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The root mean square speed (\( v_{rms} \)) is an important concept in the kinetic theory of gases. It provides a way to determine the average speed of gas particles at a given temperature. The formula used to calculate this speed is \( v_{rms} = \sqrt{\frac{3RT}{M}} \), where \( R \) represents the ideal gas constant, \( T \) is the temperature in Kelvin, and \( M \) is the molar mass of the gas.Understanding the root mean square speed is crucial, as it helps in comparing different gases at various temperatures. Since the rms speed depends on the molar mass and temperature, gases with lower molar masses and at higher temperatures move faster. This can be used to arrange or compare the speeds of different gases under particular conditions with ease.

Molar mass, denoted as \( M \), is the mass of one mole of a substance, typically expressed in grams per mole (g/mol). It plays a vital role in determining the root mean square speed. To find the molar mass, one can refer to the periodic table, where the atomic or molecular masses are listed.For gases, lighter molecules, meaning those with lower molar masses, will generally have higher rms speeds if other conditions like temperature are constant. In the exercise, helium has the lowest molar mass, making it the fastest among the given gases. Converting the molar mass from grams per mole to kilograms per mole is necessary for calculations involving the ideal gas constant, thus making the results coherent and accurate.

The ideal gas constant (\( R \)) is a key part of the ideal gas equation and is used in calculating the root mean square speed. This constant has a value of 8.314 J/mol·K. It relates the energy scale to the temperature scale, allowing for various calculations involving gas properties. In the calculation of rms speed, \( R \) helps connect temperature and molar mass to derive the speed. Having a clear grasp of this constant and its role in equations aids in understanding a variety of thermodynamic and kinetic expressions concerning gases.

Temperature conversion is an essential step when dealing with gas calculations. Most scientific calculations, including those involving the root mean square speed, require temperatures to be in Kelvin. The Kelvin scale is an absolute temperature scale starting from absolute zero, making it suitable for use in physical equations. To convert Celsius to Kelvin, you simply add 273.15 to the Celsius temperature. For example, converting 100 °C to Kelvin would result in 373.15 K. This step is crucial because it ensures all variables are in the correct units, maintaining consistency and accuracy in further calculations. Understanding this conversion ensures that you can solve problems correctly and efficiently.