest telescopes have a magnitude of 23 or so...

-Lyrae)  which was defined to be of magnitude zeroover any part of the spectrum. With modern, more practical, standards  (outlined below)  Vega'svisual magnitude is now listed to be  0.03.

As brightnessdecreases by a factor of  100 ^1/5,magnitudeincreases by one unit. This factor is known as Pogson's ratio, in honor of Norman Pogson.

100 ^0.2   = 10 ^0.4   =   2.51188643150958...

This simply means that onestar magnitude is exactly equal to  4 decibels(4 dB). However, star magnitudes are very rarely (if ever) expressed in decibels. Historically, the relation is reversed:  The idea for expressingpowers in decibels came from the stellar magnitude system !

There are 20 stars of the first magnitude (magnitude less than 1.5) 60 stars of thesecond magnitude (magnitude between 1.5 and 2.5) about 180 stars of the thirdmagnitude (between 2.5 and 3.5) etc. This tripling pattern holds for relatively bright stars but tends to be lessexplosive thereafter (it looks more like a mere doubling for starsaround magnitude 20).

Most physicists would probably prefer to base star magnitudeson theirbolometric output powers (in which all electromagnetic frequencies carry equalweight).  This is rarely done, if ever, except for the Sun itself.

Ideally, thevisual magnitude of a star should be based on thepower it emits in the visible spectrum,using the same standardphotopic response of the human retinaon which thedefinition of the lumenis based  (although the dark-adaptedscotopic response mightbe more relevant to direct telescopic observations by humans).

In practice, however, various standard filters are used instead which allow anautomated determinationof a star's magnitude in different portions of the electromagnetic spectrum. In the main,  the emission spectrum of a star is close to that of a blackbody and calibrated comparisons of the different flavorsof magnitude are used to determine a star's surface temperature (T).

Regardless of what spectrum-specific "flavor" of star magnitude is used,the absolute  magnitude of a star is defined aswhat its apparent magnitude would be if it was observed at a distanceof 10 pc  (10 parsecs is about 32.6 light-years). To determine the absolute magnitude of a star, its distance must first be estimated (using parallax or other methods)  so that the apparent magnitude can beadjusted, knowing that the observed power flux varies as the inversesquare of the distance.

Conversely, the absolute magnitude of some stars may be known fromother considerations (e.g., the absolute magnitude of a Cepheid variable star is a function of the period of its oscillation in brightness). Some distances can thus be derived from apparent magnitudes,without the need for delicate parallax measurements (which aren't possible for intergalactical distances).

(2015-07-18)  
A logarithmic scale foracidity, devised by
SørenSørensen  (1868-1939)  in 1909.

Before it was given a quantified meaning, the notion of acidity was recorded throughthe color changes it induces in a large lineup of indicator  substances,  still commonly used today (includinglitmus,  which dates back to 1300 or so).

The  pH  of a solution (at first, Sørensen wrote  p[H+] )  is the opposite of the decimallogarithm of the activity of the hydrogen ions  (H+)  in it:

pH   =   log₁₀ [ H⁺ ]

More precisely, we should talk about the activity of the hydronium ions (H₃O+)  since every bare hydrogen ion in water will instantly combine with at least one water molecule.  For non-acidic solutions or dilute  acids, this complication can be ignored,  because the number of water molecules so combined isan insignificant portion of the total number of water molecules.

Strictly speaking, the activity  of a solute is a thermodynamical quantity which need not be strictly proportional toits concentration. For dilute solution, however, this is an excellent approximation which is universallyadopted.  In particular, the acid dissociationconstants  given in all modern chemical references are for  "activities" equated to the concentrations expressed in moles per liter  (mol/L).

There's only one exception to this convention, but it's an important one: When water is used as the solvent, the concentration of undissociated water moleculesremains constant, except for extremely  concentrated solutions (for which the whole model breaks down).  With ludicrous precision:

[ H₂O ]   =   (999.9720 g/L) / (18.0152 g/mol)   =   55.5071 mol/L   (nominally)
[ H₂O ]   =   (998.2071 g/L) / (18.0152 g/mol)   =   55.4092 mol/L   (at  20°C)
[ H₂O ]   =   (997.0479 g/L) / (18.0152 g/mol)   =   55.3448 mol/L   (at  25°C)

Because it's very nearly constant and largely irrelevant, that concentrationis incorporated into the well-known ionic product  for water (at 24.9°C):

H₂O     H ⁺  +  OH      with      K_W   =   [ H⁺]  [ OH ]  =   10 ¹⁴

For pure water, electrical neutrality implies that  [ H⁺]  =  [ OH ]  so that both concentrations are equal to  10 ⁷. Thus, the pH of pure water is 7. This decreases with temperature;  at 50°C, the pH of pure water is only 6.6.

The  pH of pure water depends on temperature.

Temperature	KW / (mol/L)2	pH
0°C	0.114 14	7.472
10°C	0.293 14	7.267
20°C	0.681 14	7.083
24.9°C	1.000 14	7.000
25°C	1.008 14	6.998
30°C	1.471 14	6.916
40°C	2.916 14	6.768
50°C	5.476 14	6.631
100°C	51.3  14	6.145

Human blood is a buffered liquid of pH 7.4.  It's normally tightly regulated,chemically and organically, to a healthy pH range between 7.35 and 7.45.

Most  pH  values ordinarily encountered are between  pH 0 (e.g., hydrochloric acid,  HCl,  at a concentration of  1 mol/L) and  pH 14  (e.g., sodium hydroxide,  NaOH,  at  1 mol/L). Most meters won't go beyond those limits, but there's nothing sacred about them: Solutionstwice as concentrated as the two we just quoted can still beconsidered dilute, but their pH values are respectively -0.3 and 14.3.

In water, chemical reactions are often critically dependent on acidity.

For example, an acidicstop bathis normally used when processing photographic filmsto put an abrupt end to the action of the developer (which can only operate in an alkaline solution). That's more efficient and more precise than an amateurish plain water stop-bath whichwould merely slow down the action of the developer by diluting it greatly.

Concentrated solutions:

In concentrated solutions, water no longer plays the rôle of an overwhelmingsolvent.  Even if we keep believing that activities are proportional to concentrations (per unit of volume)  we must acknowledge that there's no longer a directproportionality between the volume of the solution and the number of water molecules in it. We must also get rid of the aforementioned simplified equation for the dissociation of water andrestore a proper mass-action  equationfor it  (which reduces to the previous one in the dilute case):

2 H₂O    H₃O ⁺  +  OH

with      K_U   =   h  [ OH ]/ [ H₂O ] ²  =   10 ^17.486

Working out the  pH  of a complicated aqueous solution:

Computing   pH = -log h   from the initial concentrations of all thereactants is a basic engineering skill that all undergraduates are expected to master.

Surprisingly enough, the tradition is to teach them a murky method which is only effectivein conjunction with simplifying assumptions made a priori  which are supposed to test the intuition of the student (that's summarized by the infamousRICE mnemonic). The only merit of that method is to quickly give good approximative answers... in academic tests designed for it!

It's important to realize, by contrast, that a simpler method yields directlythe algebraic equation satisfied by the concentration  h  of hydrogen ions (whose logarithm is the opposite of the  pH). Here's that two-step method:

1. Express all concentrations in terms of  h  and the dissociation constants.
2. The lack of a net electric charge then yields the equation satisfied by  h.

Note that only the concentrations of charged ions are needed to carry outthe second step.  So, we may as well think of the first step as the algebraicelimination of the concentrations of neutral species.

Let's apply this to the example of a weak monoacid with a dissociation constant K_A = 10 ^4.76 (this value is foracetic acid) at concentration  c:

AH     H +  +  A		KA	=	h  [ A ]  /  [ AH ]
		c	=	[ A ]  +  [ AH ]

By eliminating  [ AH ]  we obtain   [ A ]  =  c / (1+h/K_A).
Likewise, for the dissociation of water itself, we have:

H₂O     H ⁺  +  OH      and      K_W   =   h  [ OH ]   =   10 ¹⁴

Let's plug the ensuing value  [ OH ]  =  K_W / h   into the neutrality equation:

[ OH ]  +  [ A ]   =   [ H ⁺ ]      becomes      K_W / h  +  c / (1+h/K_A)   =   h

This is a cubic  equation in  h  which would turn intoa quadratic one if we knew a priori  that K_W / h  is negligible (which is the normal assumption presented with the RICE recipe for acidic solutions). This would entail:

c   =   h  (1 + h/K_A)      or      h²  +  K_Ah    K_Ac   =   0

As usual, for the sake of numerical robustness, I recommend shunning the traditional quadratic formula when solving any quadratic equationwhich may have small solutions  (in this day and age when direct and inversetrigonometric or hyperbolic functions are as readily available as the lowly square-root function). Instead, let's consider the quadratic equation in  h whose roots are  x exp(-y)  and  -x exp(y):

h²   (x exp(-y) x exp(y)) h   x ²   =   0

For historical reasons, vinegar and acetic acid are often rated by volume,in a way similar to alcoholic spirits. A molar solution  (60.052 g/L)  is  5.6881%  by volume, or  "56.881 grains"  (same thing, by definition).

Citric Acid :  (CH₂ COOH)₂ COH COOH

The above can be readily generalized to more complicated cases,without the need for a priori  simplifications. Let's illustrate this with citric acic,  second only to acetic acid in the heartof old-school photographers, who routinely measure its molar concentration  (c)  by weighing its monohydrate (C₆H₈O₇ , H₂O) at  210.14 g/mol.

Here,  H₃A  will stand for the undissociated moleculeof citric acid and the citrate anions, partially hydrogenated or not,  are: H₂A, HA²  and A³. Their respective concentrations verify the following equations :

• h [H₂A] /  [H₃A]  =   K₁   =   10 ^3.13
• h [HA²] /  [H₂A]  =   K₂   =   10 ^4.76
• h [A³] /  [HA²]  =   K₃   =   10 ^6.39

Therefore :

• [H₂A]  =   [H₃A]  K₁ / h
• [HA²]  =   [H₃A]  K₁K₂ / h²
• [A³]  =   [H₃A]  K₁K₂K₃ / h³

This yields  c   =   [H₃A]  ( 1 +  K₁ / h +  K₁K₂ / h² +  K₁K₂K₃ / h³ )
which gives  [H₃A]  and, thus, the concentrations ofall citric species.

Electrical neutrality then provides the desired relation between  h  and  c :

[H⁺]    [OH]  =  [H₂A] +  2 [HA²] +  3 [A³]

h    KW / h  =     c	K1 / h +  2 K1K2 / h2 +  3 K1K2K3 / h3
	1  +  K1 / h +  K1K2 / h2 +  K1K2K3 / h3

To plot the curve quickly,  just remark that  c is a rational function  of  h.

That function gives directly the concentration of a solution ofobserved  pH. Conversely, an algebraic equation must be solved to predict the pHobtained from a given concentration,  as in the following table:

Generalization :

In the above discussion of citric acid, the emerging expressions suggest introducing a cubic polynomialand using its derivative  as follows:

Q_A (x)   =  1  +  K₁ x +  K₁K₂ x² +  K₁K₂K₃ x³
q_A (x)   =   x Q_A' (x)  /  Q_A (x)

For a monoprotic or diprotic acid, we'd have instead a linear or quadraticpolynomial,  respectively,  but we'd always end up with this relation:

h    K_W / h  =     c  q_A (1/h)

Likewise, with any mix of weak acids at concentrations a₁,a₂,a₃...

h    K_W / h  =   a₁  q_A1 (1/h) + a₂  q_A2 (1/h) + a₃  q_A3 (1/h) +  ...

(2015-09-15, drafted in 2001)  
Hotness of peppers  (French:poivrons). Burning sensation in the mouth.

InNahuatl, the language of the Aztecs,  6 adjectives describe the hotnessof chili peppers in order of increasing pungency: coco, cocopatic,cocopetzpatic, cocopetztic, copetzquauitl, and cocopalatic.

WhenColumbussailed West in search for a more direct route to Asian spices, what hefound instead were the spices of the New World, the pungent capiscums, orchili peppers, whose piquancy was so important to the Aztecs.

Pungency  (also called piquancy, spicyness, burn, hotness, heat,warmth, bite, sting or kick) is a component of food flavor which happens to be technicallydifferent from taste and smell, as it relies on chemoreception by the free (undifferentiated)  nerve endings of thetrigeminal network, mostly in the tongue but also in the rest of the mouth, in the nose and in the eyes. That makes very pungent compounds effective in self-defense pepper sprays  which can incapacitate an aggressor.

The pungency of chili peppers is primarily due to a potent alkaloid called capsaicin [ trans-8-methyl-N-vanillyl-6-nonenamide ]  identified in 1816 by Christian-FriedrichBucholz (1770-1818).  More than a dozen related capsaicinoids  have been found in nature:

The three major capsaicinoids :  Dihydrocapsaicin  may occur inconcentrations similar to those of capsaicin itself. Both have about the same pungency (if the potency per mole was the same, dihydrocapsaicin's slightly highermolar mass would make it only 0.66% less pungent).  In most cases, 90% ofthe pungency of chili peppers is attributable to these two alkaloids. When the third major  capsaicinoid (nordihydrocapsaicin) is included as well, about 98% of the pungency is usually accounted for.

Minor capsaicinoids :  All the other capsaicinoids are considered minor. About half-a-dozen minor capsaicinoids occur in chile peppers. The most important are homodihydrocapsaicin and homocapsaicin. Others  (including norcapsaicin and nornorcapsaicin) do not occur naturally in significant concentrations.

At high concentrations, capsaicinoids are extremely painful and may be harmful. In the tongue, cells with vanilloid receptors may be damaged or destroyed byhigh levels of capsaicin.  Contrary to popular belief, however, capsaicin does not cause ulcers or any other direct damage to the stomachlining or to the rest of the digestive tract.

The Scoville Pungency Scale :

The original quantitative pungency scale was devised in 1912 and isnamed after its inventor, the American pharmacologistWilbur L. Scoville(1865-1942) who was then employed by the Parke Davis Pharmaceutical Company.

Scoville had found that the chemical methods avaible at the time wereunable to detect the presence of capsaicin at the very low concentrationswhich made the human tongue react:  Even minute amounts of capsaicin willtrigger the tongue's pain receptors  (free trigeminal nerve endings).

Ignoring the mockery of some of his colleagues, Scoville thus decidedthat the needs of the spice trade would be best served by a physiological pungencyscale directly based on human sensory perception  (a so-called organoleptic  scale)  which he described in 1912:

The ScovilleOrganoleptic Test  is carried out from one grain  (64.79891 mg)  ofpepper macerated overnight in 100 mL of ethanol (the solubility of capsaicin in water would be too low for this extraction step). Once filtered, that solution is rated by a panel of trained testers/tasters. A rating of  N Scoville heat units  (SHU) means that most panel members feel the solution to be somewhat pungent  when one volume is diluted in N volumes of sweetened water (the substance will typically still be detectable  at a dilution about twice as high).

Scoville ratings apply to other pungent chemicals besides capsaicinoids. For example, zingerone (the pungent ingredient of cooked ginger)  is 1000 times less pungent than capsaicin.

Pungency may be either measured directly by such organoleptic tests, orit may be deduced from the known concentrations of all pungent ingedientsand their previously established respective pungencies (using a table likethe one provided below). A common approximation, which is roughly valid forthe relative concentrations of capsaicinoids observed in typical chilipeppers, is to obtain the number of Scoville Heat Units (SHU) by multiplyingby 15 the total concentration of capsaicinoids expressed in ppm.  Forexample, habaneros  (rated at 300000 SHU) contain 20000 ppm (or 2%)  of capsaicinoids.

This rule of thumb begat a new unit endorsed by the American Spice Trade Association (ASTA): An ASTA unit is essentially defined to be 15 SHU, sothat you obtain the approximate pungency of any compound by forgoing themultiplication by 15 in the above rule.  Also, ASTA rating proceduresachieve better consistency in organoleptic tests by comparing differentcompounds over a whole range of perceptions, not just at the threshold ofdetection, as with the original Scoville test:  Chemicals are given ratingsin an A:B ratio if they are judged to give equally pungent solutions atrespective dilutions that are in a B:A ratio.

To obtain more consistent pungency  measurements, Paul W. Bosland  (professor of horticulture atNew MexicoState University and co-founder of the Chile PepperInstitute)  has pioneered the use of high-performance liquid chromatography  (HPLC) toobtain the concentrations of the main capsaicinoids (and/or other pungentingredients): By definition, the pungency of a given mixture is obtained by adding theknown pungencies of all its chemical components (seetable) weighed by their respective concentrations  (by weight).

That method has been endorsed bythe American Spice Trade Association  for the "ASTA units"described above, but the Scoville scale remains much more popular for publication (the ASTA unit is thus now considered to be equal to 15 SHU,de jure). The pungency of hundreds of varieties of chile pepper has been rated this way. Ratings do depend on different crops of the same variety and may even vary from pod to pod.

According to the "Guiness Book of World Records", the record in 2001  (first draft of this article) was a 1994 measurement of 577000 SHU for a "Red Savina" habanero pod. This variety is a natural mutant strain discovered in 1989 by Frank Garcia (GNSSpices Inc., of Walnut, CA) who spotted a single red pod in themiddle of his field of orange habaneros.

Since then, the record for chili peppers has been broken several times:

•    577 000 SHU : Red Savina  by Frank Garcia  (1994).
• 1 041 427 SHU : Ghost Pepper  (2007).
• 2 009 231 SHU : Trinidad Moruga Scorpion  (2012-02-13).
• 2 200 000 SHU : Carolina Reaper  by Ed Currie  (2013-08-07).
• 2 480 000 SHU : Dragon's_Breath  by Neal Price  (2017, unconfirmed).
• 3 180 000 SHU : Pepper X  by Ed Currie  (2017, unconfirmed).

Scoville Heat Units  (SHU) for some pungent chemicals and foodstuff :

Substance	Formula	g/mol	SHU
Resiniferatoxin (RTX)		628.718	16 000 000 000
Tinyatoxin (TTX)		598.692	5 300 000 000
capsaicin, mace		305.421	16 000 000
dihydrocapsaicin, nonanamide		307.436	15 000 000
nonivamide =PAVA  (synthetic)		293.409	9 200 000
nordihydrocapsaicin		293.409	9 100 000
homodihydrocapsaicin		321.462	8 100 000
homocapsaicin		319.447	6 900 000
piperine		285.345	230 000
red pepper			200 000
shogaol		276.380	160 000
birdseye  (India)			140 000
kumataka  (Japan)			110 000
Mexican tabiche			100 000
gingerol		294.183	60 000
cayenne pepper, tabasco pepper			40 000
zingerone		194.232	16 000
tabasco® sauce			2140
cubanelle =  Cuban pepper			1000
paprika			400
pimiento =  pimento  =  cherry pepper			100
ASTA unit			15
Scoville heat unit  (SHU)			1
bell peppers,  sweet peppers,  water			0

Now, could anyonepleasetell me what SHU ratings are described ascoco, cocopatic, cocopetzpatic, cocopetztic, copetzquauitl, or cocopalatic?

Some dangerous  neurotoxins have been found to bemany times more pungent than capsaicin itself.  They can inflict extreme pain and chemical burns. They could even kill  a human being,  in gram quantities:

• Resiniferatoxinis about 1000 more pungent than pure capsaicin.
• Tinyatoxin has been rated around  5 300 000 000 SHU.

Other Types of Pungency :

Some spicy foods are normally not assigned any Scoville rating at all...

Plants of the Allium genus (includinggarlic,shallot andotheronions) have a different kind of pungency, mostly due to a volatile  oil (allyl propyl disulfide) and they are rated according to thepyruvate scale.

The pungency of mustard oils  comes from the isothiocyanate chemical group  (-N=C=S). The sharp taste ofmustard,Japanesewasabi,horseradish andradish is mostly due toallyl isothiocyanate (AITC).

(2018-10-06)  
Gas marks  for kitchen ovens and traditional descriptions used by cooks.

Modern kitchen ovens increasingly indicate temperatures directly indegrees Celsius and/or degrees Fahrenheit.  So do many modern cookbooks.

Regulated gas oven originally came with numbered marks which became popular along traditional descriptions. The following formulas are all but forgotten but the indications surviveon appliances and in cookbooks  (with dubious accuracy in both cases).

The quoted formulas must be rounded to the nearest integer.

Origin	Name	Formula	Notes
British	Gas Mark	F / 25 10	F = Temperature in °F
French	Thermostat	C / 20 5	C = Temperature in °C
German	Stufe	C / 20 6	C = Temperature in °C

Marking of  ½  and  ¼ just below  "1"  may be used for low temperatures.

The above guidelines have apparently not been strictly enforced by manufacturers. Over time,  they've eroded beyond repair,  from one cookbook to the next.

The equivalences in the following table are loosely based on common Celsius ratings, which are always multiples of  5°C.  Those temperatures correspond exactly  to the wholenumber of degrees Fahrenheit indicated (which may differ slightly from common Fahrenheit equivalences, often rounded to a multiple of  10°F  or  25°F).

Gas Mark	Nominal		Traditional Description
	80°C	176°F	Drying
	110°C	230°F	Very Slow	Very Low
	120°C	248°F
1	130°C	266°F	Slow	Low
2	140°C	284°F
3	160°C	320°F	Moderately Slow	Warm
4	175°C	347°F	Moderate	Medium
5	190°C	374°F		Moderately Hot
6	200°C	392°F	Moderately Hot
7	220°C	428°F	Hot
8	230°C	446°F	Hot	Very Hot
9	250°C	482°F	Very Hot
10	260°C	500°F	Extremely Hot
	275°C	527°F

(2005-11-26)  
The seismicenergy radiated is the basis of a rationalized Richter scale.

The original Richter Scale was devised in 1935 at theCalifornia Institute of Technologyby Beno Gutenberg and Dr. Charles F. Richter.  More modern versions of that scale have beendevised which are adequate to measure the largest earthquakes while being roughly compatiblewith the traditional 1935 definition for small earthquakes.

The  1935Richter Scale  of Richter and Gutenberg (now calledlocal magnitude) was defined asa logarithmic  scale; strictly based on readings from a particular type of instrument then used at CalTech (theWood-Andersontorsion seismometer). Magnitude 0  was arbitrarily assigned to an earthquake that would cause amaximum combined horizontal displacement of 1 micron  (1 micrometer) on such an instrument at  100 km from the epicenter. (This reference level is so low that negative magnitudes are very rarely quoted.) If that amplitude increases by a factor of 10, the local magnitude  increases by one unit.

The problem with this viewpoint is that the amplitude originally considered by Richteris not a simple function of the energy released, except for the smallest earthquakes. There are nonlinearities and the duration of the earthquake is also an important factor,especially for very large quakes which may last several minutes...

• Mercalli Intensity Scale: The effects measured at a particular location.
• Wood-Anderson seismographs at Caltech.
• Charles F. Richter & Beno Gutenberg: log E = 11.8 + 1.5 R
• Seismic Moment, Hiro Kanamori: M is about 20000 E.

(2020-09-04)  
Devised by Christopher G. Newhall  and Stephen Self.

(2020-01-04)  
A short history of graphite pencil leads and their degrees of hardness.

Natural Graphite Deposit  (1555)

Ordinary coal is too fragmented for direct use in drawing or writing. For this, you need pure solid graphite  in crystal form. The only large-scale deposit of graphite ever found in this form was discovered in 1555near the hamlet of Seathwaitein the Lake District of the county of Cumbria (North West England). That mineral was originally mistaken for some form of lead (Pb) and it was dubbed plumbago  (plumbagine, in French) until correctly identified as a crystalline form of carbon,  in 1778,  by Carl Wilhelm Scheele (1742-1786). The name graphite  ("writing stone", based on the Greek rootá) was subsequently coined in 1779 by Abraham Gottlob Werner (1749-1817) along with the words molybdena and black lead for substances which had been confused with natural graphitebefore Scheele's analysis...

Artificial graphite  (1792, 1795)

Credit for the modern pencil is usually given to Nicolas-Jacques Conté (1755-1805) for patenting (1795)  the Conté process of binding under pressure powdered graphite or coal and inertclay  (kaolin) with wax  (or some other binder, including modern polymers)  to obtain a material suitable for themanufacture of round pencil cores by extrusion  (the cut leads are then baked for stiffness).  Conté did so in just a few days of experimentation,  at the request of Lazare Carnot (1753-1823) because of the shortage of the aforementionned  high-grade graphite fromEngland due to theFrench Revolutionary Wars. The process was patented in 1795.

The successful Austrian architect Josef Hardtmuth (1758-1816)had founded  (in 1790)  the  pencil company,  now calledKoh-i-Noor Hardtmuth.

In 1792, Josef Hardtmuth was confronted with the same wartime shortage of high-grade graphite as Contéand he came up independently with the same solution:  Hardtmuth's artificial graphite was produced by a method similar to Conté's process,  which is still usedto produce modern pencils.  Common powdered graphite and clay are fused ubder pressure with a binder  made from wax  (or modern polymer, now). Hardtmuth was granted a patent for this in 1802.

The company was managed by Josef's widow (Elisabeth Kiesler, 1762-1828) from 1816 to 1828,  at which time their two [youngest] sons inherited it and the name was changed to L. & C. Hardtmuth. However, Ludwig "Louis" (1800-1861) eventually lost interest and it wasCarl Hardtmuth (1804-1881) who took over the family business.

In 1846, he relocated the headquarters from Vienna to their current location in the Bohemiantown of Budweis, where the construction of the world's first purpose-built pencil factory was completed in 1848.

In 1851,  Carl's son Franz Hardtmuth(1832-1896)  had the idea to manufacture different hardnesses by varying the proportion of theingredients and to commercialize different degrees under the letter-based scalewe still use today:  HB denotes the most common medium hardness  (called #2 in the US) and F (#3) comes between HB and H (#4).  Franz Hardtmuth chose the letters F, H and B simply because they werehis own initials and the initial of the Company's new location  (Budweis, in Bohemia). The English mnemonic is that "H" is hard  and "B" is black.

In 1889,  they stamped those grades on iconic hexagonal yellow pencils and namedthe new brand after one of the most famous pieces of pure carbon ever, the Koh-I-Noor  diamond. In 1894,  the Kooh-I-Noor trademark became part of the Harstmuth company name.  Just in time for the 1900 World Fair in Paris.  Flawless marketing.

At least 28 degrees of hardness have been made commercially available:
10H, 9H, 8H, 7H, 6H, 5H, 4H, 3H, 2H, H(#4), F(#3), HB(#2), B(#1),
2B, 3B, 4B, 5B, 5B, 7B, 8B, 9B, 10B, 11B, 12B, 13B, 14B, 15B, 16B.

CurrentlyKoh-i-NoorHardtmuth  itself produces 21 degrees  (10H to 9B). Staedtler offers 24  (10H to 12B)  with the softest grades at a premium price.

At least one other brand  (Pentel Ain Stein)  offers a special gradedubbed "HB soft" or "HB1",  intermediate between  "HB" and "B".

The competing numerical scale commonly used in the US  (where "#2 pencils" have HB cores) covers only the middle of the above range. It was introduced by the pencil business of the family of the American philosopherHenry David Thoreau (born David Henry Thoreau, 1817-1862) The business was started by his father John Thoreau and his maternal uncle,  Charles Dunbar,  who had purchased a graphite source in New Hampshire in 1821. In 1844. Henry David Thoreau  rediscovered the Conté process (1795) whichmade better pencil production possible using either Dunbar's New-Hampshire deposits and anothermine exploited on the Tantiusques reservation of the Nipmuc tribe.

The Thoreau pencil was the most popular American-made pencil of its day and it financedthe publication of Thoreau's books.

Some sources disagree with the US equivalents listed above  (between parentheses) they call  "H"  #3 pencils  (which makes "F" #2½) so "2H" becomes #4.

Graphite leads now come in the following standard diameters (in mm): 0.2, 0.3, 0.35 (rare), 0.4,0.5,0.7, 0.9, 1.1 (rare), 1.3,2.0, 4.0 and 5.6. The most common size for wooden pencils is 2 mm  (4.0 mm for soft degrees or colored pencils) It's 0.5 or 0.7 mm for mechanical pencils.

Thesurface hardness of paint coatings is routinely measured on the same scale as pencils. It can be loosely defined as the hardness of the hardest pencil which can write on a painted surface without making a dent in it.

Relation between pencil hardness and Brinell hardness :

In recent years, the re-loading community  (gun enthusiasts who hand-cast theirown bullets, which they like to call boolits)  have been using pencil setsto test the hardness of their lead alloys (using mostly Staedtler's Mars Lumograph  sets of6, 12, 20  or even 24 degrees). Some reloaders have compared this common test to the better-defined Brinell hardness numbers  used by metallurgists. Conversely,  this opens up the way to a future scientific definition of the HBscale for pencils, losely based on the following table which circulates among reloaders:

Pencil Hardness  vs.  Brinell Hardness (BHN) of Lead Alloys

Pencil			HBN	Practical Alloys
		6B	4-5	100% Pb. Pure Lead. Lead Wire. Lead sheet.
		5B	7-8	40:1 lead-tin. Plumber's tin.
		4B	9	25:1 lead-tin.
		3B	10	20:1 lead-tin.  NewWheel weights.
		2B	11-12	Range scrap.  OldWW.
#1	#1	B	13	WW+2% tin.  Quenched range scrap.
#1½,		HB1	14
#2	#2	HB	14-15	Lyman #2 alloy,  1:1 linotype alloy
#2½	#3	F	16-18	Commercial cast bullets
#3	#4	H	20-22	1:1 linotype WW.  Monotype.
#4		2H	26-28	Quenched WW.  Monotype.

For alloys of lead (Pb) containing a small percentage of tin (Sn) and/or antimony (Sb) some reloaders rely on the following approximative formula giving the Brinell Hardness Number  (BHN)  as a linear function ofthe respective percentages of tin and antimony  (by weight):

BHN   =   8.60  +  0.29 × Sn%  +  0.92 × Sb%

Shooters may want a BHN of 15  (12 to 18 is fine). The above says that's achievedby a Pb-Sn alloy with 22% of tin, by a Pb-Sb alloy with 7% of antimony, or by any mix of those two  (e.g., 11% Sn and 3.5% Sb).

Hardness is slightly increased by water quenching. Namely, the practice of dropping bullets from the mould directly into a bucket of water,instead of the recommended dry soft pad.