# Avogadro's Law

For most solids and liquids it is convenient to obtain the amount of sub- stance (and the number of particles, if we want it) from the massA measure of the force required to impart unit acceleration to an object; mass is proportional to chemical amount, which represents the quantity of matter in an object.. In the section on The Molar Mass numerous such calculations using molar mass were done. In the case of gases, however, accurate measurement of mass is not so simple. Think about how you would weigh a balloon filled with helium, for example. Because it is buoyed up by the air it displaces, such a balloon would force a balance pan *up* instead of down, and a negative weightA measure of the gravitational force on an object; directly proportional to mass. would be obtained. Solids and liquids are also buoyed up, but they have much greater densities than gases. For a given mass of a solid or liquid, the volume is much smaller, much less air is displaced, and the buoyancy effect is negligible. The mass of a gas can be obtained by weighing a truly empty container (one in which there is a perfect vacuum), and then filling and reweighing the container. But this is a time-consuming, inconvenient, and sometimes dangerous procedure. (Such a container might **implode**—explode inward—due to the difference between atmospheric pressureForce per unit area; in gases arising from the force exerted by collisions of gas molecules with the wall of the container. outside and zero pressure within.)

A more convenient way of obtaining the amount of substanceA material that is either an element or that has a fixed ratio of elements in its chemical formula. in a gaseous sample is suggested by the data on molar volumes in Table 1. Remember that a molar quantity (a quantity divided by the amount of substance) refers to the same number of particles.

TABLE 1 Molar Volumes of Several Gases at 0°C and 1 atm Pressure.

Substance
| Formula
| Molar Volume/liter mol
^{–1} |

Hydrogen | H_{2}(g)
| 22.43 |

Neon | Ne(g)
| 22.44 |

Oxygen | O_{2}(g)
| 22.39 |

Nitrogen | N_{2}(g)
| 22.40 |

Carbon dioxide | CO_{2}(g)
| 22.26 |

Ammonia | NH_{2}(g)
| 22.09 |

The data in Table 1, then, indicate that for a variety of gases, 6.022 × 10^{23} molecules occupy almost exactly the same volume (the * molar volume*) if the temperatureA physical property that indicates whether one object can transfer thermal energy to another object. and pressure are held constant. We define

*for gases as 0°C and 1.00 atm (101.3 kPa) to establish convenient conditions for comparing the molar volumes of gases. The molar volume is close to 22.4 liters (22.4 dm*

**Standard Temperature and Pressure (STP)**^{3}) for virtually all gases. (Since the liter, now defined as exactly 1 dm

^{3}, is commonly used as a unitA particular measure of a physical quantity that is used to express the magnitude of the physical quantity; for example, the meter is the unit of the physical quantity, length. of volume in conjunction with the atmosphereA unit of pressure equal to 101.325 kPa or 760 mmHg; abbreviated atm. Also, the mixture of gases surrounding the earth. as a unit of pressure, we shall use it that way in this chapter.) That

*equal volumes of gases at the same temperature and pressure contain equal numbers of molecules*was first suggested in 1811 by the Italian chemist Amadeo Avogadro (1776 to 1856). Consequently it is called

**Avogadro’s law**or

**Avogadro’s hypothesis**.

Avogadro’s law has two important messages. First, it says that molar volumes of *all* gases are the same at a given temperature and pressure. Therefore, even if we do not know what gas we are dealing with, we can still find the amount of substance. Second, we expect that if a particular volume corresponds to a certain number of molecules, twice that volume would contain twice as many molecules. In other words, *doubling* the volume corresponds to *doubling* the amount of substance, *halving* the volume corresponds to *halving* the amount, and so on.

In general, if we *multiply* the volume by some factor, say *x*, then we also *multiply* the amount of substance by that same factor *x*. Such a relationship is called direct proportionality and may be expressed mathematically as

*V n* (1)

where the symbol means “is proportional to.” Any proportion, such as Eq. (1) can be changed to an equivalent equation if one side is multiplied by a proportionality constant, such a *k _{A}* in Eq.(2):

*V* = *k _{A}*

*n*(2)

If we know *k _{A}* for a gas, we can determine the amount of substance from Eq. (2).

The situation is complicated by the fact that the volume of a gas depends on pressure and temperature, as well as on the amount of substance. That is, *k _{A}* will vary as temperature and pressure change. Therefore we need quantitative information about the effects of pressure and temperature on the volume of a gas before we can explore the relationship between amount of substance and volume.

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