The ideal gas constant (R) is a universal constant, meaning it has the same value for all gases. This constancy arises from the fact that R is derived from two fundamental constants: Avogadro's number and Boltzmann's constant. Avogadro's number (approximately $( 6.022 \times 10^{23} ) mol(^{-1})$) is the number of particles in a mole of substance, and Boltzmann's constant (approximately ($ 1.38 \times 10^{-23} $) J/K) relates the average kinetic energy of particles in a gas to the temperature of the gas. The ideal gas constant is essentially the product of these two constants and thus incorporates the fundamental particle-based nature of matter. Since these constants are inherent properties of matter and do not vary from one gas to another, R remains constant across all gases. This universality makes the ideal gas law broadly applicable and simplifies calculations involving different gases, as the same value of R can be used irrespective of the gas type.

The ideal gas law finds relevance in various everyday applications. For instance, it explains how car tyres respond to temperature changes. As the temperature increases, the air inside the tyre expands, increasing the pressure, which can affect tyre performance and safety. Similarly, in colder weather, the pressure decreases, potentially leading to under-inflation. This principle is also applied in hot air balloons. The balloon rises because the heated air inside is less dense than the cooler air outside. By heating the air, the volume increases, or the pressure decreases, making the balloon buoyant. Another example is the use of aerosol cans, such as deodorants or spray paints. The contents are under high pressure, and when released, they rapidly expand to form a fine spray. This rapid expansion and cooling are principles explained by the ideal gas law. In the kitchen, pressure cookers utilise the ideal gas law; they cook food faster by increasing the internal pressure, which raises the boiling point of water, cooking food at a higher temperature. These examples illustrate how the ideal gas law is not just a theoretical concept but a practical tool that explains and enables various everyday phenomena.

The ideal gas law is an approximation and works best under conditions where gases behave ideally - typically at low pressure and high temperature. Under these conditions, the assumptions of the ideal gas law (negligible molecular volume and no intermolecular forces) are nearly true. However, at high pressures and low temperatures, gases deviate significantly from ideal behaviour. The molecules are closer together, so the volume occupied by the molecules themselves becomes significant compared to the overall volume of the gas. Additionally, intermolecular forces become more pronounced at these conditions. To account for these deviations, real gas equations, such as the van der Waals equation, are used. These equations introduce correction factors for volume and pressure, providing a more accurate representation of the behaviour of real gases under non-ideal conditions. Therefore, while the ideal gas law is a valuable tool in many scenarios, its applicability is limited and does not extend to all gases under all conditions.

When the temperature of an ideal gas is altered while keeping the pressure and volume constant, the number of moles of the gas (n) changes. This is because temperature is a measure of the average kinetic energy of the gas molecules. An increase in temperature results in an increase in the kinetic energy of the molecules. According to the ideal gas law (( pV = nRT )), if pressure (p) and volume (V) are constant, any increase in temperature (T) must lead to an increase in the number of moles (n), assuming the gas constant (R) remains constant. This can be interpreted as the gas particles moving more vigorously, leading to more frequent and energetic collisions. Conversely, a decrease in temperature would lead to a decrease in the kinetic energy of the molecules, and thus a decrease in the number of moles of the gas. This relationship is crucial in understanding how temperature affects gas behaviour, especially in processes like cooling and heating in industrial and laboratory settings.

The ideal gas law is instrumental in understanding atmospheric pressure, which is the pressure exerted by the weight of the Earth's atmosphere. Atmospheric pressure is a result of the collisions of air molecules with surfaces. According to the ideal gas law (( pV = nRT )), the pressure exerted by a gas is directly proportional to its temperature and

the number of gas molecules. Since the Earth's atmosphere can be thought of as a mixture of gases in a vast container (the atmosphere itself), the ideal gas law helps in calculating the pressure at different altitudes. As altitude increases, the number of air molecules (n) decreases, leading to a decrease in pressure. This explains why atmospheric pressure is lower at higher altitudes. Additionally, variations in temperature also affect atmospheric pressure; warmer air expands and becomes less dense, leading to lower pressure. The ideal gas law, therefore, provides a fundamental framework for meteorologists and scientists to understand and predict weather patterns, as well as for pilots and mountaineers to prepare for changes in atmospheric conditions with altitude.