Ideal gas law

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The ideal gas law is the famous equation, taught to all chemistry students:

${\displaystyle PV=NRT\,}$

that describes the behavior of an "ideal" gas. Various parts of it are also known as Charles' law, Boyle's law, Gay-Lussac's law, or Avogadro's law.

The formulation of this law played a crucial role in the development of the atomic theory (that is, the realization that matter really is made up of atoms), and in the development of thermodynamics.

This "law" was, for a long time, only an experimental observation, without any scientific insight into why it is true. It wasn't until the advent of the kinetic theory of gases, in the mid 1800's, that it was put on a firm theoretical foundation. That theoretical foundation is beyond the scope of this article.

In the formula ${\displaystyle PV=NRT\,}$:

• P is the pressure of the gas. In SI units, this is measured in Pascals, or Newtons of force per square meter of area. ("Standard atmospheric pressure at sea level" is about 101,000 Pascals, or 101 KiloPascals. 100 KPa is also known as 1 bar, and pressure is often described in millibars. On mercury-calibrated barometers, 100 KPa is 29.5 inches (750 millimeters) of mercury, so standard atmospheric pressure is 29.8 inches.)
• V is the volume, in cubic meters.
• T is the temperature, in Kelvins. That is, the absolute temperature. The Kelvin, or absolute, temperature is the Celsius temperature plus 273.15.
• N is the amount of gas, in moles. A mole is defined as Avogadro's number (6.022 x 1023) of particles. This number is chosen so that 1 mole of any atom or molecule, measured in grams, is equal to the atomic weight of all of the atoms tht the molecule contains. For example, Chlorine is a diatomic molecule, Cl2. The atomic weight of Chlorine is 35.45, so 1 mole of Cl2 molecules weighs 70.9 grams.
• R is the universal gas constant that makes it all work. That is, the ideal gas law says that
${\displaystyle {\frac {PV}{NT}}}$

is a constant, and that constant, called R is experimentally found to be

${\displaystyle 8.314\ {\frac {(Nt/m^{2})(m^{3})}{(mole)(Kelvin)}}}$

which is

${\displaystyle 8.314\ {\frac {Joules}{(mole)(Kelvin)}}}$

The various parts of the law were discovered in several steps by several people performing careful experiments in the 17th, 18th, and 19th centuries. It only applies to "ideal" gases (see Gases and gas laws for a discussion of this), but common gases are sufficiently close to ideal behavior that the experiments were fruitful.

An animation showing the relationship between pressure and volume when amount and temperature are held constant.

Boyle's law, formulated around 1662 by Robert Boyle (1627-1691), says that, at constant temperature, the volume occupied by a given quantity of gas is inversely proportional to the pressure. Intuitively, a piston on a cylinder of gas is "springy"—it will resist attempts to push it in.

${\displaystyle PV={\textrm {something}}\ \ {\textrm {that}}\ \ {\textrm {depends}}\ \ {\textrm {on}}\ \ {\textrm {temperature}}\ \ {\textrm {for}}\ \ {\textrm {a}}\ \ {\textrm {given}}\ \ {\textrm {amount}}\ \ {\textrm {of}}\ \ {\textrm {a}}\ \ {\textrm {given}}\ \ {\textrm {gas}}\,}$

An animation demonstrating the relationship between volume and temperature.

Gay-Lussac's law, formulated around 1802 by Joseph Louis Gay-Lussac (1778-1850), says that, at constant pressure, the volume occupied by a given quantity of gas is directly proportional to the temperature.

This law is sometimes called Charles' law, after an earlier formulation of the same principle around 1787 by Jacques Charles (1746-1823.)

Intuitively, as a gas is heated, it expands. This required redefining the temperature scale to get the proportionality to work out. That temperature scale is now called the Kelvin scale, or absolute temperature. Kelvin temperature is 273.15 degrees higher than Celsius or centigrade temperature. Gay-Lussac's law implies that, if a gas could be cooled to absolute zero (0° Kelvin, or -273.15° Celsius), its volume would go to zero. In practice, a substance would cease to be a gas before this point is reached.

Putting Boyle's law and Gay-Lussac's law together, one gets:

${\displaystyle {\frac {PV}{T}}={\textrm {something}}\ \ {\textrm {that}}\ \ {\textrm {depends}}\ \ {\textrm {on}}\ \ {\textrm {the}}\ \ {\textrm {type}}\ \ {\textrm {and}}\ \ {\textrm {amount}}\ \ {\textrm {of}}\ \ {\textrm {gas}}\,}$

The right hand side of this equation is clearly proportional to the amount of the gas, measured in grams, for example, since two grams will occupy twice as much volume as one gram, other things being equal.

Around 1811, Amedeo Avogadro (1776-1856) made the remarkable discovery that, if the amount of gas is measured in actual number of molecules, there is just one constant—it doesn't depend on the type of gas. This built on the atomic theory, formulated around 1803 by John Dalton (1766-1844) that matter is made of molecules and atoms, and that the mass of a given atom is proportional to its atomic weight, a measure that had been worked out by then for many elements based on their chemistry. This meant that, if the amount of gas is measured in multiples of its atomic weight, there is just one constant. An amount of gas (or any other substance) whose weight in grams is equal to its atomic or molecular weight, is called a mole (or perhaps gram-mole to make it clear how things are being weighed) and is a definite number of atoms or molecules. That number is Avogadro's number, which is now taken to be 6.022 x 1023.

Avogadro's number was not known with any certainty in the early years. It is basically a measure of how small atoms are—Avogadro's number of Hydrogen atoms weigh 1.008 grams (its atomic weight.) It was simply known that atoms are extremely small, and hence Avogadro's number is extremely large. But the ideal gas law, and the chemical laws of definite proportions and multiple proportions, which gave rise to the atomic theory, didn't depend on knowing the actual value.

Putting this all together, one gets:

${\displaystyle PV=NRT\,}$

where N is the number of moles of gas, that is, the number of molecules divided by Avogadro's number, or, equivalently, the mass in grams divided by the molecular weight. R is the universal gas constant:

${\displaystyle R=8.314\ {\frac {Joules}{(mole)(Kelvin)}}}$

In the laboratory, it is common to use "standard temperature and pressure" (STP) as a starting approximation. This is a temperature of 273.15° Kelvin, or 0° Celsius, and a pressure 101.325 KiloPascals, or 760 millimeters of Mercury. With this assumption, the volume of one mole of gas is:

${\displaystyle V={\frac {RT}{P}}={\frac {8.314\times 273.15}{101325}}=.0224\ m^{3}=22.4\ L}$

So, at STP, 39.95 grams (its atomic weight) of Argon occupies 22.4 liters.

Boltzmann's constant

This particular value of the gas constant is really just an artifact of the choice of Avogadro's number, which is an artifact of the way atomic weights are measured. We can divide the gas constant by Avogadro's number, getting a more fundamental "per molecule" value instead of a "per mole" value. This is called Boltzmann's constant, typically denoted k.

${\displaystyle k=1.38065\times 10^{-23}\ {\frac {Joules}{Kelvin}}}$

Expressed this way, the ideal gas law is

${\displaystyle PV=nkT\,}$

where n is the number of molecules and k is Boltzmann's constant. It is a convention that a lower-case n counts molecules, while an upper-case N counts moles.

Boltzmann's constant plays a prominent role in thermodynamics and statistical mechanics.

Heat capacity

The heat capacity, or specific heat of a substance is the amount of heat energy that is required to raise its temperature by some amount. It is often measured in Joules per degree (Celsius or Kelvin) per amount of substance. While the general topic of a substance's heat capacity is far beyond the scope of this article, there are a few things that can be said about it in the context of the ideal gas law.

The amount of substance can be measured in grams, meaning that the unit of heat capacity is ${\displaystyle {\frac {Joules}{(gram)(Kelvin)}}}$. For substances that have a clear notion of what constitutes a molecule, the mole is perhaps a better choice for the unit of substance. Measured this way, the unit of heat capacity is ${\displaystyle {\frac {Joules}{(mole)(Kelvin)}}}$. This is the same dimensionality as the universal gas constant R, which means that the heat capacity of a substance can be expressed as some multiple of R.

For gases, it makes a difference whether the volume is held constant ("heat capacity at constant volume", denoted Cv) or the pressure is held constant ("heat capacity at constant pressure", denoted Cp). For liquids or solids, the change in volume is insignificant, so it makes essentially no difference.

Here are a few examples, all at reasonable temperatures and pressures:

Substance C, in ${\displaystyle {\frac {Joules}{(gram)(Kelvin)}}}$ C, in ${\displaystyle {\frac {Joules}{(mole)(Kelvin)}}}$ C, as multiple of R
Helium Cp = 5.19 Cp = 20.78 Cp = 2.5 R
Cv = 3.12 Cv = 12.47 Cv = 1.5 R
Neon Cp = 1.03 Cp = 20.78 Cp = 2.5 R
Cv = .618 Cv = 12.47 Cv = 1.5 R
Argon Cp = .52 Cp = 20.78 Cp = 2.5 R
Cv = .312 Cv = 12.47 Cv = 1.5 R
Nitrogen Cp = 1.04 Cp = 29.1 Cp = 3.5 R
Cv = .742 Cv = 20.8 Cv = 2.5 R
Oxygen Cp = .918 Cp = 29.39 Cp = 3.53 R
Cv = .659 Cv = 21.1 Cv = 2.53 R
Chlorine Cp = .478 Cp = 33.9 Cp = 4.08 R
Cv = .361 Cv = 25.6 Cv = 3.08 R
Ammonia (gas, 40° C) Cp = 2.12 Cp = 36.0 Cp = 4.33 R
Cv = 1.63 Cv = 27.7 Cv = 3.33 R
Sulfur hexafluoride Cp = 5.52 Cp = 806 Cp = 97 R
Cv = 5.47 Cv = 798 Cv = 96 R
Lithium C = 3.57 C = 24.8 C = 2.98 R
Aluminum C = .897 C = 24.2 C = 2.91 R
Copper C = .385 C = 24.47 C = 2.94 R
Iron C = .412 C = 25.09 C = 3.02 R
Sulfur C = .709 C = 22.75 C = 2.73 R
Uranium C = .116 C = 27.7 C = 3.33 R
Water (liquid, room temperature) C = 4.184 C = 75.3 C = 9.06 R
Steam (100° C) Cp = 2.02 Cp = 36.4 Cp = 4.38 R
Cv = 1.56 Cv = 28.1 Cv = 3.38 R

The fact that, for many substances, the heat capacity is an uncannily simple multiple of R is explained by the kinetic theory and statistical mechanics, and is beyond the scope of this article. However, there is a simple relationship between Cp and Cv for any gas that obeys the ideal gas law.

Since R = 8.314 Joules per mole-Kelvin, one can check that the third column in the above table is always 8.314 times the fourth column. Since the ratio of a mole to a gram is the molecular weight, can check that the third column is always the molecular (not atomic) weight times the second column. For example, the molecular weight of Chlorine (Cl2) is 70.9, so 33.9 = .478 × 70.9.

The calorie

The calorie is a unit of energy equal to 4.184 Joules. This makes the heat capacity of liquid water exactly 1 calorie per gram-Kelvin. The kilocalorie (or "large Calorie", written as a capital C) is 1000 times that, or 4184 Joules. The heat capacity of liquid water is 1 Calorie per kilogram-Kelvin. The Calorie used in nutritional measurements is the large-C Calorie.

The relationship between Cp and Cv

Because the volume of a gas can easily change, heat capacity must be defined carefully. The "heat capacity at constant volume" (Cv) is the amount of heat energy that is required to raise the temperature of a gas sample by some amount, while holding the volume constant. When heat energy is introduced, the temperature and pressure will both increase. The "heat capacity at constant pressure" (Cp) is measured while holding the pressure constant. When heat energy is introduced, the temperature and volume will both increase.

Suppose some quantity Qv of heat energy is introduced into a cylinder of gas, forcing the volume to be held constant. The temperature will increase by ΔT, and the pressure will increase. No mechanical work will be done, since the piston does not move. The ratio of heat energy per mole to the temperature rise is Cv:

${\displaystyle Q_{v}=NC_{v}\Delta {}T}$

Now suppose instead that a quantity Qp of heat energy is introduced, holding the pressure constant. That is, the piston is free to move, and has a fixed weight on it. Qp is set so that the temperature rise is the same. This time the gas will do mechanical work, in the amount of the piston travel distance times the force. The force is just the pressure times the piston area, so

${\displaystyle \Delta {}E=lF=lPA}$

But ${\displaystyle lA=\Delta {}V}$, the change in volume, so

${\displaystyle \Delta {}E=P\Delta {}V}$

Now

${\displaystyle \Delta {}(PV)=P\Delta {}V+V\Delta {}P}$

But ${\displaystyle \Delta {}P=0}$ in this instance, so

${\displaystyle \Delta {}E=P\Delta {}V=\Delta {}(PV)=NR\Delta {}T}$

But conservation of energy requires that the heat energy in the second instance be ${\displaystyle \Delta {}E}$ greater that the heat energy in the first instance, so

${\displaystyle Q_{p}-Q_{v}=\Delta {}E=NR\Delta {}T=N(C_{P}-C_{v})\Delta {}T}$

So

${\displaystyle C_{p}=C_{v}+R}$

One can see from the table above that this relationship is preserved for all (ideal) gases, even those, like Sulfur hexafluoride, that have unusually large heat capacities.