Gas laws

A gas can be described by its pressure $$p$$, volume $$V$$, absolute temperature $$T$$ and amount. Gas laws describe how these quantities are related when particular conditions are kept constant.

Temperature must be measured on the absolute temperature scale, in kelvin ($$\mathrm{K}$$). A change in kelvin has the same size as a change in degrees Celsius, but $$0,\mathrm{K}$$ is absolute zero. Pressure in the gas laws is absolute pressure.

Pressure and volume at constant temperature

For a fixed amount of gas at constant temperature, pressure is inversely proportional to volume. If the volume is reduced, the pressure increases; if the volume is increased, the pressure decreases. Their product remains constant:

For example, doubling the volume halves the pressure, provided the temperature and amount of gas remain unchanged.

Volume and temperature at constant pressure

For a fixed amount of gas at constant pressure, volume is directly proportional to absolute temperature. As the temperature in kelvin increases, the volume increases in the same proportion:

For example, doubling the absolute temperature doubles the volume, provided the pressure and amount of gas remain unchanged. This proportionality does not apply to temperature measured in degrees Celsius.

Pressure and temperature at constant volume

For a fixed amount of gas at constant volume, pressure is directly proportional to absolute temperature. As the temperature in kelvin increases, the pressure increases in the same proportion:

For example, doubling the absolute temperature doubles the pressure, provided the volume and amount of gas remain unchanged.

Gas volume and mass

For a given gas at fixed pressure and temperature, volume is directly proportional to mass. For example, twice the mass of that gas occupies twice the volume under the same pressure and temperature.

Absolute zero of temperature

Absolute zero is the zero of the thermodynamic temperature scale. It is $$0,\mathrm{K}$$, equivalent to $$-273.15,^{\circ}\mathrm{C}$$.

A temperature change of one kelvin is the same size as a change of one degree Celsius. The scales have different zero points, so a temperature on the Celsius scale, $$\theta$$, is converted to a thermodynamic temperature, $$T$$, using:

For example, $$0,^{\circ}\mathrm{C}$$ corresponds to $$273.15,\mathrm{K}$$.

Estimating absolute zero from gases

For a fixed amount of gas at constant pressure, volume changes approximately linearly with Celsius temperature. If the straight-line trend is extended beyond the measured temperatures, it reaches zero volume at approximately $$-273.15,^{\circ}\mathrm{C}$$.

Similarly, for a fixed amount of gas at constant volume, pressure changes approximately linearly with Celsius temperature. Extending this trend gives zero pressure at approximately the same temperature.

These are extrapolations: they extend a measured trend beyond the range of observations. A real gas can change state before reaching the extrapolated point. The extrapolations therefore do not mean that an actual gas remains gaseous down to absolute zero or reaches zero volume there.

Ideal gas equation

The pressure, volume and temperature of an ideal gas are related to the amount of gas present. When the amount is measured in moles, the relationship is:

Here $$p$$ is pressure, $$V$$ is volume, $$n$$ is the amount of gas in moles, $$T$$ is absolute temperature, and $$R$$ is the molar gas constant.

If the amount is expressed as the number of molecules instead, the same relationship is:

In this form, $$N$$ is the number of molecules and $$k$$ is the Boltzmann constant. The pressure, volume and temperature have the same meanings in both forms.

Connecting the two forms

The Avogadro constant $$N_A$$ is the number of constituent particles per mole. Thus, for a gas of molecules, the number of molecules in $$n$$ moles is $$N=nN_A$$. The constants are related by $$R=N_Ak$$.

Substituting $$R=N_Ak$$ into $$pV=nRT$$ gives $$pV=nN_AkT$$. Since $$nN_A=N$$, this becomes $$pV=NkT$$. The two equations therefore describe the same gas state; they differ only in how the amount of gas is expressed.

SI units

Use pressure in pascals ($$\mathrm{Pa}$$), volume in cubic metres ($$\mathrm{m^3}$$), temperature in kelvin ($$\mathrm{K}$$), and amount in moles ($$\mathrm{mol}$$). Temperature must be absolute temperature in kelvin, not a value in degrees Celsius.

In SI units, $$R\approx 8.31\ \mathrm{J,mol^{-1},K^{-1}}$$ and $$k\approx 1.38\times10^{-23}\ \mathrm{J,K^{-1}}$$. The Avogadro constant is $$N_A\approx 6.02\times10^{23}\ \mathrm{mol^{-1}}$$.

Pressure–volume work

A gas does work when it expands and pushes against a surrounding pressure. If the pressure opposing the change in volume is constant, the work done by the gas is:

Here $$p$$ is the constant opposing pressure and $$\Delta V = V_{\text{final}} - V_{\text{initial}}$$ is the change in the gas’s volume.

Why pressure multiplied by volume change gives work

Imagine a gas pushing a piston of area $$A$$. Pressure is force per unit area, so the opposing force is $$F = pA$$. If the piston moves outward through a distance $$d$$, the gas does work equal to $$Fd$$. Its volume increases by $$Ad$$, so the work is $$pAd = p\Delta V$$.

Signs and units

• Expansion: $$\Delta V$$ is positive, so the work done by the gas is positive.
• Compression: $$\Delta V$$ is negative, so the work done by the gas is negative.

The work done on the gas has the opposite sign: it is positive during compression and negative during expansion.

If pressure is measured in pascals ($$\mathrm{Pa}$$) and volume change in cubic metres ($$\mathrm{m^3}$$), the work is in joules ($$\mathrm{J}$$).

Molar and molecular mass

A mole is an amount of substance containing $$N_A$$ molecules. The Avogadro constant is $$N_A = 6.022 \times 10^{23},\mathrm{mol^{-1}}$$.

Molar mass, $$M$$, is the mass of one mole of a substance. If a sample has mass $$m$$, its amount of substance $$n$$ is:

Molar mass can be expressed in $$\mathrm{kg,mol^{-1}}$$ or $$\mathrm{g,mol^{-1}}$$. The mass units must be consistent: use kilograms for $$m$$ if $$M$$ is in $$\mathrm{kg,mol^{-1}}$$, or grams for $$m$$ if $$M$$ is in $$\mathrm{g,mol^{-1}}$$. The resulting amount $$n$$ is in moles.

Mass of one molecule

Molecular mass is the mass of one molecule. One mole contains $$N_A$$ molecules, so dividing the mass of one mole by the number of molecules in it gives:

If $$M$$ is in $$\mathrm{kg,mol^{-1}}$$, the molecular mass is in kilograms per molecule. If $$M$$ is in $$\mathrm{g,mol^{-1}}$$, it is in grams per molecule.

Number of molecules in a sample

Each mole contains $$N_A$$ molecules, so a sample containing $$n$$ moles has a total number of molecules $$N$$ given by:

For example, water has a molar mass of approximately $$18.0,\mathrm{g,mol^{-1}}$$. A $$9.00,\mathrm{g}$$ sample therefore contains $$n = 9.00/18.0 = 0.500,\mathrm{mol}$$. Its number of molecules is $$N = 0.500 \times 6.022 \times 10^{23} = 3.01 \times 10^{23}$$.

The mass of one water molecule is $$18.0/(6.022 \times 10^{23}) = 2.99 \times 10^{-23},\mathrm{g}$$.