Thermodynamics: The Kinetic Theory of Gases

Transcription

1 Thermodynamics: The Kinetic Theory of Gases Resources: Serway The Kinetic Theory of Gases: 10.6 AP Physics B Videos Physics B Lesson 5: Mechanical Equivalent of Heat Physics B Lesson 6: Specific and Latent Heat Physics B Lesson 7: Heat Transfer and Thermal Expansion MIT CourseWare 3: Thermal Expansion 33: Ideal-Gas Law

2 Kinetic Theory of Gases The ideal gas law provides no insight as to how pressure and temperature are related to properties of the gas molecules themselves (such as their masses or speeds). To understand this, we have to look at the dynamics of the molecular motion.

3 The Kinetic Molecular Model The IDEAL Gas Law PV = nrt Boyle s Law P 1 V 1 = P V at constant n and T Charles s Law V 1 / T 1 = V / T at constant n and P The General Gas Law P 1 V 1 / T 1 = P V / T at constant n These are all useful law s, they accurately express observable and repeatable behaviors. But they DO NOT EXPLAIN the cause.

4 The Kinetic Molecular Model Welcome to the Kinetic Molecular Model This model attempts to explain the behavior of gas molecules at the microscopic level. The previous laws were developed at the macroscopic level. Solid carbon dioxide, known as dry ice, passes directly into a gaseous state at room temperature.

5 The Kinetic Molecular Model -Assumptions This model is built on the following assumptions: 1. The number of molecules is large, and the average separation between them is large. The molecules occupy negligible volume in the container.. The molecules obey Newton s laws of motion, but as a whole they move randomly. 3. The molecules undergo elastic collisions with each other and with the container. 4. The forces between molecules is negligible except during a collision. 5. The gas is a pure substance.

6 The Kinetic Molecular Model - Derivation Recall, momentum is the product of an object s mass and velocity, p = mv Consider a molecule of gas moving inside a container of length, L colliding with its walls: p x = p f,x p i,x = mv f,x (-mv i,x ) = mv x The wall of the container exerts a force causing p: p x = F t (Impulse-Momentum Theorem) F = p x / t F = mv x / t Remember v x, v x = x / t t = x / v x

7 Gas Molecules Simulation

8 The Kinetic Molecular Model - Derivation Substituting t, F = mv x / t = (mv x ) / ( x / v x ) Referring back to the container of length, L: x = L F = (mv x ) / (L / v x ) = mv x / L Restating, F = mv x / L Recall the definition of pressure, P = F / A P = (mv x / L) / L P = mv x / L 3 P = mv x / V

9 The Kinetic Molecular Model - Derivation Recall, P = mv x / V All N molecules in the container follow this behavior, P = Nmv x / V Expanding to all directions (x, y, z) and true randomness predicts that: v x = v y = v z And v av = v x + v y + v z Therefore, v x = 1/3 v av

10 The Kinetic Molecular Model - Derivation Recall, P = Nmv x / V Substituting, v x = 1/3 v av P = 1/3 Nmv av / V Recall, KE = 1/ mv av KE = mv av P = 1/3 N(KE) / V P = /3 NKE / V PV = /3 NKE

11 The Kinetic Molecular Model - Derivation Recall, PV = /3 NKE Recall, PV = Nk B T Combining, Nk B T = /3 NKE Algebraic magic KE = 3/ k B T A direct connection between Temperature and Kinetic Energy!!!

12 The Kinetic Molecular Model - Derivation Recall, K avg = 3/ k B T Combining, 1/ mv av = 3/ k B T v av = 3 k B T / m Taking the square root of both sides, v av = 3kT / m v av is called the Root Mean Square v av = v RMS v RMS = 3k B T / m = 3RT / M m = µ = mass of molecule M = molar mass

13 The Kinetic Molecular Model The kinetic molecular model predicts that the temperature of an object is a measurement of its average molecular kinetic energy! The kinetic molecular model also predicts that pressure is a direct result of repeated molecular collisions with a container! Root mean squared speed is used due to the

14 The Kinetic Molecular Model Important equations: v RMS = 3k B T / m = 3RT / M K avg = 3/ k B T PV = /3 NKE P = mv x / V These equations are the foundation of everything you studied in chemistry. Chemistry observes the behavior, Physics explains the cause! m = µ = mass of molecule M = molar mass

15 Summary Kinetic Molecular Theory Where, K v avg rms = 3 k T B 3RT 3kBT = = M µ K avg : average kinetic energy of a gas molecule (J) T: temperature (K) v rms : root-mean-square speed of a gas molecule (m/s) M: molar mass µ: mass of molecule k B : Boltzman s constant = 1.38 x 10-3 J/K R: universal gas constant = 8.31 J/(mol K)

16 Practice Problem What is the average kinetic energy and the average speed of oxygen molecules in a gas sample at 0 o C? Given: T= 0 o C Find: KE =? v =? Solution: To find KE we ll need to apply the kinetic molecular model, KE = 3/ k B T But first we ll have to convert the temperature to kelvins, T = 73 K Recall Boltzman s constant, k B = 1.38 x 10-3 J/K KE = (3/)(1.38 x 10-3 J/K )(73 K) = 5.65 x 10-1 J Solution: To find v, we ll need to apply the rms speed equation v RMS = (3kBT / m) = (3RT / M) We ll need either the molar mass or mass of the oxygen molecule (O ). M = x Atom Weight = x 16 = 3 g/mol = 0.03 kg/mol) v RMS = (3RT / M) = [(3)(8.31)(73)]/(0.03) v RMS = 461. m/s