A gas law is a mathematical model that describes the behavior of a gas by relating its measurable properties: pressure (\(P\)), volume (\(V\)), temperature (\(T\)), and the quantity of gas (\(n\), measured in moles). There are three historical relationships that form the basis of gas behavior, which, when combined with a fourth law concerning quantity, lead to a single expression that governs the behavior of most gases.
The Foundational Relationships Governing Gas Behavior
These initial laws focus on the relationships between pressure, volume, and temperature, assuming the amount of gas remains constant. They describe how changing one property affects another based on the kinetic energy and collision frequency of the gas molecules.
Boyle’s Law describes the inverse relationship between a gas’s pressure and its volume when temperature and the amount of gas are held constant. As volume decreases, gas molecules collide with the container walls more frequently, resulting in a proportional increase in pressure. This inverse relationship is expressed mathematically as \(P_1V_1 = P_2V_2\). A simple example is the action of a syringe: pulling the plunger out increases the volume and decreases the pressure inside.
Charles’s Law establishes the direct relationship between volume and absolute temperature, provided pressure and the amount of gas are unchanged. As temperature increases, the kinetic energy of the molecules rises, causing them to move faster. To maintain constant pressure, the volume must expand proportionally. This relationship is represented by the equation \(\frac{V_1}{T_1} = \frac{V_2}{T_2}\). A balloon shrinking when moved from a warm room into the cold outdoors illustrates this principle.
Gay-Lussac’s Law describes the direct relationship between a gas’s pressure and its absolute temperature when volume and quantity are kept constant. If a gas is heated in a rigid container, the faster-moving molecules strike the walls with greater frequency and force, leading to a direct increase in pressure. This relationship is written as \(\frac{P_1}{T_1} = \frac{P_2}{T_2}\). This principle explains why pressure inside a car tire is higher on a hot day.
Integrating the Quantity of Gas
The three foundational relationships rely on the premise that the amount of gas remains fixed. The final variable, the quantity of gas, is incorporated through Avogadro’s Law, which introduces the mole (\(n\)) as the unit for the amount of substance.
Avogadro’s Law states that for a gas held at constant temperature and pressure, the volume is directly proportional to the number of moles (\(n\)) of gas present. Increasing the number of moles requires a proportional increase in volume to keep pressure and temperature constant. The mathematical expression for this relationship is \(\frac{V_1}{n_1} = \frac{V_2}{n_2}\). This law also shows that equal volumes of any ideal gas, at the same temperature and pressure, contain the exact same number of molecules. Blowing air into a balloon demonstrates this law, as adding more moles causes the volume to expand.
The Ultimate Expression: The Ideal Gas Law
The four individual laws concerning pressure, volume, temperature, and quantity are interconnected parts of a single, comprehensive model for gas behavior. By combining Boyle’s, Charles’s, Gay-Lussac’s, and Avogadro’s laws, scientists derived the Ideal Gas Law, which is the single equation \(PV = nRT\). This equation connects all four variables simultaneously, allowing for the calculation of any one property if the other three are known.
The letter \(R\) is the Ideal Gas Constant, which acts as a proportionality factor relating the units of pressure, volume, temperature, and moles. Its value is typically \(0.08206 \text{ L}\cdot\text{atm}/\text{mol}\cdot\text{K}\) or \(8.314 \text{ J}/\text{mol}\cdot\text{K}\). The law is named for an “ideal gas,” a theoretical model assuming particles have negligible volume and no intermolecular forces. Real gases closely approximate this ideal behavior under most common conditions. However, the model deviates under extreme conditions, specifically very high pressures or very low temperatures. At high pressures, molecular volume becomes significant, and at low temperatures, attractive forces become noticeable, making \(PV=nRT\) less accurate.