Kinetic Theory of Gases Physics 313 Professor Lee Carkner Lecture 11

Exercise #10 Ideal Gas  8 kmol of ideal gas   Compressibility factors  Z m =  y i Z i   y CO2 = 6/8 = 0.75   V = ZnRT/P = (0.48)(1.33) = m 3  Error from experimental V = m 3   Compressibility factors: 1.5%  Most of the deviation comes from CO 2 

Ideal Gas  At low pressure all gases approach an ideal state  The internal energy of an ideal gas depends only on the temperature:  The first law can be written in terms of the heat capacities: dQ = C V dT +PdV dQ = C P dT -VdP

Heat Capacities  Heat capacities defined as: C V = (dQ/dT) V = (dU/dT) V  Heat capacities are a function of T only for ideal gases:  Monatomic gas  Diatomic gas  = c P /c V

Adiabatic Process  For adiabatic processes, no heat enters of leaves system   For isothermal, isobaric and isochoric processes, something remains constant  

Adiabatic Relations dQ = C V dT + PdV VdP =C P dT (dP/P) = -  (dV/V).  Can use with initial and final P and V of adiabatic process

Adiabats  Plotted on a PV diagram adibats have a steeper slope than isotherms    If different gases undergo the same adiabatic process, what determines the final properties?  

Ruchhardt’s Method  How can  be found experimentally?   Ruchhardt used a jar with a small oscillating ball suspended in a tube  

Finding    Also related to PV  and Hooke’s law    Modern method uses a magnetically suspended piston (very low friction)

Microscopic View  Classical thermodynamics deals with macroscopic properties     The microscopic properties of a gas are described by the kinetic theory of gases

Kinetic Theory of Gases  The macroscopic properties of a gas are caused by the motion of atoms (or molecules)   Pressure is the momentum transferred by atoms colliding with a container  

Assumptions  Any sample has large number of particles (N)   Atoms have no internal structure   No forces except collision   Atoms distributed randomly in space and velocity direction   Atoms have speed distribution 

Particle Motions  The pressure a gas exerts is due to the momentum change of particles striking the container wall     We can rewrite this in similar form to the ideal equation of state: PV = (Nm/3) v 2

Applications of Kinetic Theory  We then use the ideal gas law to find T: PV = nRT T = (N/3nR)mv 2   We can also solve for the velocity:  For a given sample of gas v depends only on the temperature

Kinetic Energy  Since kinetic energy = ½mv 2, K.E. per particle is:  where N A is Avogadro’s number and k is the Boltzmann constant 