Question

Assume you have a container with 100 ideal gas molecules. (a) What is the
probability that zero molecules will be found in a small volume corresponding 1% of
the container’s total volume. (b) How many molecules will you need for the
probability to be 1 in 1000 (or 0.001) for the same volume inside the container?

Answer 1

Solution

The container having ideal gas molecules =100 molecules

Ideal gas having standard

temperature- 273K

Pressure -1 atm

Ideal gas constant (R) - 0.0821L-atom mol- K

Ideal gas equation

PV = nRT

V = nRT/P

V = 100×0.0821×273/1 = 224.13liters

Then we find out 1% of total volume

= 1×224.13/100 =2.241liters  

And in 2.241 liters volume contains 0.1 molecules of ideal gas

a) Probability of volume 1% in total container volume = 2.241/224.1 = 0.001

same for the molecule 0.001 molecules found in small region.

b) probability to be 0.0001 for 224.1liters

molecules will we need for 0.0001 probability = ?

• 100 molecules having probability =0.001

In 1000 molecules having probability = 0.0001

So, 1000 molecules are needed for 0.0001 probability