Chapter 5: Phenomena Phenomena: To determine the properties of gases scientists recorded various observations/measurements about different gases. Analyze the table below looking for patterns between the gases. Use these patterns to fill in the information for the mystery gas. Gas Color Amount (mol) Volume (L) Molar Mass g mol Pressure (atm) Odor Temperature ( C) Cl 2 Yellow 1.0 24.4 70.906 1.0 Bleach Like 25 C Cl 2 Yellow 1.0 48.8 70.906 0.5 Bleach Like 25 C CH 4 Colorless 0.5 12.2 16.0426 1.0 Odorless 25 C CH 4 Colorless 3.0 12.2 16.0426 6.0 Odorless 25 C NH 3 Colorless 1.0 24.4 17.0307 1.0 Feline Urine 25 C NH 3 Colorless 1.0 48.8 17.0307 0.5 Feline Urine 25 C NO 2 Brown 0.5 12.2 46.005 1.0 Pungent 25 C NO 2 Brown 1.5 36.6 46.005 1.0 Pungent 25 C Xe Colorless 1.0 24.4 131.29 1.0 Odorless 25 C Xe Colorless 0.37 36.6 131.29 1.0 Odorless 919 C Mystery 1.0 24.4 1.0 25 C Mystery 12.2 6.0 25 C
Big Idea: The pressure, volume, number of moles, and temperature are needed to classify a gas. Knowing 3 of these quantities allows us to calculate the forth, using the ideal gas law. Generally these properties are independent of gas type. The ideal gas laws can be explained using a model of a gas in which the molecules are in ceaseless random motion and widely spread out. Chapter 5 Gases Properties of Gases Pressure Ideal Gas Law Other Equations of States Gas Stoichiometry Dalton s Law of Partial Pressures The Kinetic Model of Gases Effusion and Diffusion 2
Properties of Gases Eleven elements are gases under standard conditions. In addition, many compounds with low molar masses (HCl, CH 4, C 3 H 8 ) are also gases under standard conditions. Note: All elements that are gases at room temperature temperatures are diatomic with the exception of the 6 noble gases, which are monatomic. 3
Properties of Gases Bulk Matter: Forms of matter consisting of large numbers of molecules. The properties of a gas sample emerge from the collective behavior of vast numbers of its individual particles. Properties of Gases Compressible. Fill up the space that is available (this implies that the molecules are moving quickly in a chaotic motion). 4
Pressure 5 A barometer is used to measure the pressure of the atmosphere. The pressure of the atmosphere is balanced by the pressure exerted by the column of mercury. The height of the column is proportional to the atmospheric pressure. Therefore, by measuring the height of the column, we can monitor the pressure of the atmosphere. P=dhg
Pressure 6 (a) An open-tube manometer. The pressure inside the apparatus to which the narrow horizontal tube is connected pushes against the external pressure. In this instance, the pressure inside the system is lower than the atmospheric pressure by an amount proportional to the difference in heights of the liquid in the two arms. (b) The pressure in the adjoining apparatus is proportional to the difference in heights of the liquid in the two arms. The space inside the closed end is a vacuum.
Pressure Common Units of Pressure Units Pa* mm Hg Torr psi in Hg Conversion to atm 101325 Pa = 1 atm 760 mm Hg = 1 atm 760 torr = 1 atm 14.7 psi = 1 atm 29.92 in Hg = 1 atm *The SI derived unit of pressure is Pa. 7
Ideal Gas Law Temperature and Moles Constant Boyle s Law: For a fixed amount of gas under constant temperature, the volume is inversely proportional to pressure. Mathematically Volume α 1 Pressure PV = constant P I V I = P F V F Note: A change that occurs at constant temperature is referred to as an isothermal change. 8
Ideal Gas Law Pressure and Moles Constant Charles s Law: For a fixed amount of gas under constant pressure, the volume varies linearly with the temperature. Mathematically Volume α Temperature V = constant T V I T I = V F T F Note: A change that occurs at constant pressure is referred to as an isobaric change. 9
Ideal Gas Law Temperature Conversions Units Kelvin (K)* Conversion to Kelvin T K Celsius ( C) T K = T C + 273.15 Fahrenheit ( F) T K = 5 9 T F 32 + 273.15 *SI unit Note: Temperature conversions involve both multiplication/division and addition/subtraction, therefore, be very careful when using a unit other than Kelvin. 10
Ideal Gas Law A balloon containing 1.0 L of air at 25 C is put in liquid nitrogen which is at a temperature of -196 C. What is the final volume of the balloon? 11
Ideal Gas Law Implications of Charles s Law 12
Ideal Gas Law Pressure and Temperature Constant Avogadro s Law: At constant pressure and temperature, the volume of a gas varies linearly with the number of moles. Mathematically Volume α Number of Moles V = constant n V I n I = V F n F 13
Ideal Gas Law Ideal Gas Law PV=nRT Boyle s, Charles's, and Avogadro s Law are just special cases of ideal gas law Boyle s Law Charles s Law Avogadro s Law PV = C B V = C c T V = nc A C B = nrt C C = nr P C A = TR P 14
Ideal Gas Law How do they get R? Three pistons all with different gases. (P = 1 atm, T = 298.15 K, n = 1 mol) He N 2 O 2 R= Ideal Gas Constant 0.08206 L atm (P(atm),V(L), n(mol) and T(K)) 8.3145 mol K V He =V N2 =V O2 =24.5L J mol K (P(Pa),V(m3 ), n(mol) and T(K)) 15
Ideal Gas Law Student Question What is the volume (in L) occupied by 35.2 g of nitrogen gas at 35 C and 0.975 atm? Helpful Information: M N = 14.01 g mol a) 26.8 L b) 18.7 L c) 21.8 L d) 32.6 L e) None of the above 16
Ideal Gas Law 17
Other Equations of State Equation of State: A mathematical expression relating the pressure, volume, temperature, and amount of substance present in a sample. Note: The ideal gas law is an example of an equation of state. Limiting Law: A law that is accurately obeyed only at the limit of a property, which is when a property (the pressure of a gas, for example) is made very small. Note: The ideal gas law is a limiting law. 18
Other Equations of State A common procedure in chemistry to improve the ideal gas law is to suppose that the terms on the right-hand side of an equation (such as nrt in PV=nRT of the ideal gas law) are just the leading (and predominant) term of a more complicated expression. Virial Equation: PV = nrt 1 + B V m + C V m 2 + B,C, are called the second virial coefficient, third virial coefficient and so on. The virial coefficients are found by fitting experimental data and are temperature dependent. 19
Other Equations of State Gas a atm L2 mol 2 b L mol He 0.034 0.0237 Ne 0.211 0.0171 Ar 1.35 0.0322 Kr 2.32 0.0398 Xe 4.19 0.0511 H 2 0.244 0.0266 N 2 1.39 0.0391 O 2 1.36 0.0318 Cl 2 6.49 0.0562 CO 2 3.59 0.0427 CH 4 2.25 0.0428 Parameter a corrects for attractions (intermolecular forces) between particles; particles with strong intermolecular forces (large molecules with many electrons)have larger a values. Parameter b corrects for the size of the particle; it can be thought of as representing the volume of an individual molecule. 20
Dalton s Law of Partial Pressure Dalton wondered if he had a certain amount of oxygen in a container that has a pressure of 0.60 atm and he had another container of the same size of nitrogen that had a pressure of 0.40 atm, if he combined the two gases in a third container of the same size, what the pressure would be. He found that the pressure would be 1.00 atm. 21
Dalton s Law of Partial Pressure Partial Pressure (P x ): The pressure that a gas (X) in a mixture would exert if it alone occupied the container. Law of Partial Pressures: The total pressure of gases is the sum of the partial pressures of its components. P tot = P A + P B + Mole Fraction (χ): The amount of particles (molecules, atoms, or ions) of a substance in a mixture expressed as a fraction of the total amount of particles in the mixture. χ A = n A n tot = n A n A + n B + 22
The Kinetic Model of Gases Assumptions Made for the Kinetic Model of Gases 1) The particles are so small compared with the distances between them that the volume of the individual particles can be assumed to be negligible (zero). 2) The particles are in constant motion. The collisions of the particles with the walls of the container are the cause of the pressure exerted by the gas. 3) The particles are assumed to exert no forces on each other; they are assumed to neither attract nor repel each other. 4) The average kinetic energy of a collection of gas particles is assumed to be directly proportional to the Kelvin temperature of the gas. 23
The Kinetic Model of Gases Purpose: Want to show that ideal gas law and kinetic model of gases are consistent with each other PV n T Calculate the Pressure in a Rectangular Prism (Kinetic Model) 1) P = F A F= Force, A=Area 2) F = p t (Newton's second law) p=momentum (p = mu u=velocity) z y x A x = Area of a side on x A y = Area of a side on y A z = Area of a side on z 24
The Kinetic Model of Gases Goal: Calculate Change in Momentum 3) Calculate of Δp of 1 molecule hitting 1 of the walls. p before hits wall p x = mu x p after hits wall p x = m u x Change in p p x = m u x mu x = 2mu x 4) Calculate p for multiple molecules (traveling at the same speed) p xtot = N col p x = N col 2mu x N col = number of molecules that collide with wall 25
The Kinetic Model of Gases Goal: Calculate Change in Momentum 5) How many molecules collide with wall in time period Δt? Kinetic Model: particles in random motion, therefore, roughly evenly distributed N pos = V pos V tot N tot N pos = number of molecules that could hit wall V pos = the volume that those particles are in V tot = total volume of rectangular prism N tot = total number of molecules Note: V tot is the same volume used in the ideal gas law but we need to find expressions for the rest of the variables. 26
The Kinetic Model of Gases Goal: Calculate Change in Momentum 6) What is the volume in which particles have the potential of hitting the wall (V pos ) x max = u x t x max = max distance away from wall and still collide with side V pos = A x x max = A x u x t 27 7) Plug expression into N pos N pos = V pos N V tot = A xu x t N tot V tot tot Problem: Half of the molecules in that area are moving toward the wall and half are moving away. N col = N pos 2 = A xu x a t N tot 2V tot
The Kinetic Model of Gases Goal: Calculate Change in Momentum 8) Plug N col into Δp tot p xtot = N col 2mu x = A xu x t N 2V tot 2mu x = A xmu 2 x t tot V tot N tot Goal: Calculate Force 9) Calculate force on 1 side of rectangle prism F x = p t = A x mu x 2 t V tot t N tot = A xmu x 2 V tot N tot Note: When we originally calculated the change in momentum it was with respect to the molecules. When we calculate the force, we need the change in momentum with respect to the wall, therefore, the sign must be flipped. 28
The Kinetic Model of Gases Goal: Calculate Pressure 10) Calculate pressure on 1 side of the rectangular prism P x = F x A x = A x mu x 2 N V tot tot A x = mu 2 x N V tot tot 11) Remove requirement that all molecules are traveling at the same speed. 2 P x(rms) = mu x(rms) V tot N tot Note: A line over a variable represents the average value. 29
The Kinetic Model of Gases Goal: Calculate Pressure 12) Calculate P tot Note: The pressure inside the container is constant. P tot = P x(rms) = mu 2 x(rms) V tot N tot 12) Change to overall velocity instead of velocity in x u2 2 rms = u x(rms) 2 u x(rms) 2 = u y(rms) 2 2 u rms = 3u x(rms) 2 + u y(rms) 2 = u z(rms) 2 + u z(rms) P tot = mu x 2 N V tot = m1 3 tot u2 rms V tot N tot 30
The Kinetic Model of Gases Goal: Calculate Pressure 14) Change to number of moles n tot N A = N tot N A = Avogadro s Constant P tot = mu 2 rms N 3V tot = mu 2 rms n tot 3V tot N A = n totmu rms tot 3V tot 2 N A 31
The Kinetic Model of Gases Goal: Calculate Pressure 15) Kinetic Model: The average kinetic energy of a collection of gas particles is assumed to be directly proportional to the Kelvin temperature of the gas. KE ave = 1 2 mu2 T P tot = n 2 totmu rms N A 3V tot n tott V tot Note: The variables boxed in green are proportional to the kinetic energy and the variables boxed in red are constants. Ideal Gas Law P tot n tott V tot Kinetic Theory P tot n tott V tot 32
The Kinetic Model of Gases Maxwell Distribution ( N = Nf u u) ΔN = Number of molecules with a speeds in the narrow range between u and u+ ΔU f u = 4π M 2πRT Τ 3 2 u 2 e Τ Mu 2 2RT f(u) = Maxwell distribution of speeds James Clerk Maxwell 1831-1879 Wrote the equations that unified electricity, magnetism, and optics. 33
The Kinetic Model of Gases The heavier the atom, the slower the average speed and the closer the speeds are to the average speed. As the temperature increases, the average speed increases and the spread of speeds widens. 34
The Kinetic Model of Gases Most Probable Velocity (u mp ) df u du u mp = = 0 solve for u 2RT M Average Velocity (u ave ) u ave = തu = 0 uf u du u ave = 8RT πm Root Mean Square Velocity (u rms ) u rms = 0 u 2 f u du u rms = 3RT M 35
The Kinetic Model of Gases Student Question Consider three 1.0-L flasks at STP. Flask A contains He gas, flask B contains O 2 gas, and flask C contains H 2 gas. In which flask do the gas particles have the lowest average kinetic energy? a) Flask A b) Flask B c) Flask C d) All are the same. 36
Effusion and Diffusion Diffusion: The spreading of one substance through another substance. Effusion: The escape of a substance (particularly a gas) through a small hole. Graham s Law of Effusion: At constant temperature, the rate of effusion of a gas is inversely proportional to the square root of its molar mass. Note: This law was determined experimentally. 37
Effusion and Diffusion Determining the relationship between rates of effusion of different substances. Rate effusiona = C 1 M A Rate effusionb = C 1 M B Divide the two quantities by each other. Rate effusiona Rate effusionb = C 1 M A Rate effusiona C 1 = Rate effusionb M B M B M A Note: rate = particles time 38
Effusion and Diffusion Student Question A sample of an unknown gas effuses in 8.2 minutes. An equal volume of krypton in the same apparatus at the same temperature and pressure effuses in 4.0 minutes. What is the likely identity of the gas? Helpful Information: M Kr = 83.80 g mol, M Ar = 39.95 g, mol M UF6 = 352.03 g, M mol Ne = 20.18 g, and M mol Br 2 = 159.80 g mol a) UF 6 b) Br 2 c) Ar d) Ne e) None of the Above 39
Effusion and Diffusion NH 3 (g) +HCl(g) NH 4 Cl(s) Notice that the ammonium chloride initially forms closer to the end with HCl. 40
Take Away From Chapter 5 Big Idea: The pressure, volume, number of moles, and temperature are needed to classify a gas. Knowing 3 of these quantities allows us to calculate the forth by using the ideal gas law. Generally these properties are independent of gas type. The ideal gas law can be explained using a model of a gas in which the molecules are in ceaseless random motion and widely spread out. Properties of Gases Pressure (22) Know how to convert between different pressure units. Know the conditions of STP (1 atm and 0 C) and SATP (1 atm and 25 C). 41 Numbers correspond to end of chapter questions.
Take Away From Chapter 5 Ideal Gas Law Know how to use : Boyle s Law P 1 V 1 = P 2 V 2 Charles's Law V I T I = V F T F (41) Avogadro s Law V I n I = V F n F (42) 42 Ideal gas Law PV = nrt (11,33,34,36,37,40,43,55,56) Know the condition in which the ideal gas law is a good approximation. (93) Other Equations of State Know the advantages and disadvantages of the Virial equation. Know how to use the Van der Waal equation and what the Van der Waal constants correct for. (91,92) Gas Stoichiometry (8) Be able to use the gas laws to perform stoichiometry problems. (60,62,65,66,73,119) Numbers correspond to end of chapter questions.
Take Away From Chapter 5 Dalton s Law of Partial Pressure Be able to calculate the partial pressures of gases. (29,46,49,127) P A = χ A P tot P tot = P A + P B + The Kinetic Model of Gases Know what assumptions are made for kinetic model of gases. Know the relationship between kinetic energy, temperature, and velocity.(79,80) KE ave = 3 RT (81) 2 Know how the shape of the Maxwell distribution of speeds changes with mass and temperature. (94) Know the difference and how to calculate u rms, u mp, and u ave. (82,83,86,99) u rms = 2RT u M mp = 2RT u M ave = 8RT πm 43 Numbers correspond to end of chapter questions.
Take Away From Chapter 5 Effusion and Diffusion Know the difference between effusion and diffusion Be able to use the effusion equation: (89,90) Rate A Rate B = M B M A time for B to effuse time for A to effuse = M B M A The time equation only works if the number of particles that effused are the same Be able to use the diffusion equation: d A d B = M B M A 44 Numbers correspond to end of chapter questions.