Gas laws

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The laws describing the behaviour of gases under fixed pressure, volume and absolute temperature conditions are called Gas Laws. The basic gas laws were discovered by the end of the 18th century when scientists found out that relationships between pressure, volume and temperature of a sample of gas could be obtained which would hold to approximation for all gases. These macroscopic gas laws were found to be consistent with atomic and kinetic theory.

History

Following the invention of the Torricelli mercury barometer in mid 17th century, the pressure-volume gas law was soon revealed by Robert Boyle while keeping temperature constant. Marriott, however, did notice small temperature dependence. It took another century and a half to develop thermometry and recognise the absolute zero temperature scale before the discovery of temperature-dependent gas laws.

Boyle's law

Main article: Boyle's law

In 1662, Robert Boyle systematically studied the relationship between the volume and pressure of a fixed amount of gas at a constant temperature. He observed that the volume of a given mass of a gas is inversely proportional to its pressure at a constant temperature. Boyle's law, published in 1662, states that, at a constant temperature, the product of the pressure and volume of a given mass of an ideal gas in a closed system is always constant. It can be verified experimentally using a pressure gauge and a variable volume container. It can also be derived from the kinetic theory of gases: if a container, with a fixed number of molecules inside, is reduced in volume, more molecules will strike a given area of the sides of the container per unit time, causing a greater pressure.

Statement

Boyle's law states that:

The volume of a given mass of a dry gas is inversely related to its pressure when its temperature is kept constant.

The concept can be represented with these formulae:

• ${\displaystyle V\propto \frac{1}{P}}$, meaning "Volume is inversely proportional to Pressure", or
• ${\displaystyle P\propto \frac{1}{V}}$, meaning "Pressure is inversely proportional to Volume", or
• ${\displaystyle PV=k_1}$, or

$${\displaystyle P_1{V}_1=P_2{V}_2}$$

where P is the pressure, V is the volume of a gas, and k₁ is the constant in this equation (and is not the same as the proportionality constants in the other equations).

Charles' law

Main article: Charles's law

Charles' law, or the law of volumes, was founded in 1787 by Jacques Charles. It states that, for a given mass of an ideal gas at constant pressure, the volume is directly proportional to its absolute temperature, assuming in a closed system. The statement of Charles' law is as follows: the volume (V) of a given mass of a gas, at constant pressure (P), is directly proportional to its temperature (T).

Statement

Charles' law states that:

The volume of a given fixed mass of a dry gas is directly proportional to its absolute temperature at a constant pressure.

Therefore,

• ${\displaystyle V\propto T}$, or
• ${\displaystyle \frac{V}{T}=k_2}$, or

$${\displaystyle \frac{V_1}{T_1}=\frac{V_2}{T_2}}$$

,

where "V" is the volume of a gas, "T" is the absolute temperature and k₂ is a proportionality constant (which is not the same as the proportionality constants in the other equations in this article).

Gay-Lussac's law

Main article: Gay-Lussac's law

Gay-Lussac's law, Amontons' law or the pressure law was founded by Joseph Louis Gay-Lussac in 1808.

Statement

Gay-Lussac's law states that:

The pressure exerted by a given mass and constant volume of an ideal gas on the sides of its container is directly proportional to its absolute temperature.

Therefore,

• ${\displaystyle P\propto T}$, or
• ${\displaystyle \frac{P}{T}=k}$, or

$${\displaystyle \frac{P_1}{T_1}=\frac{P_2}{T_2}}$$

,

where P is the pressure, T is the absolute temperature, and k is another proportionality constant.

Avogadro's law

Main article: Avogadro's law

Avogadro's law, Avogadro's hypothesis, Avogadro's principle or Avogadro-Ampère's hypothesis is an experimental gas law which was hypothesized by Amedeo Avogadro in 1811. It related the volume of a gas to the amount of substance of gas present.

Statement

Avogadro's law states that:

The volume occupied by an ideal gas at a constant temperature is directly proportional to the number of molecules of the gas present in the container.

This statement gives rise to the molar volume of a gas, which at STP (273.15 K, 1 atm) is about 22.4 L. The relation is given by:

${\displaystyle V\propto n}$, or

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

where n is equal to the number of molecules of gas (or the number of moles of gas).

Combined and ideal gas laws

Main article: Ideal gas law

Relationships between Boyle's, Charles's, Gay-Lussac's, Avogadro's, combined and ideal gas laws, with the Boltzmann constant k_B = R/N_A = n R/N  (in each law, properties circled are variable and properties not circled are held constant)

The Combined gas law or General Gas Equation is obtained by combining Boyle's Law, Charles's law, and Gay-Lussac's Law. It shows the relationship between the pressure, volume, and temperature for a fixed mass of gas:

${\displaystyle PV=k_{5}T}$

This can also be written as:

${\displaystyle \frac{P_1{V}_1}{T_1}=\frac{P_2{V}_2}{T_2}}$

With the addition of Avogadro's law, the combined gas law develops into the ideal gas law:

${\displaystyle PV=nRT}$

where P is the pressure, V is volume, n is the number of moles, R is the universal gas constant and T is the absolute temperature.

The proportionality constant, now named R, is the universal gas constant with a value of 8.3144598 (kPa∙L)/(mol∙K).

An equivalent formulation of this law is:

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

where P is the pressure, V is the volume, N is the number of gas molecules, k_B is the Boltzmann constant (1.381×10⁻²³J·K⁻¹ in SI units) and T is the absolute temperature.

These equations are exact only for an ideal gas, which neglects various intermolecular effects (see real gas). However, the ideal gas law is a good approximation for most gases under moderate pressure and temperature.

This law has the following important consequences:

1. If temperature and pressure are kept constant, then the volume of the gas is directly proportional to the number of molecules of gas.
2. If the temperature and volume remain constant, then the pressure of the gas changes is directly proportional to the number of molecules of gas present.
3. If the number of gas molecules and the temperature remain constant, then the pressure is inversely proportional to the volume.
4. If the temperature changes and the number of gas molecules are kept constant, then either pressure or volume (or both) will change in direct proportion to the temperature.

Other gas laws

Graham's law

This law states that the rate at which gas molecules diffuse is inversely proportional to the square root of the gas density at a constant temperature. Combined with Avogadro's law (i.e. since equal volumes have an equal number of molecules) this is the same as being inversely proportional to the root of the molecular weight.

Dalton's law of partial pressures

This law states that the pressure of a mixture of gases simply is the sum of the partial pressures of the individual components. Dalton's law is as follows:

${\displaystyle P_{\text{total}}=P_1+P_2+P_3+\cdots +P_n\equiv \sum _{i=1}^n{P}_i,}$

and all component gases and the mixture are at the same temperature and volume

where P_total is the total pressure of the gas mixture

P_i is the partial pressure or pressure of the component gas at the given volume and temperature.

Amagat's law of partial volumes

This law states that the volume of a mixture of gases (or the volume of the container) simply is the sum of the partial volumes of the individual components. Amagat's law is as follows:

${\displaystyle V_{\text{total}}=V_1+V_2+V_3+\cdots +V_n\equiv \sum _{i=1}^n{V}_i,}$

and all component gases and the mixture are at the same temperature and pressure

where V_total is the total volume of the gas mixture or the volume of the container,

V_i is the partial volume, or volume of the component gas at the given pressure and temperature.

Henry's law

states that At constant temperature, the amount of a given gas dissolved in a given type and volume of liquid is directly proportional to the partial pressure of that gas in equilibrium with that liquid.

${\displaystyle p=k_{\mathrm{H}}c}$

Real gas law

formulated by Johannes Diderik van der Waals (1873).