2.4 Distribution of Molecular Speeds

Gases are made up of tiny particles zipping around at different speeds. The Maxwell-Boltzmann distribution describes how fast these particles move in an ideal gas. It's a key concept in understanding gas behavior at the molecular level.

This distribution helps us calculate important speeds like the most probable, average, and root-mean-square speed of gas molecules. Temperature plays a big role, affecting how fast the particles move and how their speeds are spread out.

Distribution of Molecular Speeds in Ideal Gases

Maxwell-Boltzmann distribution of speeds

• Probability distribution of molecular speeds in an ideal gas at thermal equilibrium assumes no intermolecular interactions and elastic collisions between molecules (billiard balls)
• Probability density function $f(v)$ gives the probability of finding a molecule with a specific speed $v$
  • $f(v) = 4\pi \left(\frac{m}{2\pi k_B T}\right)^{3/2} v^2 e^{-\frac{mv^2}{2k_B T}}$, where $m$ is molecular mass, $k_B$ is Boltzmann constant, and $T$ is absolute temperature (Kelvin)
• Distribution is asymmetric and positively skewed with a long tail at high speeds (right-skewed)
  • Most molecules have speeds close to the most probable speed with fewer molecules at very low or very high speeds (bell curve)

Calculation of molecular speeds

• Most probable speed $v_{mp}$ is the speed at which Maxwell-Boltzmann distribution reaches its maximum value
  • $v_{mp} = \sqrt{\frac{2k_B T}{m}}$
• Average speed $\bar{v}$ is the mean speed of all molecules in the gas
  • $\bar{v} = \sqrt{\frac{8k_B T}{\pi m}}$
• Root-mean-square (rms) speed $v_{rms}$ is the square root of the average of the squares of molecular speeds
  • $v_{rms} = \sqrt{\frac{3k_B T}{m}}$
• Relationship between these speeds: $v_{mp} < \bar{v} < v_{rms}$
  • Most probable speed is lower than average speed which is lower than rms speed
• These speeds are inversely proportional to the square root of molecular mass, affecting the distribution of speeds for different gases

Temperature effects on speed distribution

• As temperature increases, Maxwell-Boltzmann distribution shifts to the right and becomes broader (higher speeds)
  • Peak of distribution (most probable speed) moves to higher speeds
  • Proportion of molecules with higher speeds increases (more high-speed molecules)
• Average, most probable, and rms speeds all increase with square root of absolute temperature
  • Doubling absolute temperature increases these speeds by factor of $\sqrt{2}$ (41% increase)
• At higher temperatures, distribution of molecular speeds becomes more spread out with larger standard deviation (wider distribution)
• Changes in temperature do not affect shape of distribution, only its position and width (still bell-shaped)

Kinetic Theory of Gases and Thermal Equilibrium

• Kinetic theory of gases explains macroscopic properties of gases using the motion of their constituent particles
• Thermal equilibrium is achieved when a system's temperature is uniform and not changing with time
• Mean free path is the average distance a molecule travels between collisions, affecting the rate of energy transfer in the gas