Description

Week 2: kinetic-molecular theory of gases

Statistics: mean
• Average of a data set xi/probability distribution p(xi) (discrete) or f(x) where x lives in [a,b] (continuous) is defined as
b
xmean∑xi or ∑xip(xi) or ∫ xf(x)dx (1)
N i i a
where N is the size of the dataset (discrete).
• Example: given a set of numbers
1,1,3,3,3,4
Calculate the mean of this dataset.
Statistics: mean square & root mean square
• Mean square is the mean of square
b
xms or ∫ x2f(x)dx (2)
N i i a
• Example: given a set of numbers
1,1,3,3,3,4
Calculate the root mean square of this dataset.
Statistics: mode
• Mode is the value in the dataset which gives the maximum probability (i.e., the most probable value)
xmp =argmaxp(xi) or argmaxf(x) (3)
xi x i
• Example: given a set of numbers
1,1,3,3,3,4
Calculate the mode of this dataset.
Kinetic theory of gas
• Three essential statistical values for gas. vmp: most probable. vmean: average.
vrms: root-mean-square.
√
2RT √8RT √3RT
vmp =, vmean =, vrms =(4)
M πM M
Problems
Activity 1.1
Measurement on a noble gas sample at a temperature of 1000K show a discribution of molecular speeds characterized by a root mean square speed of 1.12 × 103ms−1. What noble gas is this?

Problems

Activity 1.2
Consider two Maxwell-Boltzmann speed distribution curves below
a. If the curve represent the speed distributions for argon and helium at the same temperature, which curve (1 or 2) best depicts the behavior of helium?
b. If the curves represent the speed distributions for helium gas at two different temperatures, T1 and
T2 (where T2 > T1), which curve (1 or 2) best depicts the hither temperature sample?