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The kinetic molecular theory of gases is a fundamental concept used to explain the behavior of gases. According to this theory, a gas is composed of a large number of small particles (molecules) that are in constant random motion. These gas molecules collide with each other and the walls of the container, and these collisions are assumed to be elastic, meaning kinetic energy is conserved.

The theory provides a microscopic explanation of pressure and temperature. Pressure is understood as the force exerted by gas molecules when they collide with the walls of the container, while temperature is a measure of the average kinetic energy of the gas molecules. According to this theory, an increase in temperature will result in an increase in the average kinetic energy and consequently the speed of the gas molecules. Thus, understanding the kinetic molecular theory helps in comprehending how temperature affects the root-mean-square speed of gas molecules.

Gas laws are mathematical equations that describe the behavior of gases and how they respond to changes in temperature, pressure, volume, and quantity. The most fundamental laws consist of Boyle's Law, Charles's Law, and Avogadro's Law, which, when combined, give the ideal gas law: \( PV=nRT \).

In this equation, \( P \) stands for pressure, \( V \) for volume, \( n \) for the number of moles of gas, \( R \) is the ideal gas constant, and \( T \) represents temperature. Understanding these gas laws is essential in analyzing the root-mean-square speed because the laws relate temperature (\( T \)) and the amount of substance (\( n \) - which implies molar mass when dealing with a single gas) to the behavior of gas molecules.

Molar mass is the mass of one mole of a substance and is measured in grams per mole (g/mol). It is a physical property that is characteristic of each unique substance. In the context of gases, molar mass plays a critical role in determining the behavior of the gas molecules when subjected to temperature changes or when comparing different gases.

As mentioned in the exercise solution, the root-mean-square speed of gas molecules is inversely proportional to the square root of their molar mass at a constant temperature. This is because heavier molecules (with higher molar mass) have more inertia and thus, move slower, whereas lighter molecules (with lower molar mass) move faster. The root-mean-square speed offers a way to compare the average speed of molecules in different gases, taking into account their varying molar masses.

Gas temperature is a measure of the average kinetic energy of the molecules in a gas sample. When the gas temperature increases, the average kinetic energy of the gas molecules also increases, leading to faster movement and higher collision rates.

The exercise solution shows that root-mean-square speed is directly proportional to the square root of the gas temperature. This implies that if the temperature of the gas is doubled (assuming constant molar mass), the square root of the temperature is increased by a factor of the square root of two (\( \sqrt{2} \) ), resulting in an increase in the root-mean-square speed by the same factor. Conversely, a decrease in temperature will lead to slower molecular speeds. Thus, gas temperature is directly linked to the kinetic energy and velocity of gas molecules.