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Thermal Properties of Matter Dr. Gary Grim 7/24/02 State Equations The conditions under which matter finds itself are described by equations called “state equations,” or “equations of state.” These equations are comprised of “state” variables. Example state variable include pressure, temperature, volume, mass, etc... Ideal Gas Law The state equation for an ideal gas is given by, “the ideal gas law,” PV = nRT The ideal gas law gives the relationship between pressure, temperature, volume and mass, all state variables. The quantity, n, is the mole count, I.e. the number of moles present in the gas, and therefore the mass of the gas is given by: m= nM Where M is the molar mass. Ideal Gas Law R is the gas constant, or ideal gas constant and is given by: R = 8.31 J/(mol K) In units of liters and atmospheres, R is: R = 0.082 L atm/(mol K) For a constant amount of mass, the ratio pV T must be constant if the pressure, volume or temp. change. Van Der Waals EOS The ideal gas law is just that, an ideal law. The van der Waals equation of state is an empirical improvement. 2 an p + 2 (V − nb) = nRT V The constants a & b, must be determined by experiment with each gas. pV Diagrams Molecular Properties As stated previously, all matter is made of atoms and molecules. For our purposes, we’ll not distinguish between the two for now. We will idealize the molecules and atoms as point objects for most of our discussion, but their structure gives rise to important thermodynamic properties. Molecules in all matter are always in motion and have a kinetic energy associate with them. This is true regardless of the state of the matter. Molecular Properties For a gas of molecules, the force between molecules is generally small and attractive at large distances, and large and repulsive at very small distances. At a distance which corresponds to the distance between molecules in a solid or liquid, the inter-molecular forces are roughly zero, and any deviations from this distance results in a restoring force. The molecules are said to be resting in a potential well. Molecular Properties For molecules with a kinetic energy which is larger than the height of the “potential well,” the molecules do not remain “bound” to each other, resulting in a “gas” phase of matter. For molecules with a kinetic energy which is smaller than the potential well, the molecules oscillate with some amplitude around the average inter-molecular spacing. The matter is in either a liquid or solid phase for this situation. A liquid has a higher energy than a solid, thereby allowing a molecule to escape the potential well with one molecule and be captured by the well of another. Molecular Properties For a solid the probability for a molecule to escape the potential well with another molecule is very small. At low temperatures, matter is in the solid phase. As the temperature rises, the matter melts and then vaporizes. From a molecular perspective, temperature and kinetic energy are closely related! Moles & Gophers The book defines one mole as the amount of substance that contains as many elementary entities as there are atoms in 12 g of carbon 12. Translation: there are NA, molecules in a mole, where NA, is given by Avogadro’s number, 6.02 x 1023 molecules/mole. Thus, molar mass, the mass of 1 mole of a compound, is given by: M = NAm, where m is the mass of one molecule. Kinetic Theory of Gas We’d like to use our molecular model of a gas to form a theory on the macroscopic properties of the gas. Our assumptions include: 1) A container of volume V, with a very large number, N, of identical gas molecules, each with mass m. 2) The molecules behave as point objects 3) The molecules are in constant motion, obeying the laws of mechanics. 4) The container walls are rigid and immovable. Kinetic Theory of Gas Now, let’s calculate what happens to a molecule when it elastically collides with the wall of the container. In the direction parallel to the wall, (y), the velocity remains constant after collision. In the direction normal to the wall, the velocity reverses direction. Thus the momentum change in the collision is directed along the normal to the wall is equal to: mvx − − mvx = 2mvx Kinetic Theory of Gas Now, for the collision to take place in an amount of time, dt, the molecule will have to be a distance no further than vxdt from the wall. If we now count the number of molecules that strike an area A on the surface of the wall, erroneously assuming, all molecules have an x-component of velocity vx, then the number striking the wall in a time dt, is given by: 1 N Avx dt 2 V The 1/2 comes from the fact that half the molecules are moving away from the wall at any time, while half are moving towards the wall. Kinetic Theory of Gas The N/V term is just the gas density in the volume. 1 N Avx dt 2 V The total momentum change in a time dt, then is just 2mvx times this quantity: NAmvx2 dt dPx NAmvx2 dPx = ⇒ = V dt V Employing Newton’s second law, this is just the force being exerted on the wall by the gas and therefore pressure is: F Nmv x2 p= = A V Kinetic Theory of Gas The gas is made up of a distribution of velocities, and what we’re interested in describing is the average pressure exerted by the gas on the walls not the pressure from one particular range of velocities, therefore: v → (v ) 2 x 2 x avg 1 2 = (v )avg 3 This gives: 1 2 2 1 2 pV = Nm (v )avg = N m(v )avg 3 3 2 Kinetic Theory of Gas The in square brackets represents the average translational kinetic energy of the individual gas molecules. We can rewrite the equation as: 2 pV = K tr 3 where K is the total translational kinetic energy of the gas. Comparing this with the ideal gas law, pV=nRT, 3 K tr = nRT 2 Kinetic energy is proportional to temperature! Kinetic Theory of Gas If we divide this result by N, the total number of molecules, and remember that n, the number of moles in the gas, is given by N/NA, then we get: K tr 1 3 R 2 = m(v )avg = T N 2 2 NA Thus, the average kinetic energy of a molecule is given by: 1 3 2 m(v )avg = kT 2 2 Where k = R/NA is Boltzmann’s constant and is equal to 1.381 x 10-23 J/(molecule K) Kinetic Theory of Gas The average velocity of a gas molecule, may now be determined by the temperature and mass. The root mean square of a velocity is given by: vrms = ( ) v 2 avg = 3kT = m 3RT M The reason for using rms, is that (vavg)2 will be zero for a gas confined to a container. Kinetic Theory The molecules in our container don’t just collide with walls. They also collide with themselves. Clearly, the more dense the gas, the more frequent the collisions, and the larger the molecules are, the more frequent the collision. The “mean free path” is the average distance a molecule travels between collisions and is given by the velocity of the molecule times the mean time between collisions: λ = vtmean = V 4π 2r 2 N Kinetic Theory N/V is the number of molecules per unit volume (number density), and r is the radius of a molecule. Substituting in for macroscopic quantities, we can recast the formula as: kT λ= 4π 2r 2 p Heat Capacity From our kinetic theory, we just calculated that the translational kinetic energy of the gas was given by: K tr = 3 3 nRT ⇒ dK tr = nRdT 2 2 where the last form represents the “change in energy” as the temperature changes, i.e., “heat,” or energy in motion. Thus, 3 nRdT ⇒ 2 3 Cv = R 2 nCv dT = Heat Capacity The last result was for “point” masses, meaning the only degrees of freedom for motion are translational, as in the case of “monatomic” gasses. If the molecules are “diatomic” or “polyatomic,” then the along with translational degrees of freedom the gasses have rotational degrees of freedom, at least along axes not parallel to the axis through the diatom. This means kinetic energy of the translational motion and the rotational motion must be considered, along with any vibrational motion associated with the atomic bonding. Heat Capacity For a diatomic molecule with rotational degrees of freedom, the heat capacity is given by: 5 Cv = R = 20.79J /(mol ⋅ K ) 2 Adding vibrational degrees of freedom to a diatom gives: 7 Cv = R 2 Heat Capacity For solids, we can carry out the same analysis as above. Details are in the text. The punchline is that heat capacity is given by: Cv = 3R This is known as the rule of Dulong and Petit, and pretty much fails miserably at low temperatures, but works for higher temps. Need solid state theory! Phases Phase equilibrium -- a locus of (p,V,T) point where two phases of matter may coexist. Phase diagram -- a 2-D plot of p vs. T showing the phases of matter at a given volume. Triple point -- the point on the phase diagram where all three phases may coexist. Critical Point -- the maximum pressure and temperature at which the material may exist in liquidgas phase equilibrium. Phases T c, pc -- The critical point temperature and pressure. A gas at a pressure above the critical pressure will not separate into two phases when cooled at constant pressure. Instead the gas will gradually and continuously from the gas state to the liquid state without a phase transition.