Gay-Lussac’s Law, sometimes called the Pressure-Temperature Law, is a fundamental principle used in chemistry and physics. It establishes a clear relationship between the pressure a gas exerts and its temperature. Understanding this relationship is foundational for solving practical problems involving sealed containers and temperature changes, such as aerosol cans or pressure cookers.
The Direct Relationship Between Pressure and Temperature
Gay-Lussac’s Law states that the pressure of a fixed amount of gas is directly proportional to its absolute temperature. If the temperature increases, the pressure increases proportionally, and vice versa. This relationship holds true only if the volume and the amount of gas remain unchanged.
The physical explanation lies in the kinetic energy of the gas molecules. When the temperature is raised, particles absorb thermal energy and move faster and with greater force. These energized molecules collide with the container walls more frequently and forcefully. Since pressure is a measure of the force of these collisions, increased kinetic energy results in a higher measured pressure. Conversely, cooling the gas causes the molecules to slow down, reducing the intensity of impacts and lowering the pressure.
Applying the Mathematical Formula
To solve problems using this gas law, a specific mathematical relationship connects the initial conditions of the gas to its final conditions after a temperature change. The formula \(\frac{P_1}{T_1} = \frac{P_2}{T_2}\) expresses the direct proportionality, where the ratio of pressure to temperature remains constant. \(P_1\) and \(T_1\) represent the initial pressure and temperature, while \(P_2\) and \(T_2\) represent the final pressure and temperature.
A necessary requirement when using this formula is the use of the absolute temperature scale (Kelvin). Temperature must be converted to Kelvin before any calculation, as the relationship does not hold true for Celsius or Fahrenheit scales. To convert from Celsius to Kelvin, use the formula \(T_K = T_{°C} + 273.15\). Pressure terms (\(P_1\) and \(P_2\)) must be expressed using the same units, such as atmospheres (atm) or kilopascals (kPa).
Solving an Example Problem
Consider a scenario where a sealed metal container of gas starts at a pressure of 3.00 atm and a temperature of \(25.0 \text{°C}\). If the container is heated until the temperature reaches \(125.0 \text{°C}\), the final pressure (\(P_2\)) can be determined. The first step involves identifying the known variables: \(P_1 = 3.00 \text{ atm}\), \(T_1 = 25.0 \text{°C}\), and \(T_2 = 125.0 \text{°C}\).
The second step is to convert both initial and final temperatures from Celsius to Kelvin. The initial temperature converts to \(T_1 = 25.0 + 273.15 = 298.15 \text{ K}\), and the final temperature becomes \(T_2 = 125.0 + 273.15 = 398.15 \text{ K}\). Next, the mathematical formula \(\frac{P_1}{T_1} = \frac{P_2}{T_2}\) must be algebraically rearranged to isolate the unknown variable, \(P_2\).
To solve for the final pressure, the equation is multiplied by \(T_2\) on both sides, resulting in the working equation \(P_2 = \frac{P_1 \cdot T_2}{T_1}\). Substitute the numerical values into this rearranged formula: \(P_2 = \frac{(3.00 \text{ atm}) \cdot (398.15 \text{ K})}{298.15 \text{ K}}\). The Kelvin units cancel out, leaving the pressure unit of atmospheres. Performing the calculation yields a final pressure of approximately \(P_2 = 4.01 \text{ atm}\). This result makes sense conceptually because the temperature increased, so the pressure had to increase proportionally.