When the temperature of a gas is lowered while maintaining a constant volume, the pressure of the gas decreases. This occurs due to a decrease in the kinetic energy of the gas particles. As the temperature decreases, the particles move slower, reducing the frequency and force of their collisions with the container's walls. According to the kinetic molecular theory, the pressure exerted by a gas is directly proportional to the average kinetic energy of its particles. Thus, with a lower temperature, the average kinetic energy decreases, leading to a reduction in pressure. This relationship is quantitatively described by Charles' Law, which states that the volume of a gas is directly proportional to its temperature at constant pressure. However, in this scenario, since the volume is constant, a decrease in temperature directly translates to a decrease in pressure. This phenomenon is evident in real-life situations such as deflated tyres in cold weather, where the air inside the tyre contracts, lowering its pressure.

The kinetic theory of gases provides a comprehensive explanation for the increase in pressure when a gas is heated in a fixed volume. According to this theory, gas particles are in constant, random motion and collide with each other and the walls of their container. These collisions are what cause gas pressure. When the gas is heated, the energy imparted to the gas increases the kinetic energy of its particles. As a result, the particles move faster, leading to more frequent and more forceful collisions with the container walls. The increased rate and intensity of collisions directly translate to an increase in pressure. This is because pressure is defined as the force exerted per unit area, and with more vigorous collisions, the force per collision increases. Furthermore, since the volume is fixed, the particles have the same amount of space to move in, so the increased energy cannot be dispersed by expanding the gas, leading instead to an increase in pressure. This principle is key in many applications, such as in internal combustion engines where fuel-air mixtures are compressed and heated, significantly increasing the pressure to produce mechanical work.

The behavior of gases at extremely high pressures cannot be accurately predicted using the ideal gas law. The ideal gas law, PV = nRT, assumes that gas particles have no volume and no intermolecular forces. However, these assumptions break down at extremely high pressures. At such pressures, the volume of the gas particles themselves becomes significant compared to the total volume of the gas. Additionally, intermolecular forces, which are typically negligible under standard conditions, become increasingly significant as particles are forced closer together. These forces can either be attractive or repulsive, affecting the behavior of the gas. For example, attractive forces between particles can cause the gas to exert less pressure than predicted by the ideal gas law, leading to deviations. Therefore, under conditions of high pressure (and also at very low temperatures), real gas behavior needs to be considered, and more complex equations of state, like the Van der Waals equation, are used to describe the gas behavior more accurately.

A gas cools down when it expands in a vacuum due to the principle of adiabatic expansion. In an adiabatic process, no heat is exchanged with the surroundings. When a gas expands in a vacuum, it does work on its surroundings as it increases in volume. However, since there is no external source of heat (as it's a vacuum), the energy needed for this expansion comes from the internal energy of the gas itself. As the gas expands, the kinetic energy of its particles decreases because part of their energy is used in doing the work of expansion. Since temperature is a measure of the average kinetic energy of the particles in a substance, a decrease in kinetic energy leads to a decrease in temperature. This phenomenon is a key principle in refrigeration systems, where the rapid expansion of refrigerant gases results in cooling, which is then used to lower the temperature inside the refrigerator.

Altitude significantly affects both the pressure and temperature of a gas. As altitude increases, the atmospheric pressure decreases. This is because atmospheric pressure is caused by the weight of the air above the measurement point. At higher altitudes, there is less air above a given point, resulting in lower pressure. The temperature of the gas also typically decreases with altitude, although this relationship can be more complex due to atmospheric conditions. In the lower atmosphere (troposphere), temperature generally decreases with increasing altitude. This decrease is due to the reduction in air pressure at higher altitudes, which causes the air to expand and cool (adiabatic cooling). However, in higher layers of the atmosphere, other factors come into play that can cause temperature variations. In practical terms, this relationship between altitude, pressure, and temperature is crucial for various applications, such as aviation and meteorology. For instance, aircraft cabins must be pressurized for passenger comfort and safety due to the low atmospheric pressure at cruising altitudes.