Kinetic Model for Ideal Gas Three assumptions: (I) The gas consists of molecules of mass m in ceaseless random motion. (ii) The size of the molecules is negligible, in the sense that their diameters are much smaller than the average distance traveled between collisions. (iii) The molecules do not interact, except that they make perfect elastic collisions when they are in contact. Kinetic Models of Gas, Properties of Real Gas Application questions: how fast do you think a gas molecule in the room is traveling? How often do they colloid with each other? Will my use of ideal gas law be valid to gases in the gas cylinder? 1
Derivation of ideal gas law PV = (1/3)nM c C =<s > 1/,root-meansquare speed Consider a container of Volume V, at T, contains n mole gas. Pressure is a result of molecular collisions on the wall. Every collision, the momentum change is mv x, No. of molecules that will collide in time t= (1/)(particles in greenshaded volume). Total momentum change/ t=force Pressure = Force/Area Maxwell Distribution of Speed The speed of molecules in gas actually obey Maxwell distribution of speed: f (s) 3/ ( Ms / RT ) M f ( s) = 4π s exp πrt f(s)ds: is the fraction of molecules that have speeds in the range of s to s+ds.
Maxwell Distribution of Speed Given the Maxwell Distribution of speed, we can obtain Average of speed or mean speed c = 0 8RT sf ( s) dv = πm 1/ Root-mean-square of speed c = <s > 1/ =(3RT/M) 1/ Final result--ideal Gas law Kinetic model leads to PV = (1/3)nM <s > The Maxwell distribution of speed gives c = <s > 1/ =(3RT/M) 1/ Combine these two lead to PV = nrt Note: Pressure is determined by the temperature---the higher the temperature, the larger the pressure Pressure is also determined by the number density, n/v,---the higher the number density, the higher the pressure. 3
More about root-mean-square speed The root-mean-square of speed or the average speed is proportional to (T/M) 1/. ---lighter gas molecules have higher average speed However, the mean kinetic energy of molecules, (1/)M<v >, only depend on temperature, not the molecular masses. It can be argued that thermal equilibrium implies that the mean kinetic energy of molecules must be equal regardless if it is in solid, liquid or gas Real Application questions Let s estimate how fast H and N are traveling in the air. Can we estimate how often the gas molecules will collide with each other? (concept of collision frequency, mean free path). 4
Real gases Real gases show deviations from the perfect gas law because of molecular interaction Repulsive forces assist expansion Attractive forces assist compression Repulsive forces dominant only when molecules are close together on average. Typically this would mean a high pressure. Figure: A typical intermolecular interaction potential Attractions will be significant when molecules are relatively close but not too close. Or at low temperature when molecules do not move fast enough and they can capture each other. Typically moderate pressure, low temperature. Compression factor We can use compression factor Z to quantify the deviations of real gas from ideal gas law Z=PV m /RT or Z = V m /V m0 where V m0 is the molar volume of the ideal gas (i.e,: V m0 =RT/P). According to this definition: Z=1 if the gas obeys ideal gas law Z >1 implying the gas are more difficult to compress than ideal gas (larger molar volume than the ideal gas V m0 )--- repulsive forces dominant. Z < 1 implying the gas are more compressible than ideal gas (less molar volume than V m0 ) ---attractive forces dominant. 5
Some examples of compression factor Graph on the left shows how Z varies with pressure at 0ºC for a few real gases. Observe: 1. At low pressure, all Z ~ 1. Gas behave ideally.. At very high pressure, all Z >1. Repulsive forces dominant. 3. At intermediate pressure, most gases have a range Z <1, attractive forces dominant. Some may not, but will have Z<1 range at lower temperature. Compression factor at different temperature Graph on the left shows how Z would vary with pressure at different temperature T for a given type of gas. Boyle temperature is the temperature at which dz/dp = B (T)=0 (when P 0) or dz/d(1/vm)=b(t) 0. (see the discussion on the virial expansion) 6
Virial Expansion Virial expansion is another way to describe how the real gas behavior deviate from the ideal gas law. One incorporate the deviation by including higher order terms which are absent in the idea-gas law. pv m =RT(1+B P+C P + ) the coefficients, B, C,..are called virial coefficients. They themselves are functions of temperature. According to this, Z=1+B P+C P +. Another way is to write the virial equation is in the order of (1/V m ) B C pv m = RT ( 1+ + + K) V m V m Isotherm of Real Gases Graph on the left is the experimental isotherms of CO gas at several temperature. At 0 C, there is a discontinuity in isotherm at point C-D-E. This is the gas-liquid phase transition. The critical temperature T c is the point below which there is liquidgas phase transition, above which there is not. The critical point is marked by the *. The two-coexisting phases (like C,D merge together to one point at T c that gives the critical point. 7
Van-der Walls equation In 1873, J. H. van der Walls proposed a general equation that can fit many of experimental observed equation of states of real gases. P = nrt/(v-nb) - a (n/v) or : P = RT/(V m -b) - a/v m a, b are called van-der Walls coefficients. They are independent of T, P or V, but are characteristic of molecular nature of the gases. (a/v m ) corrects for the attraction. b corrects for repulsion, can be related to the volume of the molecular spheres, b~ N a (1/6)π d 3. Table 1.5: Van der Walls coefficients Ar CO He Xe a /(atm L /mol - ) 1.337 3.610 0.0341 4.137 b /(L mol -1 ) 0.030 0.049 0.038 0.0516 Exercise: from Van-der-Walls equation, obtain second-virial coefficient B, determine Boyle temperature for Ar. 8
Features of VDW equation Isotherms from VDW equation VDW equation predicts isotherms very similar to that of observed experimental isotherms At high Temperature, isotherms are like ideal gas law. P~RT/(V m -b). At lower temperature, it has van-der-walls loop, signify the liquid-gas phase transition Van-der-walls loop The critical point in VDW equation The critical temperature T c is the temperature below which the van-der-walls develops. It is a well-defined point with specific T c, P c and V c in terms of van-derwalls constants a and b. The critical point is defined by the following two conditions dp dv m d P dv m RT = ( V b) m RT = ( V b) m 3 a + 3 V m 6a V 4 m = 0 = 0 which leads to: V c =3b, P c =a/7b, T c =8a/7Rb so Z c =P c V c /RT c =3/8. 9
The principle of corresponding states Molecular characteristics are reflected in van-der-walls constants a, and b, which in turn determines P c, T c, V c. Let s define P r =P/P c, T r =T/T c, V r =V/V c The observation that the real gases at the same reduced volume and reduced temperature exerts the same reduced pressure is called the principle of corresponding states. Experiments confirm such truth to certain extents. most gases with spherical molecules obey the corresponding states well, but not non-spherical or highly polar molecules. Examples of experimental data confirm principles of corresponding state This figure shows how Z for different gases at same reduced temperature form a common curve. 10
Other examples There are many examples where such plot with reduced variables produce a single curve regardless of molecular nature. Ref: Wang & Teraoka, Macromolecules 30, 8473 (1997) Use of Model Systems The principle of corresponding states reflects the fact that many physical laws are not governed by the molecular characteristics, but are governed by some other physical principles. --- statistical principles. This justify the use of model fluids, model polymer chains where these model systems do not need to include chemical identify. One model fluid that help to understand the liquid-gas phase transition is the Lennard-Jones fluids. One example of model of polymer chains is the selfavoiding walks on the lattice. 11
Hard sphere fluid vs. LJ fluid Many research are done with molecules modeled as purely repulsive hard spheres (no attractive interactions). For this type of fluid, there is no liquid-gas transition. The collections of spheres either are in gas phase, or are in solid phase directly. Adding an attractive term in molecular interactions turns on the liquid-gas phase transition. Study of these model fluid through computer simulations help to illustrate the phase transition greatly. Summary Simple Kinetic model of gas can lead to the ideal gas law. The major assumption in simple kinetic model is that molecules do not attract or repel each other except making perfect collision. Kinetic model also shows that average speed of gas molecule is proportional to (T/M) 1/. Real gas molecules however attract or repel each other at appropriate conditions. This makes the properties of many real gas to deviate from ideal gas law behavior. Compression factor Z can be used to check how the real gas deviates from ideal gas law. 1
Sample problem Solving At what pressure does the mean free path of argon at 5C become comparable to the diameter of a spherical vessel of volume 1.0L that contains it? Take σ=0.36nm How does the mean free path in a sample of gas vary with temperature in a constant-volume container? Express the van-der-walls equation of state as a virial expansion in powers of 1/V m, and obtain expression of B and C in terms of a and b. 13