You have a sample of gas at \(-33^{\circ} \mathrm{C}\) . You wish to increase the rms speed by a factor of \(2 .\) To what temperature should the gas be heated?

Understanding gas temperature conversion is essential when working with gas laws and their associated equations. The Kelvin scale is the standard unit of measure in these cases because it starts at absolute zero, the point where theoretically no more thermal energy can be extracted from a substance.

To convert from Celsius to Kelvin, which is a typical task in gas-related problems, you simply add 273.15 to the Celsius temperature. This adjustment allows for temperature to be accurately reflected in equations that are based on absolute temperature scales. It's crucial because the behavior of gases relates directly to their kinetic energy, which is a function of absolute temperature.

For instance, in the solution provided, the initial gas temperature of (-33 degrees Celsius) was converted to Kelvin ((240.15 K) by adding 273.15. When dealing with gas properties and dynamic reactions, converting to the Kelvin scale ensures that all temperature inputs reflect the absolute thermal energy present in the system, allowing for proper application of the kinetic molecular theory.

Root-mean-square (RMS) velocity is a way to express the average velocity of gas molecules in a sample. It gives us insight into the speed of gas particles and how they change with temperature. The RMS velocity is derived from the Kinetic Molecular Theory and is directly proportional to the square root of the absolute temperature. Mathematically, the RMS speed of a particle in a gas is given by the formula: \(v_{rms} = \sqrt{\frac{3kT}{m}}\).

Where \(v_{rms}\) is the root-mean-square velocity, 'T' is the absolute temperature in Kelvins, 'm' is the mass of a particle, and 'k' is the Boltzmann constant. This formula tells us that if you want to increase the RMS speed of gas particles, as in the exercise given, you need to increase the absolute temperature of the gas. Doubling the RMS speed requires quadrupling the absolute temperature, because of the square root relationship. This is why, when increasing the initial temperature of our gas sample by a factor of four, we achieve the desired increase in RMS speed.

The Kinetic Molecular Theory of gases provides a conceptual framework for understanding the properties of gases and how they react to changes in the environment. This theory posits that gases are composed of a large number of particles, which are in constant, random motion. The assumptions of the theory include that these particles are point masses with no volume and do not interact with each other except during elastic collisions.

The temperature of a gas is a measure of the average kinetic energy of its particles. This means that as the temperature of a gas increases, the average speed of the particles also increases. This theory lays the foundation for why temperature plays a key role in gas dynamics. For example, as highlighted in the solution of the exercise, to double the RMS speed of the gas, which is linked to the kinetic energy of its particles, we must significantly increase the temperature. The exercise illustrates that understanding the kinetic molecular theory is integral in determining the behavior of gases under various conditions.