Imagine filling a sealed container attached to a pressure gauge with gas (Figure 1) so that both the volume and number of moles of gas remain constant. If the container is cooled, the gas inside likewise gets colder and its pressure is observed to decrease.  If we heat the sphere, the gas inside gets hotter and the pressure increases.

We find that temperature and pressure are linearly related, and if the temperature is on the kelvin scale, then [latex]P[/latex] and [latex]T[/latex] are directly proportional (again, when volume and moles of gas are held constant); if the temperature on the kelvin scale increases by a certain factor, the gas pressure increases by the same factor.

where [latex]k[/latex] is a proportionality constant that depends on the identity, amount, and volume of the gas.

If we fill a balloon with air and seal it, the balloon contains a specific amount of air at constant pressure. If we put the balloon in a refrigerator, the gas inside gets cold and the balloon shrinks (although both the amount of gas and its pressure remain constant). If we make the balloon very cold, it will shrink a great deal, and it expands again when it warms up.  The relationship between the volume and temperature of a given amount of gas at constant pressure is known as Charles’s law, which states that the volume of a given amount of gas is directly proportional to its temperature on the kelvin scale when the pressure is held constant.

Volume and Pressure: Boyle’s Law

If we partially fill an airtight syringe with air, the syringe contains a specific amount of air at constant temperature. If we slowly push in the plunger while keeping temperature constant, the gas in the syringe is compressed into a smaller volume and its pressure increases; if we pull out the plunger, the volume increases and the pressure decreases.  Unlike the [latex]P-T[/latex] and [latex]V-T[/latex] relationships, pressure and volume are not directly proportional to each other. Instead, [latex]P[/latex] and [latex]V[/latex] exhibit inverse proportionality: increasing the pressure results in a decrease of the volume of the gas. Mathematically this can be written:

[latex]P=k\times \frac{1}{V}[/latex] or [latex]P\times V=k[/latex]

with [latex]k[/latex] being a constant.

The Italian scientist Amedeo Avogadro determined that equal volumes of all gases, measured under the same conditions of temperature and pressure, contain the same number of molecules. This is now known as Avogadro’s law: For a confined gas, the volume ([latex]V[/latex]) and number of moles ([latex]n[/latex]) are directly proportional if the pressure and temperature both remain constant.

In equation form, this is written as:

[latex]V=k\times n[/latex].

The Ideal Gas Law

• Boyle’s law: [latex]PV=k[/latex] at constant [latex]T[/latex] and [latex]n[/latex];
• Amontons' law: [latex]{P}{T}=k[/latex] at constant [latex]V[/latex] and [latex]n[/latex];
• Charles's law: [latex]{V}{T}=k[/latex] at constant [latex]P[/latex] and [latex]n[/latex];
• Avogadro's law: [latex]{V}{n}=k[/latex] at constant [latex]P[/latex] and [latex]T[/latex].

Combining these four laws, we see that pressure varies jointly with amount and temperature, and inversely with volume. Combining these four laws yields the ideal gas law, a relation between the pressure, volume, temperature, and number of moles of a gas:

[latex]PV=nRT[/latex]

where [latex]P[/latex] is the pressure of a gas, [latex]V[/latex] is its volume, [latex]n[/latex] is the number of moles of the gas, [latex]T[/latex] is its temperature on the kelvin scale, and [latex]R[/latex] is a constant called the ideal gas constant or the universal gas constant. The units used to express pressure, volume, and temperature will determine the proper form of the gas constant. One commonly encountered value is 0.08206 L atm mol^–1 K^–1.