A) At what temperature do oxygen molecules have the same...

a) At what temperature do oxygen molecules have the same average speed as helium atoms $(M=4.00 \mathrm{g} / \mathrm{mol})$ have at $300 \mathrm{K}$ ? b) What is the answer to the same question about most probable speeds? $\mathrm{c}$ ) What is the answer to the same question about rms speeds?

Key Concepts

Temperature Dependence of Molecular Speeds

In the kinetic theory of gases, the speeds of particles are found to be dependent on the temperature and molecular mass. All key speed measures—average, most probable, and RMS speeds—scale as the square root of the temperature over mass ratio. An increase in temperature leads to higher particle speeds, while a higher molecular mass results in lower speeds. This interdependence is critical when comparing gases of different masses at various temperatures.

Most Probable Speed

The most probable speed is the speed at which the Maxwell-Boltzmann distribution attains its maximum value. It represents the speed that the largest number of particles in the gas have. This measure is also proportional to the square root of the temperature divided by the molecular mass, but its derivation stems from identifying the peak of the distribution function, offering a different perspective from the average speed.

RMS Speed

The root mean square (RMS) speed is defined as the square root of the average of the squares of the speeds of the gas particles. Being directly related to the kinetic energy per molecule, the RMS speed is proportional to the square root of the temperature divided by the mass of the particles. It emphasizes the contribution of faster-moving particles more than the average speed does and directly connects with the thermodynamic temperature.

Maxwell-Boltzmann Distribution

The Maxwell-Boltzmann distribution describes the statistical spread of speeds among particles in an ideal gas. It provides the probability that a particle will have a certain speed at a given temperature, forming the basis for understanding various statistical speed parameters such as average, most probable, and root mean square speed. This concept is fundamental in kinetic theory, linking microscopic particle motion with macroscopic thermodynamic properties.

Average Speed

The average speed of gas particles is defined as the arithmetic mean of all particle speeds in the distribution. It is calculated from the Maxwell-Boltzmann distribution and is proportional to the square root of the temperature divided by the mass of the particles. This measure incorporates contributions from all speeds and provides insight into the overall kinetic energy of the gas molecules.

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