Gases are made up of tiny particles zipping around at different speeds. The 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
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Maxwell–Boltzmann distribution - Wikipedia View original
Probability distribution of molecular speeds in an ideal gas at assumes no intermolecular interactions and elastic collisions between molecules (billiard balls)
f(v) gives the probability of finding a molecule with a specific speed v
f(v)=4π(2πkBTm)3/2v2e−2kBTmv2, where m is , kB 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 with fewer molecules at very low or very high speeds (bell curve)
Calculation of molecular speeds
vmp is the speed at which reaches its maximum value
vmp=m2kBT
vˉ is the mean speed of all molecules in the gas
vˉ=πm8kBT
Root-mean-square (rms) speed vrms is the square root of the average of the squares of molecular speeds
vrms=m3kBT
Relationship between these speeds: vmp<vˉ<vrms
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 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
is achieved when a system's temperature is uniform and not changing with time
is the average distance a molecule travels between collisions, affecting the rate of energy transfer in the gas