Kinetic Theory of Gases Physics 102 Professor Lee Carkner Lecture 4

PAL: Quenching a Dagger  Suppose a silver dagger of mass m s at T s is immersed in a mass m w of water at T w. What is the final temperature of the water? Q silver + Q water = 0 c s m s  T + c w m w  T = 0 c s m s (T f -T i ) + c w m w (T f -T i ) = 0 c s m s (T f - T s ) + c w m w (T f - T w ) = 0 c s m s T f -c s m s T s + c w m w T f - c w m w T w = 0 c s m s T f + c w m w T f = c s m s T s + c w m w T w (c s m s + c w m w )T f = c s m s T s + c w m w T w T f = (c s m s T s + c w m w T w )/(c s m s + c w m w )

A certain amount of heat Q is applied to a 1 gram sample of 3 different materials, producing a different temperature increase  T in each. Which has the greatest specific heat? a) Material A:  T = 1 C b) Material B:  T = 2 C c) Material C:  T = 3 C d) All have the same specific heat e) We can’t tell from the information given

Through which material will there be the most heat transfer via conduction? a) solid iron b) wood c) liquid water d) air e) vacuum

Through which 2 materials will there be the most heat transfer via convection? a) solid iron and wood b) wood and liquid water c) liquid water and air d) vacuum and solid iron e) vacuum and air

Through which 2 materials will there be the most heat transfer via radiation? a) solid iron and wood b) wood and liquid water c) liquid water and air d) vacuum and solid iron e) vacuum and air

Gases  m =  1 mole = X molecules  X = Avogadro’s Number = N A  M =  n = number of moles  Why do we care about moles?   Can do experiments to find relationships between P, V, T and n  Such relationship called equation of state

Ideal Gas  Different gases have different equations of state  PV = nRT  Where R is universal gas constant = 8.31 J/mol K   Can also write as:  Where N is number of molecules and k in the Boltzmann constant = 1.38 X J/K

Ideal Gas Law Units  SI units:  P is Pascals (Pa)  1 Pa =  1 kPa = 1000 Pa  1 atmosphere =  V in cubic meters (m 3 )   T in Kelvin (K)  T K =  You must use Pa, m 3 and K (if you use R = 8.31)!

Other Laws  Boyle’s Law  PV = constant   Called an isothermal process  Charles’s Law  V/T = constant   Called an isobaric process  Gay-Lussac’s Law  T/P = constant   Called an isochoric process

Using the Ideal Gas Law   For fixed amounts of gas, n is constant and we have relationship between P, V and T   Whenever you see P, V, T, think ideal gas law

What is Temperature?  Need to understand the microscopic properties to understand the macroscopic properties   If you change the temperature you change the ways in which the molecules move   How do the moving molecules produce a pressure?   From our knowledge of force and momentum (Ch. 4 and 7) we can say:   Lots of molecules with lots of energy produce lots of force

Temperature and Energy  If an increase in T increases P, then increasing T must increase the KE of the molecules   High T =  Low T = Temperature is a measure of the average kinetic energy of the molecules   So we use the root-mean-squared velocity, v rms  A sort of average velocity

Relations  We can derive: KE = (1/2) mv 2 rms = (3/2)kT  KE =  m = mass of molecule  v rms =  k = Boltzmann constant = 1.38 X J/K  For a given gas, m and k are constant so:   Note: T must be in Kelvin 

Planetary Atmospheres  Why do some planets have atmospheres and others do not?    In order to have an atmosphere: v escape > 5v rms (2GM planet /R planet ) > 5(3kT/m molecule )  What properties are conducive to retaining an atmosphere? 

Velocities  A gas with a fixed value of T has a certain average KE and velocity but   some molecules are moving slower or faster than the mean   Sometimes slowing down   While a given molecule can have any velocity, some velocities are more probable than others  Velocities follow the Maxwellian probability distribution

Maxwell’s Distribution

Next Time  Read: 13.12, 14.5  Homework: Ch 14, P: 21, 23, Ch 13, P: 33, 55  Help Sessions start this week:  Tuesday and Thursday, 6-7 pm, 120 Science