Pressure Pressure Units Pressure Units •! Pressure is defined as force •! Pressure may be measured in a number of Pressure conversion factors: per unit area. different units: 760 Torr = 1 atm • Pressure is measured with a ! atmosphere (atm): barometric pressure at sea device called a barometer. 760 mm Hg = 1 atm level •! A mercury barometer uses 14.7 psi = 1 atm the weight of a column of Hg Torr: mm of Hg—comes from use of Hg to determine the pressure of barometers 101,325 Pa = 1 atm gas pushing on a reservoir— atmospheric pressure psi: pounds per square inch corresponds to a column Pascal (Pa): official SI units for pressure—1 height of 760 mm. pascal = 1 N m-2 Behavior of Gases Molecular Speed and Energy Molecular Speed and Energy •! Gas-phase molecules and atoms are free to •! Gas molecules travel at a range of speeds •! Average speed can be defined several ways for move about their container—they fill the —some molecules move much faster than molecules: entire volume of the container unlike a others. •! The most probable speed liquid or a solid. •! The average speed corresponds to the speed at the maximum in a plot of of a gas depends on molecules vs speed—if we its molar mass—the could measure the speed of lighter the mass, the individual gas molecules, faster the average more of them would have speed. this value than any other value. 1 Molecular Speed and Energy Molecular Speed and Energy Molecular Speed and Energy -1 Kinetic energy is given by ump(CH4) = 557 m s •! The average kinetic energy of a gas is determined 2 E (CH ) = !(2.664 x 10-26 kg) (557 m s-1)2 by its temperature: ET = !mu T 4 -21 = 4.13 x 10 J E (T) = 3/2 RT/N m = mass u = velocity (speed) T A -1 -1 23 -1 ump(CO2) = 337 m s R is the gas constant mH2 = (2.0158 g mol ) / (6.022 x 10 H2 mol ) E (CO ) = !(7.308 x 10-26 kg) (337 m s-1)2 -1 -1 -1 -1 = 3.347 x 10-24 g = 3.347 x 10-27 kg T 2 R = 8.314 J mol K = .08206 L atm mol K = 4.15 x 10-21 J u = 1.57 x 103 m s-1 •! The kinetic energy of the gas depends only on its mp • Even though the three gases (H , CH , and CO ) temperature, not the identity of the gas. -27 3 -1 2 ! 2 4 2 ET = !(3.347 x 10 kg) (1.57 x 10 m s ) have different speeds, they all possess the same = 4.13 x 10-21 kg m2 s-2 = 4.13 x 10-21 J amount of kinetic energy. Molecular Speed and Energy Molecular Speed and Energy Molecular Speed and Energy •! We can equate the two expressions for kinetic •! The average speed of a gas increases with The average speed of a gas is important because it energy: increasing temperature: determines a number of properties of a gas: ! 2 pressure exerted by a gas—pressure depends ET,avg = ! muavg = 3/2 RT/NA on the rate of collision with the walls of a vessel 2 and the force of those collisions. 1/2 uavg = 3/2 RT/mNA 2 ! collision rate—how frequently gas molecules uavg = 3 RT/mNA collide, and for reactive collisions, have the 1/2 opportunity to undergo reaction. uavg = [3 RT/mNA] ! rate of diffusion—how fast one gas mixes mNA = M (molecular weight) with another 1/2 uavg = [3 RT/M] root-mean-square speed 2 Ideal Gases Ideal Gas Equation Ideal Gas Equation An Ideal Gas has two unique properties that •! The properties of an ideal gas lead to an equation What volume would 2.00 mol of an ideal gas with a distinguish it from real gases that relates the temperature, pressure, and volume pressure of 1000 Torr and a temperature of -25.0 of the gas: o 1. An ideal gas particle has no volume—it is C occupy? simply a point moving through space. PV = nRT (pivnert) (1000 Torr) (1 atm/760 Torr) = 1.32 atm P = pressure (atm) 2. An ideal gas has no intermolecular attractive -25.0 oC + 273.2 oC = 248.2 K V = volume (L) forces—collisions with other ideal gas molecules V = nRT/P or the walls of a container are perfectly elastic— n = number of moles of gas = (2.00 mol)(.08206 L atm mol-1 K-1)(248.2 K) no energy is lost in collisions. T = temperature (K) (1.32 atm) R = 0.08206 L atm mol-1 K-1 V = 30.9 L Variations of the Ideal Gas Law Variations of the Ideal Gas Law Variations of the Ideal Gas Law •! If the system is closed (no input or removal of •! At constant volume: •! At constant pressure: gas from the container), the number of moles of gas cannot change—n is fixed. nR/V = a constant nR/P = a constant ! P /T = P /T ! V /T = V /T Charles’ Law •! At constant temperature: 1 1 2 2 1 1 2 2 nRT = a constant If the pressure [temperature] is increased, the If the volume [temperature] is increased, the temperature [pressure] will increase. temperature [volume] will increase. ! P1V1 = P2V2 Boyle’s Law If the pressure [volume] is increased, the volume If the pressure [temperature] is decreased, the If the volume [temperature] is decreased, the [pressure] will decrease. temperature [pressure] will decrease. temperature [volume] will decrease. If the pressure [volume] is decreased, the volume [pressure] will increase. 3 Variations of the Ideal Gas Law Variations of the Ideal Gas Law Determination of Molecular Weight •! At constant temperature and pressure: Determination of molar mass: A sample of hydrocarbon with m = 1.65 g exerts a V/n = a constant n = m/M M = molecular weight pressure of 1.50 atm in a 945 mL container at 21.5 oC. What is the chemical formula of the ! V1/n1 = V2/n2 Avagadro’s Law Substituting into the Ideal Gas Law: hydrocarbon? If the volume of a gas at constant T and P is PV = mRT/M T = 21.5 oC + 273.2 oC = 294.7 K increased, the amount of gas must increase. Rearranging gives: M = (1.65 g) (.0821 L atm mol-1 K-1) (294.7 K) If the volume of a gas at constant T and P is M = mRT/PV decreased, the amount of gas must decrease. (1.50 atm) (.945 L) -1 = 28.1 g mol -1 formula: C2H4 (ethylene—M = 28.1 g mol ) Dalton’s Law of Partial Pressures Dalton’s Law of Partial Pressures Partial Pressure and Mole Fraction •! When a container is filled with a mixture of •! The total pressure of a gas mixture is given by: •! The mole fraction of a mixture component is gases, Dalton hypothesized that each individual Ptot = ntotRT/V defined as the number of moles of that gas behaved as if it were in a vacuum, i.e., there ntot = total number of moles of all gases in component divided by the total number of moles: is no interaction between different types of gas container molecules that would affect the resulting pressure XA = nA/ntot within the container. Also •! For a gas mixture: Ptot = PA + PB + PC + … •! Each gas in the mixture exerts a pressure equal to the pressure it would exert if no other gases were Dalton’s Law of Partial Pressures nART P n in the container PA = nART/V partial pressure of A A V A = = = XA Ptot ntotRT ntot •! Partial pressure is the pressure exerted by a gas PB = nBRT/V partial pressure of B V in a mixture as if it were in a vacuum. PC = nCRT/V partial pressure of C 4 Real Gases vs Ideal Gases The van der Waals’ Equation The van der Waals’ Equation •! We made the following assumptions in •! To better describe real gases, we can use a Example: Compare the pressure of oxygen defining an ideal gas: different equation of state to predict their determined by the ideal gas law and the van der behavior: Waals’ equation under the following conditions: " no volume—gas behaves as a point n = 25.0 mol T = 375 K V = 20.0 L Van der Waals’ Equation " perfectly elastic collisions 2 -2 -1 a(O2) = 1.36 L atm mol b(O2) = .0318 L mol •! Real gases do not follow these rules—they where a and b are measured constants occupy a volume of space, and there are The vdw b constant is a measure of the volume of Ideal gas: P = 38.5 atm intermolecular forces attracting colliding the gas molecules Van der Waals’: P = 29.7 atm gas molecules The vdw a constant is a measure of the internuclear attractive forces Real Gases vs Ideal Gases •! Gases tend to behave ideally under low pressure conditions " The time between collisions is much longer so there is less relative time for attractive forces to affect pressure (minimizes effect of a constant) " The volume occupied by the gas molecules is much smaller than the total volume of the container (minimizes effect of b constant) 5 .