Department of Chemistry Seminar Series Professor Jin Zhong Zhang Department of Chemistry University of California @ Santa Cruz Tuesday, February 23rd Duncan Hall 135 4:30 p.m. Unique optical properties and applications of nanomaterials in solar energy conversion and cancer therapy.
van der Waals equation EOS Real Gases Under many conditions, real gases do not follow the ideal gas law ... -- Intermolecular forces of attraction cause the measured pressure of a real gas to be less than expected -- When molecules are close together, the volume of the molecules themselves becomes a significant fraction of the total volume of a gas TexPoint fonts used in EMF. Read the TexPoint manual before you delete this box.: AAAAAA
The van der Waals model of real gases For real gases – both quantitative and qualitative deviations from the ideal gas model U(r) U(r) U(r) Electric interactions between electro-neutral molecules : r van der Waals attraction repulsion attraction The van der Waals equation of state for real gases
EOS van der Waals Equation Corrections for real gas behavior are made using the parameters a and b a – accounts for intermolecular attractions in real gases b – accounts for the real volumes of gases
U(r) U(r) U(r) Compression Factor The compression factor (Z) shows deviation from ideal gas behavior. Z=1 for an ideal gas. Z<1 means the gas is more easily compressed. Z>1 means it’s more difficult to compress. • The molar volume Vm=V/n is the volume taken up per mole of gas r • At very low densities (pressures) the gases approach ideal behavior • At Intermediate densities(pressures) attractive contributions dominate and the gases are more compressible (generally) than ideal • At high densities (pressures) the excluded volume effects dominate.
Dalton’s Law • In a mixture of gases, the partial pressure of each gas is the pressure it would exert if it were alone in the container. • The total pressure of a gas mixture is equal to the sum of the partial pressures of its components. • The partial pressure of component A is PA=nART/V. Therefore the total pressure is: • XA Is the Mole Fraction TexPoint fonts used in EMF. Read the TexPoint manual before you delete this box.: AAAAAAAA
Partial Molar Quantities and Extensive Quantities Consider volume of a system to be a function of the amounts of each substance What would happen if we increased the amount of each component in the system by a factor of ¸ ? (For example, ¸ =2 would be doubling the size of the system) The volume of the system were doubled. Now consider the same effect Temperature:
Extensive Quantities Volume is considered an Extensive property of the system because when the amount (the extent) of the system is doubled, the quantity doubles. Temperature is considered an intensive property of the system because when the amount of the system is doubled (leaving everything else the same), the Temperature is the same. (Consider pouring a full glass or half of glass of water from the same container). Which are extensive and which are intensive? U, P, N
Partial Molar Quantities According to Euler’s Theorem, if the following equation applies: Then: HW: Do this for an ideal gas P,V,T For Volume:
Partial Molar Quantities The total volume is the volume per mole of component 1 times the number of moles of component 1 plus the volume per mole of component 2 times the number of moles of component 2 plus … If we divide by n (the total number of moles in the system) then: Where xn is the mole fraction of component n
Is Volume a conserved quantity? Every good bartender knows that if you add alcohol to water the total volume is not the sum of the two volumes being added! That is why we had to keep the amounts of all the other components fixed when we defined a partial molar quantity.
The Ideal Gas Law or Avogadro’s number NA 6.0220451023 • The assumptions made in the kinetic theory of gases for an ideal gas were that: • The particles were non-interacting • No repulsive forces means that each particle takes up no volume. isotherms The P-V diagram – the projection of the surface of the equation of state onto the P-V plane. Note the isothermals and isobars
Isotherms for Ideal Gas Ideal Gas
Real FluidsA fluid is a gas or a liquid • If I have a certain amount of gas and I cool it down (at constant Pressure) what happens? • If I have a certain amount of liquid at atmospheric pressure and I put it under a vacuum what happens? • What about if I have a gas and increase the pressure?
Isotherms for van der Waals Eqn. of State Real Fluid
Realer Isotherms and Condensation Imagine taking a box containing a gas and reducing it’s volume while keeping its temperature constant. • AB is close to ideal behavior • BC shows deviations from this • CDE is condensation to liquid where there is a jump in the line (called a tie line) • CDE (and blue area) Vapor and Liquid coexist. The vapor pressure of a liquid is the pressure at which the vapor and liquid coexist. If the volume is lowered here, the pressure does not go up it just forces more gas to the liquid state • At E the entire sample is liquid. And compression EF is very difficult.
Real Isotherms and Condensation At the critical point marked by a star: The isothermal compressibility is: Is the slope in the figure to the left What happens to the isothermal compressibility at the phase transition? What happens to at the phase transition? (Imagine cooling water vapor to the boiling point then to right below the boiling point)
Thought Experiments Gedankenexperiment. • Is it easier to compress the gas or the liquid? • At the condensation point (dew point) what is the compressibility of the system? • Can you understand this from a molecular perspective? In terms of excluded volume and interatomic attractions? • Molecules tend to attract eachother at long distances how would you expect a real gas to vary from and ideal gas at low densities? • At high densities what molecular effects dominate the compressibility?
Thought Experiments Gedankenexperiment. • How do you tell that a liquid has condensed? • Where is this in the figure? • Look at each isotherm and mark down where the fluid condenses? • Do all of the gasses condense to a liquid? • At 40 celsius where is it?
Compression Factor The compression factor (Z) shows deviations from ideal gas behavior. Z=1 for an ideal gas. Z<1 means the gas is more easily compressed. Z>1 means it’s more difficult to compress. • The molar volume Vm=V/n is the volume taken up per mole of gas • At very low densities (pressures) the gases approach ideal behavior • At Intermediate densities(pressures) attractive contributions dominate and the gases are more compressible (generally) than ideal • At high densities (pressures) the excluded volume effects dominate.