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# Gases (Chapter 10)

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1. Gases (Chapter 10) • Rather than considering the atomic nature of matter we can classify it based on the bulk property: gaseous, liquid or solid. • Gases are the most easily understood form of matter (we shall see why). Air is an example of a complex mixture of gases: gases form homogeneous mixtures regardless of identities or proportions (unlike liquids and solids). Gases expand to fill any container, and are highly compressible (unlike liquids and solids) These characteristics arise because the molecules of gas are very far apart and don’t (mostly) interact. Different gases thus behave similarly.

2. Pressure • Pressure is the force that acts on a given area (P=F/A). • Gravity on earth exerts a pressure on the atmosphere: atmospheric pressure. • We can evaluate this by calculating the force due to acceleration (by gravity) of a 1m2 column of air extending through the atmosphere (this has a mass of ~10,000kg). This unit is a Newton (N) This unit is a Pascal (Pa)

3. Units of Pressure S.I. unit of pressure is the N/m2, given the name Pascal (Pa). A related unit is the bar (1x105 Pa) used because atmospheric pressure is ~ 1x105 Pa (100 kPa, or 1bar). Torricelli (a student of Galileo) was the first to recognise that the atmosphere had weight, and measured pressure using a barometer Standard atmospheric pressure was thus defined as the pressure sufficient to support a mercury column of 760mm (units of mmHg, or torr). Another popular unit was thus introduced to simplify things, the atmosphere (atm = 760mmHg).

4. Pressure • Atmospheric pressure and relationship between units 1 atm = 760 mmHg = 760 torr = 101.325kPa = 1.01325 bar) Measuring Pressure: the manometer Exercise: On a certain day a barometer gives the atmospheric pressure as 764.7 torr. If a metre stick is used to measure a height of 136.4mm in the open arm, and 103.8mm in the gas arm of a manometer, what is the pressure of the gas sample? (give in torr, atm, kPa and bar).

5. Result • Difference in height is 32.6 mm. Gas inside has greater pressure than prevailing atmospheric pressure: 764.7 + 32.6 mmHg = 797.3 mmHg (Torr) • Convert to atm: divide by 760 = 1.049 atm • Convert to kPa: multiply by 101.325 = 106.3 kPa • Convert to bar: divide by 100 = 1.063 bar

6. Gas Laws • A large number of experiments have determined that 4 variables are sufficient to define the physical condition (or state) of a gas: the gas laws. Boyle’s Law, Charles’ Law, Avogadro’s hypothesis Robert Boyle: (1627-1691) the first modern chemist, known as the father of chemistry. His 1661 book The Sceptical Chymist marks the introduction of the scientific method, a definition of elements and compounds and a refutation of alchemy and magic potions. Boyle biography

7. Boyle’s Law • Boyle investigated the variation of the volume occupied by a gas as the pressure exerted upon it was altered and noted that the volume of a fixed quantity of gas, at constant temperature is inversely proportional to the pressure

8. Charles’ Law • A century later, a French scientist, Jacques Charles discovered that the volume of a fixed amount of gas, as constant pressure, is proportional to the absolute temperature. Cool a balloon, or a sealed plastic bottle, to verify this! It was recognised (by William Thomson, Lord Kelvin, a Belfast born physicist) that if the graph was extrapolated to zero volume, an absolute zero of -273.15 oC is obtained.

9. Avogadro’s Law • Relationship between quantity of gas and volume established by Gay-Lussac (balloon science!) and Avogadro in the 19th Century. Result was Avogadro’s hypothesis: equal volumes of gases at the same temperature and pressure contain equal numbers of molecules Experiments show that 22.4L of gas at 0oC and 1atm (STP), or 24.8L of gas at 298.15 K and 1 bar (SATP), contains 6.022 x 1023 molecules (Avogadro’s number, NA) Avogadro’s law: volume of gas at constant temperature and pressure is proportional to the number of moles of gas (n) Remember: 1 mole = Avogadro’s number of objects

10. Putting it all together Boyle, Charles, Avogadro Combine Call proportionality constant R (gas constant) Ideal Gas Equation

11. A note on units and dimensional analysis SI unit for R is J/mol.K or m3.Pa/mol.K (R=8.315 of these units) Need to use the units of Pa for pressure and m3(=1000L) for volume in any calculation. Alternatively you can use units of kPa and L. If you wish to use atm and L (as in USA and Textbook) R=0.0826 L.atm/mol/K. Always use absolute temperature scale (K)

12. Exercises • What is the volume of 1 mole of an ideal gas under standard temperature and pressure (STP)? • How many moles (g) of CO2 is liberated into a 250mL flask when a pressure of 1.3atm is found upon heating calcium carbonate to 31oC? • If a metal cylinder holds 50L of oxygen at 18.5atm and 21oC, what volume will the gas occupy at 1atm and same T?

13. Gases in chemical reactions More Exercises If the pressure of a gas in an aerosol can is 1.5atm at 21oC, what would the pressure be if can is heated to 450oC? What is the density of carbon tetrachloride vapour at 714torr and 125oC? See student activities If an air bag has a volume of 36L and is to be filled with nitrogen gas at a pressure of 1.15atm and 26oC, how many grams of NaN3 must be decomposed?

14. Gas mixtures • Dalton’s Law of partial pressures The total pressure of a mixture of gases equals the sum of the pressures that each would exert if it were present alone PT=P1+P2+P3+….Pn Exercise: A gaseous mixture is made from 6.00g oxygen and 9.00g methane placed in a 15L vessel at 0oC. What is the partial pressure of each gas and the total pressure in the vessel?

15. Aside: A wronged chemist? John Dalton, is credited with the formulation of the atomic theory. This was disputed by William Higgins, an Irish chemist from Colloney, Sligo, who claimed to have been the first to postulate the theory in his book Comparative view of the phlogistic and anti-phlogistic theories (1789) a work very critical of the Galway chemist Richard Kirwan. (Atkinson, E. R. The Atomic Hypothesis of William Higgins, J. Chem. Ed. 1940, 17(1), 3-11).

16. Mole Fractions • The ratio n1/nT is called the mole fraction (denoted x1), a dimensionless number between 0 and 1. Mole fraction of N2 in air is 0.78, therefore if the total barometric pressure is 760 torr, the partial pressure of N2 is (0.78)(760) = 590 torr.

17. Kinetic –Molecular Theory Theory describing why gas laws are obeyed (explains both pressure and temperature of gases on a molecular level). • Complete form of theory, developed over 100 years or so, published by Clausius in 1857. • Gases consist of large numbers of molecules that are in continuous, random motion • Volume of all molecules of the gas is negligible, as are attractive/repulsive interactions • Interactions are brief, through elastic collisions (average kinetic energy does not change) • Average kinetic energy of molecules is proportional to T, and all gases have the same average kinetic energy at any given T. Because each molecule of gas will have an individual kinetic energy, and thus individual speed, the speed of molecules in the gas phase is usually characterised by the root-mean-squared (rms) speed, u,(not the same though similar to the average speed). Average kinetic energy є = ½mu2

18. Application to Gas Laws • Increasing V at constant T: Constant T means that u is unchanged. But if V is increased the likelihood of collision with the walls decreases, thus the pressure decreases (Boyle’s Law) • Increasing T at constant V: Increasing T increases u, increasing collisional frequency with the walls, thus the pressure increases (Ideal Gas Equation).

19. Molecular speeds and mass • The average kinetic energy of gases has a specific value at a given temperature. The rms speed of gas composed of light particles, He, is higher than that for heavier particles, Ne, at the same temperature. • Can derive an expression for the rms speed (from kinetic theory) M is the molar mass See student activities This gives rise to interesting consequences: effusion

20. Effusion • Thomas Graham (1846) discovered that effusion is inversely proportional to the square root of molar mass. Derived from comparison of rms speeds

21. REAL GASES 1. Molecules have volume 2. Molecules have attractive forces (intermolecular) Deviations from ideal gas law WHY? • V-nb • -a(n/V)2 Van der Waals Equation of State

22. Van der Waal’s constants a correlates with boiling point (see later) b can be used to estimate molecular radii