0.450 mole of Fe contains how many atoms? Solution: Start from the box labeled 'Moles of. Avogadro's number is the number of particles in one mole of any substance. Its numerical value is 6.02225 × 10 23. One mole of oxygen gas contains 6.02 × 10 23 molecules of oxygen, while one mole of sodium chloride contains 6.02 × 10 23 sodium ions and 6.02 × 10 23 chloride ions. The number of representative particles in that sample. The coefficient of the substance when in a chemical reaction. What does Avogadro's number represent? The number of representative particles in a mole of a substance. The number of atoms in a mole of a compound. 6.02 x 1023 representative particles.

• How To Calculate Avogadro's Constant
• Avogadro's Number Of Particles
• Avogadro's Number Of Molecules
• Avogadro's Constant Is Defined As The Number Of Particles In
• Avogadro's Number Of Representative Particles
• Avogadro's Number Equation

Avogadro's law (sometimes referred to as Avogadro's hypothesis or Avogadro's principle) or Avogadro-Ampère's hypothesis is an experimental gas law relating the volume of a gas to the amount of substance of gas present.^[1] The law is a specific case of the ideal gas law. A modern statement is:

Avogadro's law states that 'equal volumes of all gases, at the same temperature and pressure, have the same number of molecules.'^[1]

For a given mass of an ideal gas, the volume and amount (moles) of the gas are directly proportional if the temperature and pressure are constant.

The law is named after Amedeo Avogadro who, in 1812,^[2][3] hypothesized that two given samples of an ideal gas, of the same volume and at the same temperature and pressure, contain the same number of molecules. As an example, equal volumes of gaseous hydrogen and nitrogen contain the same number of atoms when they are at the same temperature and pressure, and observe ideal gas behavior. In practice, real gases show small deviations from the ideal behavior and the law holds only approximately, but is still a useful approximation for scientists.

Mathematical definition[edit]

The law can be written as:

${\displaystyle V\propto n}$

or

${\displaystyle \frac{V}{n}=k}$

where

V is the volume of the gas;

n is the amount of substance of the gas (measured in moles);

k is a constant for a given temperature and pressure.

This law describes how, under the same condition of temperature and pressure, equal volumes of all gases contain the same number of molecules. For comparing the same substance under two different sets of conditions, the law can be usefully expressed as follows:

${\displaystyle \frac{V_1}{n_1}=\frac{V_2}{n_2}}$

The equation shows that, as the number of moles of gas increases, the volume of the gas also increases in proportion. Similarly, if the number of moles of gas is decreased, then the volume also decreases. Thus, the number of molecules or atoms in a specific volume of ideal gas is independent of their size or the molar mass of the gas.

Derivation from the ideal gas law[edit]

The derivation of Avogadro's law follows directly from the ideal gas law, i.e.

${\displaystyle PV=nRT}$,

where R is the gas constant, T is the Kelvin temperature, and P is the pressure (in pascals).

Solving for V/n, we thus obtain

${\displaystyle \frac{V}{n}=\frac{RT}{P}}$.

Compare that to

${\displaystyle k=\frac{RT}{P}}$

which is a constant for a fixed pressure and a fixed temperature.

How To Calculate Avogadro's Constant

An equivalent formulation of the ideal gas law can be written using Boltzmann constantk_B, as

${\displaystyle PV=N{k}_{\mathrm{B}}T}$,

where N is the number of particles in the gas, and the ratio of R over k_B is equal to the Avogadro constant.

In this form, for V/N is a constant, we have

.

If T and P are taken at standard conditions for temperature and pressure (STP), then k′ = 1/n₀, where n₀ is the Loschmidt constant.

Historical account and influence[edit]

Avogadro's hypothesis (as it was known originally) was formulated in the same spirit of earlier empirical gas laws like Boyle's law (1662), Charles's law (1787) and Gay-Lussac's law (1808). The hypothesis was first published by Amadeo Avogadro in 1811,^[4] and it reconciled Dalton atomic theory with the 'incompatible' idea of Joseph Louis Gay-Lussac that some gases were composite of different fundamental substances (molecules) in integer proportions.^[5] In 1814, independently from Avogadro, André-Marie Ampère published the same law with similar conclusions.^[6] As Ampère was more well known in France, the hypothesis was usually referred there as Ampère's hypothesis,^{[note 1]} and later also as Avogadro–Ampère hypothesis^{[note 2]} or even Ampère–Avogadro hypothesis.^[7]

Experimental studies carried out by Charles Frédéric Gerhardt and Auguste Laurent on organic chemistry demonstrated that Avogadro's law explained why the same quantities of molecules in a gas have the same volume. Nevertheless, related experiments with some inorganic substances showed seeming exceptions to the law. This apparent contradiction was finally resolved by Stanislao Cannizzaro, as announced at Karlsruhe Congress in 1860, four years after Avogadro's death. He explained that these exceptions were due to molecular dissociations at certain temperatures, and that Avogadro's law determined not only molecular masses, but atomic masses as well.

Ideal gas law[edit]

Boyle, Charles and Gay-Lussac laws, together with Avogadro's law, were combined by Émile Clapeyron in 1834,^[8] giving rise to the ideal gas law. At the end of the 19th century, later developments from scientists like August Krönig, Rudolf Clausius, James Clerk Maxwell and Ludwig Boltzmann, gave rise to the kinetic theory of gases, a microscopic theory from which the ideal gas law can be derived as an statistical result from the movement of atoms/molecules in a gas.

Avogadro constant[edit]

Avogadro's law provides a way to calculate the quantity of gas in a receptacle. Thanks to this discovery, Johann Josef Loschmidt, in 1865, was able for the first time to estimate the size of a molecule.^[9] His calculation gave rise to the concept of the Loschmidt constant, a ratio between macroscopic and atomic quantities. In 1910, Millikan'soil drop experiment determined the charge of the electron; using it with the Faraday constant (derived by Michael Faraday in 1834), one is able to determine the number of particles in a mole of substance. At the same time, precision experiments by Jean Baptiste Perrin led to the definition of Avogadro's number as the number of molecules in one gram-molecule of oxygen. Perrin named the number to honor Avogadro for his discovery of the namesake law. Later standardization of the International System of Units led to the modern definition of the Avogadro constant.

Molar volume[edit]

Taking STP to be 101.325 kPa and 273.15 K, we can find the volume of one mole of gas:

Avogadro's Number Of Particles

${\displaystyle V_{\mathrm{m}}=\frac{V}{n}=\frac{RT}{P}=\frac{(8.314\text{J}\cdot {\text{mol}}^{-1}{\mathrm{K}}^{-1})(273.15\text{K})}{101.325\text{kPa}}=22.41{\text{dm}}^3{\text{mol}}^{-1}=22.41\text{liters}/\text{mol}}$

Avogadro's Number Of Molecules

For 101.325 kPa and 273.15 K, the molar volume of an ideal gas is 22.4127 dm³⋅mol⁻¹.

See also[edit]

• Boyle's law – Relationship between pressure and volume in a gas at constant temperature
• Charles's law – Relationship between volume and temperature of a gas at constant pressure
• Gay-Lussac's law – Relationship between pressure and temperature of a gas at constant volume.
• Ideal gas – Mathematical model which approximates the behavior of real gases

Notes[edit]

1. ^First used by Jean-Baptiste Dumas in 1826.
2. ^First used by Stanislao Cannizzaro in 1858.

References[edit]

1. ^ ^abEditors of the Encyclopædia Britannica. 'Avogadro's law'. Encyclopædia Britannica. Retrieved 3 February 2016.CS1 maint: extra text: authors list (link)
2. ^Avogadro, Amedeo (1810). 'Essai d'une manière de déterminer les masses relatives des molécules élémentaires des corps, et les proportions selon lesquelles elles entrent dans ces combinaisons'. Journal de Physique. 73: 58–76.English translation
3. ^'Avogadro's law'. Merriam-Webster Medical Dictionary. Retrieved 3 February 2016.
4. ^Avogadro, Amadeo (July 1811). 'Essai d'une maniere de determiner les masses relatives des molecules elementaires des corps, et les proportions selon lesquelles elles entrent dans ces combinaisons'. Journal de Physique, de Chimie, et d'Histoire Naturelle (in French). 73: 58–76.
5. ^Rovnyak, David. 'Avogadro's Hypothesis'. Science World Wolfram. Retrieved 3 February 2016.
6. ^Ampère, André-Marie (1814). 'Lettre de M. Ampère à M. le comte Berthollet sur la détermination des proportions dans lesquelles les corps se combinent d'après le nombre et la disposition respective des molécules dont les parties intégrantes sont composées'. Annales de Chimie (in French). 90 (1): 43–86.
7. ^Scheidecker-Chevallier, Myriam (1997). 'L'hypothèse d'Avogadro (1811) et d'Ampère (1814): la distinction atome/molécule et la théorie de la combinaison chimique'. Revue d'Histoire des Sciences (in French). 50 (1/2): 159–194. doi:10.3406/rhs.1997.1277. JSTOR23633274.
8. ^Clapeyron, Émile (1834). 'Mémoire sur la puissance motrice de la chaleur'. Journal de l'École Polytechnique (in French). XIV: 153–190.
9. ^Loschmidt, J. (1865). 'Zur Grösse der Luftmoleküle'. Sitzungsberichte der Kaiserlichen Akademie der Wissenschaften Wien. 52 (2): 395–413.English translation.

Note: this document will print in an appropriately modified format (6 pages)

The chemical changes we observe always involve discrete numbers of atoms that rearrange themselves into new configurations. These numbers are HUGE— far too large in magnitude for us to count or even visualize, but they are still numbers, and we need to have a way to deal with them. We also need a bridge between these numbers, which we are unable to measure directly, and the weights of substances, which we do measure and observe. The mole concept provides this bridge, and is central to all of quantitative chemistry.

Owing to their tiny size, atoms and molecules cannot be counted by direct observation. But much as we do when 'counting' beans in a jar, we can estimate the number of particles in a sample of an element or compound if we have some idea of the volume occupied by each particle and the volume of the container.

Once this has been done, we know the number of formula units (to use the most general term for any combination of atoms we wish to define) in any arbitrary weight of the substance. The number will of course depend both on the formula of the substance and on the weight of the sample. But if we consider a weight of substance that is the same as its formula (molecular) weight expressed in grams, we have only one number to know: Avogadro's number, 6.022141527 × 10²³, usually designated by N_A.

Amadeo Avogadro (1766-1856) never knew his own number!
Avogadro only originated the concept of this number, whose actual value was first estimated by Josef Loschmidt, an Austrian chemistry teacher, in 1895.

You should know it to three significant figures:
N_A = 6.02 × 10²³

6.02 × 10²³ of what? Well, of anything you like: apples, stars in the sky, burritos. But the only practical use for N_A is to have a more convenient way of expressing the huge numbers of the tiny particles such as atoms or molecules that we deal with in chemistry. Avogadro's number is a collective number, just like a dozen.

Think of 6.02 × 10²³ as the 'chemist's dozen'.

The basic idea (CurtisWang, 4 min) ****
The Mole Explained (utaustinchem, 9 min) ****
The Mole, Avogadro's no., counting by weight (dcaulf, 13½ min) *****
Simple mole calculations (bozemanscience, 5m) ****
Mole and Avogadro's number (Khan, 9½ m) **
The mole concept (IsaacsTeach, 5 min) ****
Mole problems review (LindaHanson, 23 min) ****

Before we get into the use of Avogadro's number in problems, take a moment to convince yourself of the reasoning embodied in the following examples.

The atomic weights of oxygen and of carbon are 16.0 and 12.0, respectively. How much heavier is the oxygen atom in relation to carbon?

Solution: Atomic weights represent the relative masses of different kinds of atoms. This means that the atom of oxygen has a mass that is 16/12 = 4/3 ≈ 1.33 as great as the mass of a carbon atom.

The absolute mass of a carbon atom is 12.0 unified atomic mass units (What are these?). How many grams will a single oxygen atom weigh?

Solution: The absolute mass of the carbon atom is 12.0 u,
or 12 × 1.6605 × 10^–27 g = 19.9 × 10^–27 kg. The mass of the oxygen atom will be 4/3 greater, or 2.66 × 10^–26 kg.

Alternatively: (12 g/mol) ÷ (6.022 × 10²³ mol^–1) × (4/3) = 2.66 × 10^–23 g.

Suppose that we have N carbon atoms, where N is a number large enough to give us a pile of carbon atoms whose mass is 12.0 grams. How much would the same number, N, of oxygen atoms weigh?

Solution: The mass of an oxygen atom (16 u) is 16/12 = 4/3 that of a carbon atom (12 u), so the collection of N oxygen atoms would have a mass of
4/3 × 12 g = 16.0 g.

Things to understand about Avogadro's number N_A

• It is a number, just as is 'dozen', and thus is dimensionless.

• It is a huge number, far greater in magnitude than we can visualize; see here for some interesting comparisons with other huge numbers.

Avogadro's Constant Is Defined As The Number Of Particles In

• Its practical use is limited to counting tiny things like atoms, molecules, 'formula units', electrons, or photons.

• The value of N_A can be known only to the precision that the number of atoms in a measurable weight of a substance can be estimated. Because large numbers of atoms cannot be counted directly, a variety of ingenious indirect measurements have been made involving such things as Brownian motion and X-ray scattering.

• The current value was determined by measuring the distances between the atoms of silicon in an ultrapure crystal of this element that was shaped into a perfect sphere. (The measurement was made by X-ray scattering.) When combined with the measured mass of this sphere, it yields Avogadro's number. But there are two problems with this: 1) The silicon sphere is an artifact, rather than being something that occurs in nature, and thus may not be perfectly reproducible. 2) The standard of mass, the kilogram, is not precisely known, and its value appears to be changing. For these reasons, there are proposals to revise the definitions of both N_A and the kilogram. See here for more, and stay tuned!

Wikipedia has a good discussion of Avogadro's number

The mole (abbreviated mol) is the the SI measure of quantity of a 'chemical entity', which can be an atom, molecule, formula unit, electron or photon. One mol of anything is just Avogadro's number of that something. Or, if you think like a lawyer, you might prefer the official SI definition:

Some useful mole links
Wikipedia article on the mole
Mystified by the Mole? Stop it!
A short history of Avogadro's Number
National Mole Day site

Avogadro's number N_A = 6.02 × 10²³, like any pure number, is dimensionless. However, it also defines the mole, so we can also express N_A as
6.02 × 10²³ mol^–1; in this form, it is properly known as Avogadro's constant. This construction emphasizes the role of Avogadro's number as a conversion factor between number of moles and number of 'entities'.

How many moles of nickel atoms are there in 80 nickel atoms?

Solution: (80 atoms) / (6.02E23 atoms mol^–1) = 1.33E–22mol

Is this answer reasonable? Yes, because 80 is an extremely small fraction of N_A.

Molar mass and its uses (IsaacsTeach, 6½ m) ****

Avogadro's Number Of Representative Particles

The atomic weight, molecular weight, or formula weight of one mole of the fundamental units (atoms, molecules, or groups of atoms that correspond to the formula of a pure substance) is the ratio of its mass to 1/12 the mass of one mole of C¹² atoms, and being a ratio, is dimensionless. But at the same time, this molar mass (as many now prefer to call it) is also the observable mass of one mole (N_A) of the substance, so we frequently emphasize this by stating it explicitly as so many grams (or kilograms) per mole: g mol^–1.

Don't let this confuse you; it is very important always to bear in mind that the mole is a number and not a mass. But each individual particle has a mass of its own, so a mole of any specific substance will always have a mass unique to that substance.

Borax is the common name of sodium tetraborate, Na₂B₄O₇. In 20.0 g of borax,
(a) how many moles of boron are present?
(b) how many grams of boron are present?

Solution: The formula weight of Na₂B₄O₇ is (2 × 23.0) + (4 × 10.8) + (7 × 16.0) = 201.2.

a) 20 g of borax contains (20.0 g) ÷ (201 g mol^–1) = 0.10 mol of borax, and thus 0.40 mol of B.

b) 0.40 mol of boron has a mass of (0.40 mol) × (10.8 g mol^–1) = 4.3 g.

The plant photosynthetic pigment chlorophyll contains 2.68 percent magnesium by weight. How many atoms of Mg will there be in 1.00 g of chlorophyll?

Solution: Each gram of chlorophyll contains 0.0268 g of Mg, atomic weight 24.3.
Number of moles in this weight of Mg: (.0268 g) / (24.2 g mol^–1) = 0.00110 mol
Number of atoms: (.00110 mol) × (6.02E23 mol^–1) = 6.64E20

Avogadro's Number Equation

Is this answer reasonable? (Always be suspicious of huge-number answers!) Yes, because we would expect to have huge numbers of atoms in any observable quantity of a substance.

This is the volume occupied by one mole of a pure substance. Molar volume depends on the density of a substance and, like density, varies with temperature owing to thermal expansion, and also with the pressure. For solids and liquids, these variables ordinarily have little practical effect, so the values quoted for 1 atm pressure and 25°C are generally useful over a fairly wide range of conditions. This is definitely not the case with gases, whose molar volumes must be calculated for a specific temperature and pressure.

Methanol, CH₃OH, is a liquid having a density of 0.79 g per milliliter. Calculate the molar volume of methanol.

Solution: The molar volume will be the volume occupied by one molar mass (32 g) of the liquid. Expressing the density in liters instead of mL, we have

V_M = (32 g mol^–1) / (790 g L^–1) = 0.0405 L mol^–1

Problem Example 7: Radius of a strontium atom

The density of metallic strontium is 2.60 g cm^–3. Use this value to estimate the radius of the atom of Sr, whose atomic weight is 87.6.

Solution: The molar volume of Sr is (87.6 g mol^–1) / (2.60 g cm^–3) = 33.7 cm³ mol^–1

The volume of each 'box' is (33.7 cm³ mol^–1) / (6.02E23 mol^–1) = 5.48E–23 cm³
The side length of each box will be the cube root of this value, 3.79E–8 cm. The atomic radius will be half this value, or 1.9E–8 cm = 1.9E–10 m = 190 pm.

Note: Your calculator probably has no cube root button, but you are expected to be able to find cube roots; you can usually use the x^y button with y=0.333. You should also be able estimate the magnitude of this value for checking. The easiest way is to express the number so that the exponent is a multiple of 3. Take 54.8E–24, for example. Since 3³=27 and 4³ = 64, you know that the cube root of 55 will be between 3 and 4, so the cube root should be a bit less than 4 × 10^–8.

So how good is our atomic radius? Standard tables give the atomic radius of strontium is in the range 192-220 pm, depending on how it is defined.

Make sure you thoroughly understand the following essential ideas which have been presented above. It is especially important that you know the precise meanings of all the highlighted terms in the context of this topic.

• Define Avogadro's number and explain why it is important to know.
• Define the mole. Be able to calculate the number of moles in a given mass of a substance, or the mass corresponding to a given number of moles.
• Define molecular weight, formula weight, and molar mass; explain how the latter differs from the first two.
• Be able to find the number of atoms or molecules in a given weight of a substance.
• Find the molar volume of a solid or liquid, given its density and molar mass.
• Explain how the molar volume of a metallic solid can lead to an estimate of atomic diameter.