Graphical Representations of the Gas Laws (College Board AP Chemistry)

• The relationship between any two of the variables which describe the properties of an ideal gas can be established by keeping the other variables constant
  • These variables are:
    • Pressure
    • Volume
    • Temperature
    • Amount of gas (moles)
  • These relationships expressed mathematically and graphically are known as Gas Laws

Pressure-Volume Relationship: Boyle’s Law

• This relationship was first investigated by the British chemist, Robert Boyle, using a J-tube, showing that the volume of gas decreased as the pressure increased
• This observation is summarized by the statement by Boyle’s Law which states that: the volume of a fixed amount of gas at constant temperature is inversely proportional to the pressure:

V ∝ P ; PV = constant

• Simplifying the mathematical equation gives Boyle’s equation:

P₁V₁ = P₂V₂

Pressure-Volume Relationship

Boyle’s experiment uses a J-tube to demonstrate the volume-pressure relationship of a gas. Gas volume decreases on increasing the pressure in the tube

• This relationship can also be visualized using two graphs:
  • A plot of V against P gives a curve for a given quantity of gas at a fixed temperature
  • A straight line is obtained when V is plotted against 1/P or P against 1/V

Graphical Representation of Pressure-Volume Relationship

Graphical representations of Boyle’s Law

Temperature-Volume Relationship: Charles’s Law

• The earliest investigator of the effect of temperature on volume was French scientist, Jacques Charles
• He observed that at constant pressure, the volume of a gas sample expands when heated and contracts when cooled
• At any given pressure, the plot of volume versus temperature yields a straight line
  • By extrapolating the line to zero volume, we find the intercept on the temperature axis to be -273.15 ℃

Temperature-Volume Relationship on Celsius Scale

A graph showing the linear relationship between temperature (in degrees Celsius) and volume. The line intercepts the temperature axis at a value of 273.15 ℃

• This implies that at a temperature of 273.15 ℃, the volume of a substance, if it remains gaseous will be zero
  • However, this could not happen, given that gases liquefy before this temperature, and the relationship does not apply to liquids

Kelvin Temperature Scale

• In 1848, British physicist William Thomson, whose title was Lord Kelvin, realized the significance of the volume-temperature relationship using the Celsius temperature scale
• He identified the lowest possible temperature as absolute zero
  • He then set up an absolute temperature scale, now called the Kelvin temperature scale (K) with absolute zero as the starting point
  • On the absolute temperature scale, 0 K is equivalent to -273.15 ℃

• In terms of the Kelvin scale, the temperature-volume relationship known as Charles’s Law can be stated as: the volume of a fixed amount of gas maintained at constant pressure is directly proportional to its absolute temperature:

V ∝ T or V/T = constant

• Simplifying the relationship gives:

V₁/T₁ = V₂/T₂

• A plot of volume against temperature in Kelvin also gives a straight line which intercepts the volume and temperature axis at zero

Temperature-Volume Relationship on the Kelvin Scale

A graph showing the linear relationship between absolute temperature and volume. The line intercepts the temperature axis at a value of 0 K

Quantity-Volume Relationship: Avogadro’s Law

• The relationship between the amount in moles of a gas and its volume follows from the work of Joseph Louis Gay-Lussac and Amedeo Avogadro
  • Gay-Lussac observed that at a given pressure and temperature, the volumes of gases that react with one another are in the ratios of small whole numbers
  • For example, two volumes of hydrogen gas react with one volume of oxygen gas to form two volumes of water vapor:

2H₂ (g) + O₂ (g) → 2H₂O (g)

• A few years later, Amedeo Avogadro interpreted Gay-Lussac’s observation by proposing an important hypothesis which set the foundation for demonstrating the mole-volume relationship and Avogadro’s Law
  • The hypothesis states that: Equal volumes of gases at the same temperature and pressure contain equal numbers of molecules
  • For example, at 0 ℃ and 1 atm, 22.4 L of a gas contains 6.022 × 10²³ gas molecule which is equivalent to one mole
  • Avogadro’s law follows from this hypothesis: the volume of a gas maintained at constant temperature and pressure is directly proportional to the number of moles of the gas:

V ∝ n ; V/n = constant

• Simplifying the relationship gives:

V₁/n₁ = V₂/n₂

Quantity-Volume Relationship

Graphical representation of Avogadro’s Law