Gay-Lussac's law Knowledge-Based Study

Study notes

1. Gay-Lussac's law, also known as the pressure-temperature law, describes the relationship between the pressure and temperature of an ideal gas when its volume and chemical amount are held constant.

   Consider a gas sealed inside a rigid container. Because the walls of the vessel cannot move, the volume $V$ remains fixed. Furthermore, because the container is airtight, the amount of gas $n$ cannot change. When heat is added to such a system, the gas particles gain kinetic energy, causing them to collide more vigorously and frequently with the container walls, which leads to an increase in pressure.

   Gay-Lussac's law states that at constant volume and constant amount of substance, the pressure $P$ of a gas is directly proportional to its absolute temperature $T$:

   $$P \propto T \quad \text{or} \quad \frac{P}{T} = k$$

   where $k$ is a constant determined by the volume and amount of the gas. For comparing the same gas sample under two different sets of conditions at constant volume, the relationship is expressed as:

   $$\frac{P_1}{T_1} = \frac{P_2}{T_2}$$

   Crucial Requirement: Temperature must always be expressed on the absolute scale (Kelvin). Calculations using degrees Celsius will yield incorrect values because the Celsius scale does not have an absolute zero reference.

   The direct proportionality predicted by Gay-Lussac's law can be represented graphically in two common ways:

   • Pressure vs. Absolute Temperature ($P$ vs. $T$ in K): This yields a straight line that passes directly through the origin $(0, 0)$. An isochoric (constant volume) process is represented by this linear curve.
   • Pressure vs. Celsius Temperature ($P$ vs. $t$ in °C): This plot also gives a straight line, but it intersects the temperature axis at $-273.15\text{ °C}$ rather than the origin. Extrapolating the pressure of an ideal gas to zero yields absolute zero.

   Comparing isochoric lines for different volumes shows that a smaller volume produces a steeper slope, because a given temperature change produces greater pressure in a more confined space.

   The microscopic behavior of gas particles explains why pressure increases with temperature at constant volume:

   • Average Kinetic Energy: Absolute temperature is directly proportional to the average kinetic energy of gas molecules ($E_k = \frac{3}{2}kT$). Raising the temperature increases the root-mean-square speed of the particles.
   • Collision Frequency: Because particles move faster across the same fixed volume, they hit the walls more times per second.
   • Collision Force: Each collision imparts greater momentum to the wall.

   Because gas pressure is defined as the force exerted per unit area, both the increased rate and greater force of wall collisions combine to raise the pressure proportionally.

   Example Problem: A rigid steel cylinder contains neon gas at a pressure of $2.40\text{ atm}$ and a temperature of $27.0\text{ °C}$. If the cylinder is heated to $127.0\text{ °C}$, what is the final pressure inside the cylinder?

   Step 1: Convert temperatures from degrees Celsius to Kelvin:

   • $T_1 = 27.0 + 273.15 = 300.15\text{ K}$
   • $T_2 = 127.0 + 273.15 = 400.15\text{ K}$

   Step 2: Apply Gay-Lussac's law:

   $$\frac{P_1}{T_1} = \frac{P_2}{T_2} \implies P_2 = P_1 \times \frac{T_2}{T_1}$$

   Step 3: Solve for $P_2$:

   $$P_2 = 2.40\text{ atm} \times \frac{400.15\text{ K}}{300.15\text{ K}} \approx 3.20\text{ atm}$$

   Common Mistake: Using Celsius values directly ($\frac{2.40 \times 127}{27} \approx 11.3\text{ atm}$) produces a major error because $127\text{ °C}$ is not roughly four times hotter than $27\text{ °C}$ in terms of thermal energy.