# Chapter 5 The Gaseous State

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1 Chapter 5 The Gaseous State I) Pressure Pressure is the force exerted per unit area. A) Devices used to measure pressure 1) barometer used to measure the atmospheric pressure at seal level and 0 o C, P atm 760 mmhg 2) manometer used to measure the pressure of a gas in a laboratory a) closed end manometer mercury filled U-tube h gas pressure (P) b) open-end manometer compare pressure of a gas to the atmospheric pressure P gas P atm + h B) Units of Pressure SI units: pascal Pa Kg/m s 2 commonly used units: 1 mmhg 1 torr 1 atm 760 mm Hg 760 torr 1 atm x 10 5 Pa 1 atm kpa

2 II) Empirical Gas Laws Relationship between volume (V), pressure (P), and temperature (T) of a given amount of gas (moles) A) Boyle's Law relationship between volume and pressure at constant temperature and amount of gas definition: At constant temperature, the volume occupied by a finite mass of gas is inversely proportional to the applied pressure. V 1/P at a given temperature PV constant comparing two sets of V and P at a given temperature P 1 V 1 P 2 V 2 Example: A sample of argon gas occupied 105 ml at atm. If the temperature remains constant, what is the volume (in L) at 197 mmhg? P 1 V 1 P 2 V 2 V 2 P 1V 1 P 2 (0.871 atm)(105 ml) (0.259 atm) V ml

3 B) Charles' Law relationship between volume and absolute temperature at constant pressure and given amount of gas definition: At constant pressure, the volume of a fixed amount of gas is directly proportional to its absolute temperature. V Tabs at a fixed P and amt. of gas V/T abs constant T abs absolute temp. (Kelvin) Law expressed in terms of two pairs of variables V 1 V 2 T 1 T 1 Example: A 425 ml sample of fluorine gas is heated from 22 o C to 185 o C at constant pressure. What is the final volume? V 1 T 1 V 2 T 2 V 2 V 1T 2 T 1 (425 ml)(458 K) (295 K) V ml C) Gay-Lussac's Law relationship between pressure and absolute temperature at constant volume and amount of gas definition: At constant volume, the pressure exerted by a fixed amount of gas is directly proportional to absolute temperature. P T abs at fixed V and amt. of gas P/T abs constant for two sets of variables P 1 T 1 P 2 T 2

4 example: In an autoclave, steam at 100 o C is generated at 1.00 atm. After the autoclave is closed, the steam is heated at constant volume until the pressure gauge indicated 1.13 atm. What is the new temperature in the autoclave? P 1 T 1 P 2 T 2 T 2 P 2T 1 P 1 P 1 T 2 P 2 T 1 (1.13 atm)( K) (100 atm) T 2 421K o C D) Combined Gas Law expressing Boyle's, Charles', and Gay-Lussac's Laws in 1 statement PV/T constant for a given amount of gas for two sets of data P 1 V 1 T 1 P 2V 2 T 2 example: What is the new volume of a sample of oxygen (in L) if 2.75 L at 742 torr and 25.0 o C is heated to 37.0 o C under conditions that let the pressure change to 760 torr? P 1 V 1 T 1 P 2V 2 T 2 V 2 P 1V 1 T 2 T 1 P 2 V 2 (742 torr)(2.75 L)(310 K) (298 K)(760 torr) V L Practice: Divers working from a North Sea drilling platform experiences pressures of 5.0 x 10 1 atm at a depth of 5.0 x 10 2 m. If a balloon is inflated to a volume of 5.0 L (volume of a lung) at that depth at a water temperature of 4.0 o C, what would the volume of the balloon be on the surface (1.0 atm pressure) at a temperature of 11 o C?

5 answer: (2.6 x 10 2 L) E) Avogadro's Law relationship between volume and amount of gas fixed temperature and pressure Definition: equal volumes of gas contain an equal number of molecules or moles of gas V n at constant T, P n moles of gas 3 H 2 (g) + N 2 (g) 2 NH 3 (g) n: 3 moles 1 mole 2 moles V: 3 volumes 1 volume 2 volumes F) Standard Conditions a set of conditions for basic reference in comparing properties of gases STP - Standard Temperature and Pressure T std O o C (273 K) P std 1 atm At STP, one mole of a gas occupies a volume of 22.4 L The above statement is true for any gas. Std. molar volume V m 22.4 L/mol

6 III) Ideal Gas Law A) Ideal Gas Law equation so far we discussed the effects of isolated variables of P, V, T, and n now lets combine these relations into one equation 1) Ideal Gas Law know: V 1/P (Boyle's Law); V T abs (Charles' Law); V n (Avogadro's Law) combine all three laws into one statement V nt P set up equality VP nt cons tan t R PV n R is the ideal gas law constant Ideal Gas Law Ideal gas assumptions: molecules that do not attract of repel one another (no interactions between molecules) volume of a gas molecules is negligible compared to the volume of the container (molecules have negligible volumes) The ideal gas law equation describes a gas fairly well within reasonable conditions: low to moderate pressures and not too low temperatures. This equation breaks down at high pressures and low temperatures. 2) Gas Law constant What is the value of the gas law constant, R? use 1 set of conditions to solve the ideal gas law equation 1 mole of gas at STP PV n P 1 atm; T 273 K; n 1 mole; V 22.4 L R PV nt (1 atm)(22.4 L) (1 mol)(273 K) atm L mol K

7 It is possible to determine a value for R using a different set of units i.e. using P 760 mmhg; T 273 K; n 1 mole; V 22.4 L R PV nt (760 mmhg)(22.4 L) (1 mol)(273 K) 62.4 mmhg L mol K example: A tank with fixed volume of 438 L is filled with kg O 2. Calculate the pressure of oxygen at 21 o C. PV n n 885 g O 2 1 mol O g O mol P n (27.7 mol) V 438 L atm L mol K (294 K) 1.53 atm

8 B) Density and the Ideal Gas Law recall: d m V The ideal gas law can be rearranged to give the density. 1) define mole (n) mole mass molar mass n m M M 2) substitute n m/m m into the ideal gas law PV n m M m 3) rearrange equation to solve for the density, m/v PV m M m PM m m V PM m m V d example: Calculate the density of ammonia (NH 3 ) in grams per liter (g/l) at 752 mmhg and 55 o C. d m V PM (752 mmhg) 1 atm m 760 mmhg 17 g mol g atm L mol K (328 K) L C) Molar mass of an unknown gas The ideal gas law can be used to find the molar mass of an unknown gas The ideal gas law can be rearranged to give the density. 1) define mole (n) n m M m 2) substitute n m/m m into the ideal gas law PV n m M m

9 3) rearrange equation to solve for the molar mass, M m PV m M m M m PV m M m m PV example: An organic chemistry isolates a colorless liquid with properties of cyclohexane (Mm 84.2 g/mol) from a petroleum sample. She obtains the following data to determine molar mass: V 213 ml, T o C, P 54 torr, mass of the flask and gas g, mass of flask g. Is the molar mass consistent with the liquid being cyclohexane? P 754 torr 1 mmhg 1 atm 1 torr 760 mmhg atm T o C K m g g g M m m (0.582 g) atm L PV mol K (373 K) (0.992 atm)(0.213 L) 84.3 g mol D) Stoichiometry problems involving gas volumes The ideal gas law is useful for calculating the volume of gas from a given amount of gas at specified conditions. Many reactions involve solid or liquid reactants and produce a gaseous product. In these types of reactions it is useful to know the volume of gas produced from a given amount of reactant. example: The equation for the metabolic breakdown of glucose is the same as that for the combustion of glucose in air. C 6 H 12 O 6 (s) + 6 O 2 (g) 6 CO 2 (g) + 6 H 2 O (l) Calculate the volume of CO 2 produced at 37 o C and 1.00 atm when 5.60 g of glucose is used up in the reaction. First need to calculate the amount of CO 2 produced. Then using the ideal gas law, the moles of gas can be converted into the volume of gas g C 6 H 12 O 6 1 mol C 6H 12 O g C 6 H 12 O 6 6 mol CO 2 1 mol C 6 H 12 O mol CO 2

10 PVn atm L mol K V n (0.187 mol) P (1.00 atm) (310 K) 4.75 L Practice: A ml flask containing a sample of octane, a component of gasoline, is placed in a boiling water bath in Denver, where the atmospheric pressure is 634 mm Hg and water boils at 95.0 o C. The mass of the vapor required to fill the flask is 1.57 g. What is the molecular weight of octane? If the empirical formula is C 4 H 9, what is the molecular formula? IV) Gas Mixtures A) Dalton's Law of Partial Pressures In a mixture of unreacting gases, the total pressure is the sum of the partial pressures of the individual gases. P T P A + P B + P C +... partial pressure - pressure of the individual gas component in the mixture Unreacting gases behave independently. Let's look at a mixture of N 2 and H 2 gases, where the volume and temperature are the same for both gases. P N2 n N 2 V Dalton's Law P T P N2 + P H2 and P H2 n H 2 V Substitute in the partial pressures of N 2 and H 2 gases and simplify P T n N 2 V + n H 2 V (n N 2 + n H2 ) V n T V P T n T V

11 It would be useful to have a simple relationship between total pressure of a gas (P T ) and partial pressure of a gas (P A ). We need to define a new term: mole fraction (χ) each component contributes a fraction of the total number of moles in the mixture in general: χ A n A n T n A (n A + n B + ) Note: the sum of the mole fractions equals one χ A + χ B + n A n + n B T n + (n A + n B + ) T n n T T n 1 T n A + n B + n T The mole fraction will be useful in defining a simple relationship between P T and P A. Set up a ratio of P A to P T P A P T n A V n T V simplify P A P T n A n T χ A rearrange P A χ A P T relationship between P T and P A B) Collecting gases over water refer to figure 5.16 in Ebbing A gas is bubbled through water which carries some of the water into the vapor phase The amount of water vapor depends on the temperature Table 5.6 p203 in Ebbing gives a list of P H2O at different temperatures vapor pressure of water partial pressure of water vapor If gas is collected over water, the vapor pressure of water must be subtracted from the total pressure. P T P gas + P H2O P gas P T P H2O

12 Example: Oxygen gas generated in the decomposition of potassium chlorate is collected over water. The volume of the gas collected at 24.0 o C and atmospheric pressure of 762 mmhg is 128 ml. Calculated the mass (in grams) of oxygen gas obtained? At 24.0 o C, P H2O 22.4 mmhg; molar mass of O 2 32 g/mol P O2 P T P H2 O 762 mmhg 22.4 mmhg 740 mmhg PV m M m m PVM (740 mmhg)(0.128 L) g m 32.0 mol g O mmHg L mol K (297 K) Practice: How many moles of O 2 would be obtained from g KClO 3? 2 KClO 3 (s) 2 KCl (s) + 3 O 2 (g) If this amount of O 2 were collected over water at 23 o C and a total pressure of 745 mmhg, what volume would this gas occupy? answer: V L

13 V) Kinetic Molecular Theory A) Postulated of Kinetic Theory Kinetic energy: energy in motion KE 1 2 mu2 mass speed 1) Particle Size (postulate 1) size of gas particles is negligible compared to average distance between them 2) Particle Motion (postulates 2-4) molecules are in constant random straight-line motion in all directions various speeds collisions are elastic no lost of KE molecules do not influence each other no forces, attractive or repulsive, between molecules 3) Particle Speed, Temperature, Energy (postulate 5) at given temperature, all gases have same average KE T, molecular KE (Note: increase in) T KE 1 2 mu2

14 B) How theory explains behavior of gas samples 1) Origin of Pressure molecules collide with the surface; this exerts a force number of molecules; number of collisions P (postulates 2-4) 2) Boyle's Law (V α 1/P, const. T) P; number of collisions (distance between molecules ) V; frequency of collisions P gases are compressible due to large spaces between molecules (postulate 1) V 3) Charles' Law (V α T, const. P) T; molecular speed, KE frequency of collisions and energetically creating internal pressure; result V to restore constant P (postulate 5)

15 VI) Graham's Law of Diffusion and Effusion A) Diffusion the gradual mixing of molecules of one gas with molecules of another by virtue of their kinetic properties. (movement of one gas through another) If molecular speeds are fast, then why does diffusion takes a long time? reason: molecules experience numerous collisions B) Grahams' Law of diffusion rate 1 M m r 1 r 2 M m2 r 1, r 2 - diffusion rates of gases 1 and 2 M m1 M, M m1 m2 - molar mass of gases 1 and 2 This law is deduced from Kinetic Theory two gases at the same Temperature have same KE Derivation: KE 1 2 m 1u 1 2 KE 1 2 m 2u m 2 1u m 2 2 2u 2 m 1 u 1 2 m 2 u 2 2 u 1 2 u m 2 2 m 1 2 u - speed of gas molecules m - mass of gas multiply both sides by 2 rearrange by gathering common variable take square root of both sides u 1 m 2 u 2 m 1 Graham's Law of diffusion

16 C) Effusion process by which gas under pressure escapes one compartment of a container to another by passing through small openings (process in which a gas flows through a small hole in a container) effusion of gas into a vacuum rate of effusion of a gas also given by Grahams' Law of diffusion lighter gas effuse faster than heavier gases at given T & P r 1 r 2 M m2 M m1 Note: rate 1 time therefore r 1 r 2 t 2 t 1 substituting time into Grahams' Law of diffusion or effusion for rate t 2 t 1 M m2 M m1 Example: Both H 2 and He have been used as the buoyant gas in blimps. If a small leak were to occur, which gas would effuse more rapidly and by what factor? M He 4.00 g/mol M H g/mol r H2 r He M He 4.00g/mol M H2 2.02g/mol H 2 effuses 1.41 times faster than He Practice: If it takes 1.25 min. for 0.01 mol He to effuse, how long will it take for the same amount of ethane (C 2 H 6 ) to effuse? answer: t C2H min.

17 VII) Real Gases: Deviation from Ideal Behavior A) Differences between ideal and real gases 1) Ideal gases assume no interaction between molecules (either attractive or repulsive) volume of molecules is negligibly small compared to container 2) Real gases know molecules have a finite molecular volume interactions between molecules exist B) van der Waals Equation van der Waals (1873) interpretation of real gas behavior at the molecular level 1) Pressure approach of a molecule toward the wall of a container intermolecular attractions soften collision impact overall effect lowers gas pressures more than expected P ideal P real + an2 V 2 a - proportionality constant n - number of moles of gas V - volume of gas (interaction between molecules depends on frequency that two molecules approach once another) 2) Volume volume occupied by a gas molecules occupy finite (although small) amount of space volume inside the container available to gas molecules is reduced V ideal V real nb n - number of moles of gas b - constant recall ideal gas Law PV n

18 substitute in van der Waals' corrections for P and V P + an2 V 2 adjusts the P up to account for the decreased impact of collisions by attractive interactions (V nb) n adjusts the V down to include only the free volume not the volume of the container taken up by gas molecules van der Waals' equation Table 5.7, page 214 in Ebbing lists van der Waals constants (a & b) for some gases Example A quantity of 3.50 mol NH 3 occupied 5.20 L at 47 o C. Calculate the pressure of the gas (in atm) using a) ideal gas Law b) van der Waals equation (van der Waals constants are a 4.17 atm L 2 /mol 2 and b L/mol) a) PV n P n (3.50 mol) V 5.20 L P atm atm L mol K (320K) b) P + an2 (V nb) n V 2 P n (V nb) an2 V 2 P (3.50 mol) atm L mol K (320 K) 5.20 L (3.50 mol) L mol L2 4.17atm mol 2 (3.50 mol) 2 (5.20 L) 2 P atm

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