QuickCheck. Collisions between molecules. Collisions between molecules

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1 Collisions between molecules We model molecules as rigid spheres of radius r as shown at the right. The mean free path of a molecule is the average distance it travels between collisions. The average time between collisions is the mean free time. In a time dt a molecule with radius r will collide with any other molecule within a cylindrical volume of radius 2r and length v dt. Collisions between molecules The average distance traveled between collisions is called the mean free path. In our model, this is just the molecule s speed v multiplied by mean free time: The more molecules there are and the larger the molecule, the shorter the mean distance between collisions. The temperature of a rigid container of oxygen gas (O 2 ) is lowered from 300ºC to 0ºC. As a result, the mean free path of oxygen molecules A. Increases. B. Is unchanged. C. Decreases Pearson Education, Inc. Slide

2 Example 1 What is the mean free path of a nitrogen molecule at 1.0 atm pressure and room temperature (20ºC)? Also, calculate the mean time between collisions Pearson Education, Inc. Slide 20-4 In-class Activity #1 What is the mean time between collisions for a monatomic gas with a molar mass of M = 40.0 g/mol at STP? Assume the diameter of the molecules to be 4.0 x m Pearson Education, Inc. Slide 20-5 The Equipartition Theorem Atoms in a monatomic gas carry energy exclusively as translational kinetic energy (3 degrees of freedom). Molecules in a gas may have additional modes of energy storage, for example, the kinetic and potential energy associated with vibration, or rotational kinetic energy. We define the number of degrees of freedom as the number of distinct and independent modes of energy storage: 2017 Pearson Education, Inc. Slide

3 A mass on a spring oscillates back and forth on a frictionless surface. How many degrees of freedom does this system have? A. 1 B. 2 C. 3 D Pearson Education, Inc. Slide 20-7 A mass on a spring oscillates back and forth on a frictionless surface. How many degrees of freedom does this system have? A. 1 B. 2 C. 3 D. 4 It can hold energy as kinetic energy or potential energy Pearson Education, Inc. Slide 20-8 Thermal Energy and Specific Heat The thermal energy of a system is E th = K micro + U micro. The figure shows a monatomic gas such as helium or neon. The atoms in a monatomic gas have no molecular bonds with their neighbors, hence U micro = 0. Since the average kinetic energy of a single atom in an ideal gas is K tr = 3/2 k B T, the total thermal energy is 2017 Pearson Education, Inc. Slide

4 Thermal Energy and Specific Heat If the temperature of a monatomic gas changes by ΔT, its thermal energy changes by Earlier we found that the change in thermal energy for any ideal-gas process is related to the molar specific heat at constant volume by Combining these equations gives us a prediction for the molar specific heat for a monatomic gas: This prediction is confirmed by experiments Pearson Education, Inc. Slide Diatomic Molecules In addition to the 3 degrees of freedom from translational kinetic energy, a diatomic gas at commonly used temperatures has 2 additional degrees of freedom from end-over-end rotations. This gives 5 degrees of freedom total: 2017 Pearson Education, Inc. Slide You have 1.00 mol of an ideal monatomic gas and 1.00 mol of an ideal diatomic gas whose molecules can rotate. Initially both gases are at room temperature. If the same amount of heat flows into each gas, which gas will undergo the greatest increase in temperature? A. the monatomic gas B. the diatomic gas C. Both will undergo the same temperature change. D. The answer depends on whether or not the diatomic molecules can also vibrate Pearson Education, Inc. Slide

5 Compare theory with experiment Table below shows that the calculated values for C V for monatomic gases and diatomic gases agree quite well with the measured values. Experimental values of C V for hydrogen gas (H 2 ) The molar heat capacity at constant volume of diatomic hydrogen gas (H 2 ) is 5R/2 at 500 K but only 3R/2 at 50 K. Why is this? A. At 500 K the molecules can vibrate, while at 50 K they cannot. B. At 500 K the molecules cannot vibrate, while at 50 K they can. C. At 500 K the molecules can rotate, while at 50 K they cannot. D. At 500 K the molecules cannot rotate, while at 50 K they can Pearson Education, Inc. 5

6 Example 2 How much heat does it take to increase the temperature of 1.80 mol of an ideal gas by 50.0 K near room temperature if the gas is held at constant volume and is diatomic? How does the answer change is the gas is monatomic? 2017 Pearson Education, Inc. Slide Example 3 Calculate the specific heat capacity at constant volume of water vapor, assuming the nonlinear triatomic molecule has three translational and three rotational degrees of freedom. The molar mass of water is 18.0 g/mol. Compare to the actual specific heat capacity of water vapor at low pressures which is about 2000 J / kg K and consider whether vibrational motion needs to be considered Pearson Education, Inc. Slide In-class Activity #2 Perfectly rigid containers each hold n moles of ideal gas, one being hydrogen (H 2 ) and other being neon (Ne). If it takes 300 J of heat to increase the temperature of the hydrogen by 3.50ºC, by how many degrees will the same amount of heat raise the temperature of the neon? 2017 Pearson Education, Inc. Slide

7 Thermal Energy of a Solid The figure reminds you of the bedspring model of a solid with particle-like atoms connected by spring-like molecular bonds. There are 3 degrees of freedom associated with kinetic energy + 3 more associated with the potential energy in the molecular bonds = 6 degrees of freedom total. The energy stored in each degree of freedom is ½ Nk B T, so 2017 Pearson Education, Inc. Slide Compare theory with experiment Experimental values of C V for lead, aluminum, silicon, and diamond are given in the figure. At high temperatures, C V for each solid approaches about 3R.. At low temperatures, C V is much less than 3R. Molecular Speeds and Collisions A gas consists of a vast number of molecules, each moving randomly and undergoing millions of collisions every second. Shown is the distribution of molecular speeds in a sample of nitrogen gas at 20ºC. The micro/macro connection is built on the idea that the macroscopic properties of a system, such as temperature or pressure, are related to the average behavior of the atoms and molecules Pearson Education, Inc. Slide

8 Molecular speeds The Maxwell-Boltzmann distribution f (v) gives the distribution of molecular speeds. Molecular speeds The most probable speed for a given temperature is at the peak of the curve. Molecular speeds The function f (v) describing the actual distribution of molecular speeds is called the Maxwell Boltzmann distribution. It can be derived from statistical mechanics considerations, but that derivation is beyond our scope. Here is the result: 8

9 A quantity of gas containing N molecules has a speed distribution function f(v). How many molecules of this gas have speeds between v 1 and v 2 > v 1? A. B. C. D Pearson Education, Inc. Example 4 Find the most probably speed of the Maxwell-Boltzman Distribution function Pearson Education, Inc. Slide Example 5 For diatomic carbon dioxide gas (M = 44.0 g/mol) at 300 K calculate the most probably speed, the average speed and the rms speed Pearson Education, Inc. Slide

10 In-class Activity #3 For water vapor (M = 10.0 g/mol) at 400 K calculate the most probably speed, the average speed and the rms speed Pearson Education, Inc. Slide