Gay-Lussac's law states that the pressure of a fixed mass of gas is directly proportional to its temperature when volume is held constant. The document provides an example of using this law to calculate the new pressure of oxygen in a container if the temperature is increased from 40°C to 100°C, given the original pressure is 21.5 atmospheres. It outlines the steps of first converting temperatures to Kelvin and then multiplying the original pressure by the ratio of the new and original Kelvin temperatures to determine the increased pressure.