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## Chapter 16 Thermal properties of matter

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**Chapter 16 Thermal properties of matter**• Equations of state • Molecular properties of matter • Kinetic-molecular model of an ideal gas • Heat capacity of gas • Molecular speeds • Phase of matter**Equation of state**• The variables which can describe the states of material are called “ state variables” • The relationship among state variables is called “equations of state” • Simplified by equation • If complicated, we can describe by graph or tables.**The ideal gas equation**• important state variables of gas are P, V, T, n, r, etc. • The relationship arises from gas laws such as Boyle’s law, Charle’s law and Guy Lussac’s law. P1 V1 n1 T1 state 1 P2 V2 n2 T2 state 2 2 states PV = nRT = NkBT**Other equation of states**2 states m = total mass m= nM M = Molar mass P1 V1 m1 N1, r1, T1 state 1 N = total number of molecules state 2 P2 V2 m2 N2, r2, T2 r =density (kg/m3)**The atmospheric pressure at different height**• From • Ideal gas equation M = Molar mass (kg/mol) y = height from sea level (m) If y1 = 0, then P1 = P0 =1.013 x 105 Pa**The van der Waals Equation**• Ideal gas ignores the volume of the molecules and the attractive forces between them. • van der Waals equation take these two omissions into account. • The interaction between gas atoms is called “ van der Waals Interaction”. The van der Waals Equation a and b are empirical constants**The van der Waals Equation**• Term V-nb is the net volume available for the molecules to move around. • The constant a depends on the attractive intermolecular forces. • If gas is diluted, n/V is not significant, the van der Waals equation becomes the ideal gas equation, If n/V is very small**The PV-diagram of a non-ideal gas**• Below TC, the isoterm shows the flat regions in which we can compress the material without an increasing in pressure. • This shaded region is called “Liquid-vapor phase equilibrium region”. • At point a, gas begin to liquefy, the volume is then decreased. • At point b , all gas change phase to liquid. • All temperature over TC, no phase transition occur. TC = Critical temperature**Molecular propertied of matter**• All familiar matter is made up of identical molecules behaving in the same manner. • The interaction of each molecule describes by a force and potential. • Molecules are always in motion, either vibrate or move, causing the kinetic energy. The kinetic energy is generally much less than the potential energy, which mean the molecules are bounded.**Molecular Properties of matter**• solid • molecules vibrate about the fixed point. • arranges in periodic called “ crystal lattice”. • the vibration may be nearly simple harmonic. • liquid • intermolecular distance are slightly greater than in solid phase. • molecules have greater freedom of movement. • liquid shows regularity of structure in the short range.**Molecular Properties of matter**• gas • molecules are widely separated. • molecules have only small attractive forces. • molecules move in a straight line until they collides with another molecule or a wall of container. • ideal gas has no attractive force and has no potential energy.**Kinetic molecular model of an ideal gas**We try to understand the macroscopic properties of gas in terms of its atomic or molecular structure and behavior. Assumptions • A container with contains a very large number N of identical molecules with mass m. • The molecules behave as point particles, their size is very small comparing to distance between particles. • Molecules are in constant motions and collides perfectly elastic. • The container wall is rigid and do not move.**Kinetic molecular model of an ideal gas**number of molecules = v vy vx v vy A vx -vx vxdt I = 2mvx number of molecules moving toward =**Kinetic molecular model of an ideal gas**The force acting on the wall is, v vy vx v vy -vx I = 2mvx where Pressure :**Kinetic molecular model of an ideal gas**Other relations : Ktr = average translational kinetic energy of gas. Ek= kinetic energy of one molecule.**Molecular speeds**We define the root mean square speed, vrms M =Molar mass • vrms relates to the kinetic energy of the gas system.**Collisions between molecules**In the time dt a molecule with radius r will collide with any other molecule within a cylindrical volume of radius 2r and length vdt. 2r The number dN with centers in this cylinder is, vdt number of collision per unit time assume one molecule is moving If all molecules are moving**Collisions between molecules**The mean free time, tmean Time that molecule can move freely The mean free path, l The distance that molecule can move freely.**Heat capacity of gases**T By adding energy dQ into the system of gas, the temperature increases by dT and the kinetic energy increase by dKtr , Ktr V and, T+dT Ktr+dKtr CV = heat capacity at constant volume dQ V J/mol K For ideal gas point particle**Degree of Freedom**3 translational df 2 rotational df 2 vibrational df**Heat capacity of gases**one df energy = Equipartition of energy • Each molecule has 3 degrees of freedom (df) of translations, give • Diatomic with 2 rotations df • Diatomic with 2 rotations df and 2 vibrations df**Heat capacity of solid**• Each atom has 3 df. • solid has both kinetic and potential energy. CV 3R CV =3R “Rule of Dulong and Petit” T**Phases of matter**Triple point = the only condition under which all three phases can coexist. Critical point = the top of the liquid-vapor equilibrium region.