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Chapter 1. the Properties of Gases

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Chapter 1. the Properties of Gases Chapter 1. The Properties of Gases 2011 Fall Semester Physical Chemistry 1 (CHM2201) Contents The Perfect Gas 1.1 The states of gases 1.2 The gas laws Real Gases 1.3 Molecular interactions 1.4 The van der Waals equation States of Matter Gases – fill container, random rapid motion, never coming to rest or clumping together. Motion is mainly translational Liquids – fixed volume, flow and assume shape of container, only slightly compressible, stronger forces hold molecules together. Motion is mainly translational Solids – fixed volume, definite shape, generally less compressible than liquids, forces hold particles in a fixed shape. Motion is mainly vibrational 1.1 The states of gases Key points 1. Pressure provides a criterion of mechanical equilibrium for systems free to change their volume 2. Pressure is measured with a barometer 3. The Zeroth Law of thermodynamics 1.1 The states of gases Physical State 1. Two samples of a substance that have the same physical properties are in the same state 2. Physical Properties : V, n, p, and T 3. It is sufficient to specify only three of these variables 4. p = f(T,V,n) : an equation of state (e.g.) p = nRT/V for a perfect gas 1.1 The states of gases (a) Pressure • Defined as force, F, divided by the area, A • p = F/A Mechanical equilibrium 1.1 The states of gases (b) The measurement of pressure • barometer • Hydrostatic pressure : P = ρgh ü ρ : the mass density of liquid ü g : gravitational acceleration ü h : height 1.1 The states of gases (c) Temperature • Indicates the direction of the flow of energy through a thermally conducting rigid wall • Types of boundary between two objects at different temperatures ü Diathermic : a change of state is observed ü Adiabatic : no change occur • Thermal equilibrium : No change of state occurs when two objects are in contact through a diathermic boundary 1.1 The states of gases (c) Temperature Thermal equilibrium 1.1 The states of gases The Zeroth Law of thermodynamics • If A is in thermal equilibrium with B, and B is in thermal equilibrium with C, then C is also in thermal equilibrium with A • Justifies the concept of temperature and the use of a thermometer • Celcius scale : °C • Thermodynamic temperature scale : K (kelvins) T/K = θ/°C + 273.15 The perfect gas : 1.2 The gas laws Key points 1. The perfect gas law summerizes Boyle’s and Charles’s laws and Avogadro’s principle 2. The kinetic theory of gases provides a model for the gas laws 3. A mixture of perfect gases behaves like a single perfect gas 1.2 The gas laws (a) The perfect gas law • Boyle’s law : pV = constant at constant n, T • Charles’s law : V = constant × T at constant n, P P = constant × T at constant n, V • Avogadro’s principle : V = constant × n at constant p, T isotherm isobars isochores 1.2 The gas laws (a) The perfect gas law • pV = nRT : perfect gas law • R : gas constant • Perfect (ideal) gas vs. Real gas 1.2 The gas laws (a) The perfect gas law • pV = nRT : perfect gas law • R : gas constant • Perfect (ideal) gas vs. Real gas • The perfect gas law is important because it is used to derive a wide range of relations that are used throughout thermodynamics 1.2 The gas laws (b) The kinetic model of gases 1. The gas consists of molecules of mass m in ceaseless random motion 2. The size of the molecules is negligible, in the sense that their diameters are much smaller than the average distance travelled between collisions 3. The molecules interact only through brief, infrequent, and elastic collisions 1.2 The gas laws (b) The kinetic model of gases 1 pV = nMc2 3 M : the molar mass of the molecules c : the root mean-square speed of the molecules 1/2 1/2 !3RT $ c = v2 = # & " M % pV = nRT 1.2 The gas laws (c) Mixtures of gases • Dalton’s law : The pressure exerted by a mixture of gases is the sum of the pressures that each one would exert if it occupied the container alone. pJ = xJ p : partial pressure n x = J : mole fraction J N pA + pB += (xA + xB +)p = p The real gas : 1.3 Molecular Interactions Key points 1. The extent of deviations from perfect behavior is summarized by the compression factor 2. The virial equation is an empirical extension of the perfect gas equation 3. The isotherms of a real gas introduce the concept of vapor pressure and critical behavior 4. A gas can be liquefied by pressure alone only if T ≤ TC. 1.3 Molecular interactions • Repulsive forces and attractive forces assist expansion and compression, respectively 1. At low pressure, the intermolecular forces play no significant role 2. At moderate pressures, the attractive forces dominate 3. At high pressures, the repulsive forces dominate 1.3 Molecular interactions (a) Compression factor Vm pVm Z = o = : Compression factor Vm RT 1. At low pressure, Z ≈ 1 2. At moderate pressures, Z < 1 3. At high pressures, Z > 1 1.3 Molecular interactions (b) Virial coefficients The virial equation of state 2 pVm = RT(1+ B' p +C' p +) ! B C $ = RT #1+ + 2 +& " Vm Vm % = RTZ dZ = B'+ 2pC'+→ B as p → 0 dp The virial coefficients depend on T dZ/dp⟶0 as p⟶0 at Boyle temperature 1.3 Molecular interactions (c) Condensation • A liquid appears • The pressure is vapor constant pressure • Two phases exist Deviate from the perfect gas law Relatively ideal Experimental Isotherms of carbon dioxide 1.3 Molecular interactions (c) Critical constants • At TC, a surface separating two phases does not appear • Critical Point : the volumes at each end of the horizontal part of the isotherm merge to a critical point • TC : critical temperature • pC : critical pressure • VC : critical molar volume • At T > TC, the supercritical fluid Experimental Isotherms of carbon dioxide 1.4 The van der Waals equation Key points 1. The van der Waals equation is a model equation of state with two parameters 2. The van der Waals equation captures the general features including the critical behavior 3. The properties of real gases are coordinated by expressing their equations of state in terms of reduced variables 1.4 The van der Waals equation (a) Formulation of the equation nRT n2 RT a p = − a 2 = − 2 V − nb V Vm − b Vm • a : the strength of attractive interactions • b : the volume taken by molecules 1.4 The van der Waals equation (b) The features nRT n2 RT a p = − a 2 = − 2 V − nb V Vm − b Vm • Perfect gas isotherms are obtained at high T and large molar volumes • Liquid and gases coexist when cohesive and dispersing effects are in balance • The critical constants are related to a and b The unrealistic Van der Waals loops need be treated by the Maxwell construction 1.4 The van der Waals equation (b) The features nRT n2 RT a p = − a 2 = − 2 V − nb V Vm − b Vm • The critical constants are related to a and b dp RT 2a = − 2 + 3 = 0 dVm (Vm − b) Vm d 2 p 2RT 6a 2 = 3 − 4 = 0 dVm (Vm − b) Vm a 8a V = 3b, p = , T = C C 27b2 C 27Rb pCVC 3 ZC = = RTC 8 1.4 The van der Waals equation (b) The principle of corresponding states nRT n2 RT a p = − a 2 = − 2 V − nb V Vm − b Vm • Reduced variables Vm p T Vr = , pr = , Tr = VC pC TC 8Tr 3 pr = − 2 3Vr −1 Vr • Real gases at the same reduced volume and reduced temperature exert the same reduced pressure .
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