1. Kinetic Molecular Model of an Ideal Gas
Prof. Marlon Flores Sacedon
Department of Mathematics and Physics
College of Arts and Sciences
Visayas State University, Visca
Baybay City, Leyte, Phiippines

2. Kinetic-Molecular Model of an Ideal Gas
Recall the Molecular theory:
Matter is made up of Molecules.
All molecules are identical for any specific chemical compound.
smallest molecules contains 1 atom, m in size.
largest molecules many atoms contains many atom, 10,000 times smallest molecules.
Crystal structure of sodium chloride
In gases the molecules move nearly independent.
In liquids and solids they are held together by molecular forces that electrical in nature.
Molecules are always in motion; their kinetic energies usually increase with temperature.
Crystal structure of sodium chloride

3. Kinetic-Molecular Model of an Ideal Gas
Assumptions of the model:
1. A container with volume V contains a very large number N of identical molecules, each with mass m..
2. The molecules behave as point particles that are small compared to the size of the container and to the average distance between molecules.
3. The molecules are in constant motion. Each molecule collides occasionally with a wall of the container. These collisions are perfectly elastics.
4. The container walls are rigid and infinitely massive and do not move.

4. Kinetic-Molecular Model of an Ideal Gas
The molar mass 𝑀 of a compound is the mass of 1 mole. It is equal to the mass 𝑚 of a single molecule multiplied by Avogadro’s number 𝑁 𝐴 :
Where:
M = molar mass (g/mol)
NA = 6.022x1023 (molecules/mol) Avogadro’s number
m = mass of single molecule (g/molecule)
The ratio 𝑅 𝑁 𝐴 is called the Boltzmann constant, k:
Where:
k = Boltzmann constant
R = ideal gas constant

5. Kinetic-Molecular Model of an Ideal Gas
(average translational kinetic energy of an ideal gas)
Where:
Ktr = ave. translational kinetic energy (J)
n = number of moles (mols)
R = ideal gas constant per mole
R= J/mols.K
R= L.atm/mol.K
T = temperature (K)
N = total no. of molecules
m = mass of single molecule (g/molecule)
vrms = root-mean-square speed of a gas molecules
NA = 6.022x1023 molecules/mol (Avogadro’s number
(average translational kinetic energy of a gas molecules)
k = x10-23 J/molecule.K
Boltzmann constant
M = molar mass (g/mol)
(root-mean-square speed of a gas molecules)

7. Kinetic-Molecular Model of an Ideal Gas
Derivation of formulas

8. Kinetic-Molecular Model of an Ideal Gas
Derivation of formulas

9. Kinetic-Molecular Model of an Ideal Gas
Example 1:. (a) What is the total translational kinetic energy of the air in an empty room that has dimensions 8𝑥12𝑥4 𝑚 if the air is treated as an ideal gas at 1 atm? (b) what is the speed of a 2000kg automobile if its kinetic energy equals the translational kinetic energy calculated in part (a)?
5.83x107 J, 242 m/s
𝐾 𝑡𝑟 = 3 2 𝑝𝑉
= 𝑎𝑡𝑚∗1.013𝑥 10 5 𝑁/ 𝑚 2 8𝑥12𝑥4 𝑚 3
=5.83𝑥107 𝐽
𝐾 𝑡𝑟 = 1 2 𝑚 𝑣 𝑎𝑣 2
𝑣 𝑎𝑣 = 2 𝐾 𝑡𝑟 𝑚
= 𝑥 10 7 𝐽 𝑘𝑔
= 𝑚/𝑠

10. Kinetic-Molecular Model of an Ideal Gas
Example 2: Calculating molecular kinetic energy and vrms : a) what is the average translational kinetic energy of an ideal gas at a temperature of 27oC b) what is the total random translational kinetic energy of the molecules in one mole of this gas? c) what is the root-mean-square speed of oxygen molecules at the temperature?
= 𝑥 10 −23 𝐽/𝐾 27 𝑜 𝐶
=6.21𝑥 10 −23 𝐽
ANSWER
𝐾 𝑡𝑟 = 3 2 𝑛𝑅𝑇
= 𝑚𝑜𝑙 𝐽/𝑚𝑜𝑙∙𝐾
=3,740 𝐽
ANSWER
= 𝐽/𝑚𝑜𝑙∙𝐾 𝐾 32𝑥 10 −3 𝑘𝑔/𝑚𝑜𝑙
ANSWER
=484 𝑚/𝑠

11. Kinetic-Molecular Model of an Ideal Gas
Example 3:. (a) A deuteron, 1 𝟐 𝑯 , is the nucleus of a hydrogen isotope and consists of one proton and one neutron. The plasma of deuterons in a nuclear fusion reactor must be heated to about 300 million K. What is the rms speed of the deuterons? Is this a significant fraction of the speed of light in vacuum (c=3.0x108 m/s)? (b) What would the temperature of the plasma be if the deuterons had an rms speed equal to 0.10c?
𝑚= 𝑀 𝑁 𝐴 = 2∗1.008𝑥 10 −3 𝑘𝑔/𝑚𝑜𝑙 𝑥 𝑚𝑜𝑙𝑒𝑐𝑢𝑙𝑒𝑠/𝑚𝑜𝑙 =3.348𝑥 10 −27 g/molecule
𝑣 𝑟𝑚𝑠 = 𝑥 10 −23 3𝑥 10 8 𝐾 𝑥 10 −27 𝑘𝑔 =1.93𝑥 10 6 𝑚/𝑠
Not significant fraction
𝑇= 𝑚 𝑣 𝑟𝑚𝑠 2 3𝑘 = 𝑥 10 −27 𝑘𝑔 𝑥 10 8 𝑚/𝑠 𝑥 10 −23 =7.3𝑥 𝐾

12. Kinetic-Molecular Model of an Ideal Gas
Example 4:. We have two equal-size boxes, A and B. Each box contains gas that behaves as an ideal gas. We insert a thermometer into each box and find that the gas in box A is at 50oC while the gas in box B is at 10oC. This is all we know about the gas in the boxes. Which of the following statements must be true? Which could be true? Explain your reasoning. (a) The pressure in A is higher than in B. (b) There are more molecules in A than in B. (c) A and B do not contain the same type of gas. (d) The molecules in A have more average kinetic energy per molecule in B. (e) The molecules in A are moving faster than those in B.

13. PROBLEMS

14. PROBLEMS

15. PROBLEMS

16. PROBLEMS
The deuteron is composed of a proton and a neutron, is a stable particle. As an atom, it is called deuterium and as an isotope of hydrogen it has an abundance of 1.5 x10-4 compared to for ordinary hydrogen.
Mass of electron = me = 9.11x10-31 kg
Mass of proton = mp = 1.673x10-27 kg
Mass of neutron = mn = 1.675x10-27 kg

17. PROBLEMS

18. PROBLEMS

19. PROBLEMS

20. ANSWERS

21. eNd