حرارة وديناميكا حرارية المحاضرة الثالثة د/ محرز لولو

Kinetic Theory Three basic assumptions of the kinetic theory as it applies to gases: 1. Gas is composed of particles- usually molecules or atoms Small, hard spheres Insignificant volume; relatively far apart from each other No attraction or repulsion between particles

2. Particles in a gas move rapidly in constant random motion Kinetic Theory 2. Particles in a gas move rapidly in constant random motion Move in straight paths, changing direction only when colliding with one another or other objects Average speed of O2 in air at 20 oC is an amazing 1660 km/h! (1.6km=1mile)

Kinetic Theory 3. Collisions are perfectly elastic- meaning kinetic energy is transferred without loss from one particle to another- the total kinetic energy remains constant

Kinetic Theory Kinetic Theory of the Ideal Gas The gas consists of a very large number of molecules, each with mass but with negligible size and no internal structure. The molecules do not exert any force on each other except during a collision. This means that there is no potential energy. The molecules are moving in random directions with a distribution of speeds that is independent of direction. Collision with each other and the walls are elastic.

Pressure and Temperature Pressure: Results from collisions of molecules on the surface Force Pressure: Area Force: Rate of momentum given to the surface Momentum: momentum given by each collision times the number of collisions in time dt

Kinetic Theory Kinetic Theory of the Ideal Gas So we need the change in momentum

Kinetic Theory Kinetic Theory of the Ideal Gas Thus, we find for the force on the wall Now using this in the pressure we find

Kinetic Theory Kinetic Theory of the Ideal Gas But now

Kinetic Theory Kinetic Theory of the Ideal Gas

van der Waals Equation The ideal gas equation is based upon the model that: Gases are composed of particles so small compared to the volume of the gas that they can be considered to be zero-volume points in space. There are no interactions, attractive or repulsive, between the individual gas particles. In 1873, Johannes H. van der Waals (1837-1923), a Dutch physicist, developed an approximate equation of state for real gases that takes these factors into account: A semiempirical equation, based upon experimental evidence, as well as thermodynamic arguments. Awarded 1910 Nobel Prize in Physics for his work.

van der Waals Equation Assuming that the particles behave as small, impenetrable hard spheres. If b is the volume of one mole of gas molecules n is the number of moles nb is the volume of the molecules The actual volume is given by:

van der Waals Equation Correcting for the volume of the particles into the ideal gas equation: Since Taking into account the actual volume of the gas particle results in an increase in pressure relative to that predicted by the ideal gas law.

van der Waals Equation the weak long-range attraction: the long-range attractive forces between the molecules tend to keep them closer together; these forces have the same effect as an additional compression of the gas. The attractive forces are proportional to the number density of the gas molecules in the container:

van der Waals Equation The pressure produced by the attractive forces is proportion to the square of the number density. Taking this into account, the corrected pressure is: which is known as the van der Waals equation. In terms of the molar volume, Vm, the van der Waals equation is:

van der Waals Equation a b The van der Waals constants: Pressure correction Represents the magnitude of attractive forces between gas particles Does not specify any physical origin to these forces b Volume correction Related to the size of the particles These constants: Are unique to each type of gas. Are not related to any specific molecular properties.

Substance a’ (J. m3/mol2) b’ (x10-5 m3/mol) Pc (MPa) Tc (K) Air .1358 3.64 3.77 133 K Carbon Dioxide (CO2) .3643 4.27 7.39 304.2 K Nitrogen (N2) .1361 3.85 3.39 126.2 K Hydrogen (H2) .0247 2.65 1.30 33.2 K Water (H2O) .5507 3.04 22.09 647.3 K Ammonia (NH3) .4233 3.73 11.28 406 K Helium (He) .00341 2.34 0.23 5.2 K Freon (CCl2F2) 1.078 9.98 4.12 385 K

van der Waals (1837-1923), Dutch

van der Waals Isotherms The van der Waals equation Generates ideal gas isotherms at high temperatures and at large molar volumes. At high temperature, the first term may be much greater than the second term. At large molar volumes, and the ideal gas law is achieved:

Real Gases: Deviations from Ideal Behavior The van der Waals Equation General form of the van der Waals equation: Corrects for molecular volume Corrects for molecular attraction

2 Critical constants of real gas critical temperature Tc Critical pressure pc Critical volume Vm,c

1.4 Molecular interactions Critical point

Critical Constants For the van der Waals Equation The critical constants are related to the van der Waals coefficients. The critical constants are found by solving for the critical point, which occurs with both the slope and curvature of the van der Waals isotherms are zero. The slope is found from taking the first derivative of the pressure with respect to the molar volume: The curvature is found from taking the second derivative of the pressure with respect to the molar volume:

Critical Constants For the van der Waals Equation Solving these two equations in two unknowns (temperature and molar volume) gives the critical temperature and critical molar volume: The critical pressure may be calculated by substituting the expressions for the critical temperature and critical molar volume into the van der Waals equation:

Example The critical temperature, Tc, and critical pressure, pc, for methane are 191K and 46.4×105Pa respectively. Calculate the van der Waals constants and estimate the radius of a methane molecule. Solution

Boyle’s temperature The Boyle temperature TB is defined as:

Compressibility Factor Z Compressibility Z is a measure of the deviation from ideal gas Z = 1 ideal gas Z can be either positive or negative For real gas:

We have critical compression factor Zc: For most gases, the values of Zc are approximately constant, at 0.26～0.31.