# 1.6 Mathematical treatment of measurement results

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By the end of this section, you will be able to:
• Explain the dimensional analysis (factor label) approach to mathematical calculations involving quantities
• Use dimensional analysis to carry out unit conversions for a given property and computations involving two or more properties

It is often the case that a quantity of interest may not be easy (or even possible) to measure directly but instead must be calculated from other directly measured properties and appropriate mathematical relationships. For example, consider measuring the average speed of an athlete running sprints. This is typically accomplished by measuring the time required for the athlete to run from the starting line to the finish line, and the distance between these two lines, and then computing speed from the equation that relates these three properties:

$\text{speed}=\phantom{\rule{0.2em}{0ex}}\frac{\text{distance}}{\text{time}}$

An Olympic-quality sprinter can run 100 m in approximately 10 s, corresponding to an average speed of

$\frac{\text{100 m}}{\text{10 s}}\phantom{\rule{0.2em}{0ex}}=\text{10 m/s}$

Note that this simple arithmetic involves dividing the numbers of each measured quantity to yield the number of the computed quantity (100/10 = 10) and likewise dividing the units of each measured quantity to yield the unit of the computed quantity (m/s = m/s). Now, consider using this same relation to predict the time required for a person running at this speed to travel a distance of 25 m. The same relation between the three properties is used, but in this case, the two quantities provided are a speed (10 m/s) and a distance (25 m). To yield the sought property, time, the equation must be rearranged appropriately:

$\text{time}=\phantom{\rule{0.2em}{0ex}}\frac{\text{distance}}{\text{speed}}$

The time can then be computed as:

$\frac{\text{25 m}}{\text{10 m/s}}\phantom{\rule{0.2em}{0ex}}=\text{2.5 s}$

Again, arithmetic on the numbers (25/10 = 2.5) was accompanied by the same arithmetic on the units (m/m/s = s) to yield the number and unit of the result, 2.5 s. Note that, just as for numbers, when a unit is divided by an identical unit (in this case, m/m), the result is “1”—or, as commonly phrased, the units “cancel.”

These calculations are examples of a versatile mathematical approach known as dimensional analysis    (or the factor-label method ). Dimensional analysis is based on this premise: the units of quantities must be subjected to the same mathematical operations as their associated numbers . This method can be applied to computations ranging from simple unit conversions to more complex, multi-step calculations involving several different quantities.

## Conversion factors and dimensional analysis

A ratio of two equivalent quantities expressed with different measurement units can be used as a unit conversion factor    . For example, the lengths of 2.54 cm and 1 in. are equivalent (by definition), and so a unit conversion factor may be derived from the ratio,

$\frac{\text{2.54 cm}}{\text{1 in.}}\phantom{\rule{0.2em}{0ex}}\text{(2.54 cm}=\text{1 in.) or 2.54}\phantom{\rule{0.2em}{0ex}}\frac{\text{cm}}{\text{in.}}$

Several other commonly used conversion factors are given in [link] .

Common Conversion Factors
Length Volume Mass
1 m = 1.0936 yd 1 L = 1.0567 qt 1 kg = 2.2046 lb
1 in. = 2.54 cm (exact) 1 qt = 0.94635 L 1 lb = 453.59 g
1 km = 0.62137 mi 1 ft 3 = 28.317 L 1 (avoirdupois) oz = 28.349 g
1 mi = 1609.3 m 1 tbsp = 14.787 mL 1 (troy) oz = 31.103 g

what is the main reaction between sodium and chlorine
when a sodium atom is transferred an electron to a chloride atom forming a sodium cation and a chlorde anoin both ions are compltely valence shells and a energetically more stable this reaction is extremely more exothermic producing a bright yellow light abd a great deal of heat energy
sani
well what kind of sodium
coland
sodium chloride
coland
Tyropanoic acid and its salt sodium tyropanoate are radiocontrast agents used in cholecystography (X-ray diagnosis of gallstones). Trade names include Bilopaque, Lumopaque, Tyropaque, and Bilopac. The molecule contains three heavy iodine atoms which obstruct X-rays in the same way as the calcium in
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Sodium tetradecyl sulfate (STS) is a commonly used synonym for 7-ethyl-2-methyl-4-undecanyl sulfate sodium salt which is anionic surfactant that is the active component of the sclerosant drug Sotradecol. It is commonly used in the treatment of varicose and spider veins of the leg, during the procedu
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Sodium stibogluconate, sold under the brand name Pentostam among others, is a medication used to treat leishmaniasis. This includes leishmaniasis of the cutaneous, visceral, and mucosal types. Some combination of miltefosine, paramycin and liposomal amphotericin B; however, may be recommended due to
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Sodium picosulfate (INN, also known as sodium picosulphate) is a Contact stimulant laxative used as a treatment for constipation or to prepare the large bowel before colonoscopy or surgery. It is sold under the trade names Sodipic Picofast, Laxoberal, Laxoberon, Purg-Odan, Picolax, Guttalax, Namilax
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Ipodate sodium (sodium iopodate) is an iodine-containing radiopaque contrast media used for X-rays. The drug is given orally and the resulting contrast allows for easy resolution of the bile duct and gall bladder. Other uses Although not FDA approved, ipodate sodium has been used to treat Graves'
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Sodium ferric gluconate complex (brand name ferrlecit by Sanofi), is an intravenously administered iron product indicated in the treatment of iron deficiency anemia. It is frequently used in patients undergoing hemodialysis, those undergoing erythropoietin therapy, and/or patients who have chronic k
coland
Dehydrocholic acid is a synthetic bile acid, manufactured by the oxidation of cholic acid. It acts as a hydrocholeretic, increasing bile output to clear increased bile acid load.
coland
Ethylenediaminetetraacetic acid (EDTA), also known by several other names, is an aminopolycarboxylic acid and a colourless, water-soluble solid. Its conjugate base is ethylenediaminetetraacetate. It is widely used to dissolve limescale. Its usefulness arises because of its role as a hexadentate ("s
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Sodium aurothiomalate (INN, known in the United States as gold sodium thiomalate) is a gold compound that is used for its immunosuppressive anti-rheumatic effects. Along with an orally-administered gold salt, auranofin, it is one of only two gold compounds currently employed in modern medicine. Med
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Sodium ascorbate is one of a number of mineral salts of ascorbic acid (vitamin C). The molecular formula of this chemical compound is C6H7NaO6. As the sodium salt of ascorbic acid, it is known as a mineral ascorbate. It has not been demonstrated to be more bioavailable than any other form of vitamin
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Diatrizoic acid (or its anionic form, diatrizoate), also known as amidotrizoic acid, or 3,5-diacetamido-2,4,6-triiodobenzoic acid, is a radiocontrast agent containing iodine. Trade names include Hypaque, Gastrografin, Iothalmate and Urografin, the latter being a combination of the sodium and meglum
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Sodium acetrizoate (rINN, trade names Urokon, Triurol and Salpix), the sodium salt of acetrizoic acid, is a high-osmolality, water-soluble, iodine-based radiographic contrast medium no longer in clinical use. Acetrizoate was developed by V.H. Wallingford of Mallinckrodt, and introduced in 1950; it
coland
i think some was missed
coland
what chlorine L-Alpha glycerylphosphorylcholine (alpha-GPC, choline alfoscerate) is a natural choline compound found in the brain. It is also a parasympathomimetic acetylcholine precursor which may have potential for the treatment of Alzheimer's disease and other dementias. Alpha-GPC rapidly delive
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Choline Theophyllinate , also known as oxtriphylline, is a cough medicine derived from xanthine that acts as a bronchodilator to open up airways in the lung. Chemically, it is a salt of choline and Theophylline. It classifies as an expectorant. The drug is available under the brand names Choledyl an
coland
Choline (/ˈkoʊlin/) is a water-soluble vitamin. It is usually grouped within the B-complex vitamins. The term cholines refers to the class of quaternary ammonium salts containing the N,N,N-trimethylethanolammonium cation (X− on the right denotes an undefined counteranion). The cation appears in the
coland
thanks
sani
hydrogen reacting with water
what is matter
apart from those device there is there any device
how many elements do we have
Modern Electronic Theory
a new way or an improvement in modern electrical products.
Andrew
find the volume of oxygen produced from the electrolysis of acidified water of a current of a 2A was passed through the electrolysis of acidified water for 2 hours
14400J
Ese
wrong question, go through it. oxygen cannot be produced.
Andrew
A group of atoms that are connected by chemical bounds
molecule
NARCOS_
the atoms are found in the shell, it is called atomic bond.
Andrew
what is molecules
what atoms
Simon
Moecules are groups of atoms
atom is the smallest part of an element or matter that indivisible
Ese
What is collision theory
hello guys am new here
what is the si unit for energy
joules
udoh
what is Charles law
Ogbu
how do mass of solvent be gram or kilogram ?
by converting
Andrew
its a force a gas exerts on the wall of the container
compressed it is
Andrew
what is gas pressure
the pressure is the amount of Gas that is compressed in the container. in other words because of the gas that is compressed in the container there is pressure on it.
Andrew