Download Ideal Gas Behavior: Macroscopic and Microscopic Concepts - Prof. Stefan Franzen and more Study notes Physical Chemistry in PDF only on Docsity! Chemistry 331 Lecture 2 Ideal Gas Behavior NC State University Macroscopic variables P, T Pressure is a force per unit area (P= F/A) Temperature derives from molecular motion (3/2RT = 1/2Mu2 ) The force arises from the change in momentum as particles hit an object and change direction. Greater average velocity results in a higher temperature. u M is molar mass Thermal Energy Thermal energy can be defined as RT. Its magnitude depends on temperature. R = 8.31 J/mol-K or 1.98 cal/mol-K At 298 K, RT = 2476 J/mol (2.476 kJ/mol) Thermal energy can also be expressed on a per molecule basis. The molecular equivalent of R is the Boltzmann constant, k. R = NAk NA = 6.022 x 1023 molecules/mol Extensive and Intensive Variables Extensive variables are proportional to the size of the system. Intensive variables do not depend on the size of the system. Extensive variables: volume, mass, energy Intensive variables: pressure, temperature, density Equation of state relates P, V and T The ideal gas equation of state is PV = nRT An equation of state relates macroscopic properties which result from the average behavior of a large number of particles. P Macroscopic Microscopic Transit time The time between collision is ∆t = 2a/ux. velocity = distance/time. time = distance/velocity. a b c ux area = bc Round trip distance is 2a The pressure on the wall force = rate of change of momentum The pressure is the force per unit area. The area is A = bc and the volume of the box is V = abc F = ∆p∆t = 2mux 2a/ux = mux 2 a P = Fbc = mux2 abc = mux2 V Average properties Pressure does not result from a single particle striking the wall but from many particles. Thus, the velocity is the average velocity times the number of particles. P = Nm ux2 V PV = Nm ux2〈 〉 〈 〉 RT is a natural energy scale We can rewrite the ideal gas law in terms of the molar volume The ideal gas law has the form The molar volume at standard T and P V = V/n PV = RT V = RTP = 8.31 J /mol–K 298 K 1.013 × 105 N /m2 = 0.0244 m3 = 24.4 L Microscopic variables Monatomic gases: translation Pressure and temperature can be described solely in terms of the ballistic motion of the gas. Diatomic gases: translation, vibration, rotation Center of mass Quantized energy levels The constant h, known as Planck’s constant gives the scale for quantized energy levels. h = 6.626 x 10-34 J Translation – particle in a box Vibration – harmonic oscillator Rotation – rigid rotator The energy levels for each of these is obtained by solution of the Schrödinger equation. Key points regarding the microscopic view Translational energy levels are so densely spaced that these can be treated using classical methods. We can treat particles as ideal even though they have vibrations and rotations. The dynamics of the gas are not affected. We will see that the heat capacity of the gas is affected by the “internal” degrees of freedom. Key points regarding the microscopic view The kinetic energy of a large number of individual particles is proportional to the temperature of the system. As the system heats up we can picture the molecules moving more rapidly. Pressure results from the net momentum transfer between the particles and wall of the container. Pressure of a dense fluid For a dense fluid (or a liquid) such as water we can think of the pressure arising from the weight of the column of fluid above the point where the measurement is made. The force is due to the mass of water m (kg) accelerated by gravity (g = 9.8 m/s2). P = = = = = ρgh where ρ is the density ρ = m/V. F mg mgh mgh A A Ah V The barometric pressure formula Then we integrate (assuming P=1 at h=0): dP P1 P = – MgRT dh0 h ln P1 = – Mgh RT P = exp – MghRT Isotherms We can plot the pressure as a function of the volume as shown below. Each of the curves on the plot has a constant temperature. Partial pressure For any gas in a mixture of gases the partial pressure is defined as: Pj = xjP where xj is the mole fraction of component j and P is the total pressure. The mole fraction is defined as: xj = nj Σni i
