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p = RT (11.1) [R=R univ /m mole ] du = c V dT (11.2) dh = c p dT (11.3)

p = RT (11.1) [R=R univ /m mole ] du = c V dT (11.2) dh = c p dT (11.3) c p + c v = R (11.4) c p = kR/(k-1) (11.6a) [k=] c v = R/(k-1) (11.6b) [k=] p v k = p/ k = constant (11.12c). EQUATION OF STATE FOR IDEAL GAS p = RT (11.1). [units of Kelvin].

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p = RT (11.1) [R=R univ /m mole ] du = c V dT (11.2) dh = c p dT (11.3)

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  1. p = RT (11.1) [R=Runiv/mmole] du = cVdT (11.2) dh = cpdT (11.3) cp + cv = R (11.4) cp = kR/(k-1) (11.6a) [k=] cv = R/(k-1) (11.6b) [k=] pvk = p/k = constant (11.12c)

  2. EQUATION OF STATE FOR IDEAL GAS p = RT (11.1) [units of Kelvin] = unique constant for each gas Good to 1% for air at 1 atm and temperatures > 140 K (-130 oC) or for room temperature and < 30 atm

  3. Daniel Bernoulli ~ Hydrodynamics, 1738 PV = const (system 1) If Ldoubled (system 2) but same v, then (# of collisions/sec)1 = v x (1 sec)/L (# of collisions/sec)2 = v x (1 sec)/2L (# of collisions/sec)1= ½ (# of collisions/sec)2

  4. Daniel Bernoulli PV = const p = F/A F  {# collisions / sec} p1(# of collisions/sec)1/(L)2 p2 (# of collisions/sec)2/(2L)2 p2½ (# of collisions/sec1)/(2L)2 p2= 1/8 p1 Vol2= 8Vol1 p2Vol2 = p1Vol1QED

  5. Daniel Bernoulli ~ PV = const Hydrodynamics, 1738 . . . . 2L L . . . . . . (# of collisions/sec)1 p1, n1, m1, vx1, T1, L1 . . (# of collisions/sec)2 p2, n1 m1, vx1, T1, L2=2L1

  6. “ The elasticity of air is not only increased by compression but by heat supplied to it, and since it is admitted that heat may be considered as an increasing internal motion of the particles, it follows that … this indicates a more intense motion of the particles of air.” Daniel Bernoulli Here was the recipe for quantifying the idea that heat is motion – two generations before Count Rumford, but it came too early.

  7. IDEAL GAS: p = RT (eq. 1.11) R = Runiv/mmole pV = N(# of moles)RunivT

  8. What is Pressure ? Assume perfect elastic reflections so: - 2mvx is change of x-momentum per collision. Initially assume vx is same for all particles.

  9. Force of one particle impact = Magnitude of momentum change per second due to one particle: = (mvx)/t =2mvx/(2L/vx) = mvx2/L Time between collisions, t, of particle with same wall is equal to: t = 2L/vx L

  10. Magnitude of momentum change per second due to n molecules: nmvx2/L <vx2> = <vy2> = <vz2>; <vx2> + <vy2> + <vz2> = <v2> <vx2> = 1/3 <v2> 1/3nm<v2>/L

  11. Pressure = F/A = [1/3nm<v2>/L]/L2 P = 1/3nm<v2>/L3 PV = 1/3nm<v2> = 2/3n (1/2m<v2>) average kinetic energy per particle Empirically it is found that : PV = nkBT n=#of particles; kB=1.38x10-23 J/K

  12. PV = 2/3n (1/2m<v2>) Empirically it is found that: PV = nkBT T(Ko) = [2/(3kB) ] [avg K.E.]

  13. pV = nkBT pV = (2/3) n <mv2/2> Uinternalfor monotonic gas Uint = f(T) depending if p or V held constant uint, v,… designate per unit mass duint/dT = cv(11.2) duint/dT=cp (# of particles)

  14. pV = nkBT n = [Nm][NAvag] 6.02x1023 nkBT = Nmx NAvagkBT = Nm x NAvag[Runiv/Navag.]T pV= NmRunivT

  15. pV= NmRunivT p=(1/V)Nmmmole{Runiv/mmole}T p=(m/V){Runiv/mmole}T p= {Runiv/mmole}T = RT (11.1)

  16. IDEAL GAS pV = NmRunivT p = {Runiv/mmole}T p v = {Runiv/mmole}T pV  Uint = f(T)

  17. The differential work dW done on the gas in compressing it by moving it –dx is –Fdx. dW on gas = F(-dx) = -pAdx = -pdV goes into dT

  18. ASIDE: Want to derive important relation between p and V for adiabatic condition, i.e.  Q = 0 pV = 2/3 U for monotonic gas pV = (k - 1) U in general k = cp/cv = 5/3 for monotonic gas U = pV/(k - 1) dU = (pdV+Vdp)/(k - 1) – eq. of state

  19. Cons. of energy dU = W + Q Compression of gas under adiabatic conditions means all work goes into increasing the internal energy of the molecules, so: dU = W = -pdV for adiabatic (Q = 0) Equation of state dU = (Vdp + pdV) / (k - 1)

  20. dU = W + Q dU = (Vdp + pdV) / (k - 1) -pdV = (Vdp + pdV) / (k - 1) -(pdV)(k - 1) = Vdp + pdV -(pdV)k + pdV = Vdp + pdV -(pdV)k - Vdp = 0

  21. -(pdV)k - Vdp = 0 • (divide by -pV) • (dV/V) + (dp/p) = 0 (integrate) kln(V) +ln(p) = ln(C) ln(pVk) = ln(C) • pVk = C or pvk = c (11.12c)

  22. IDEAL GAS dU = W + Q Q/m = cvdT + pdv pv = RT (R=Runiv/mmole) pdv + vdp = RdT Q/m = cvdT + RdT - vdp Divide by dT [Q/m]/dT = cv + R – vdp/dT If isobaric, i.e. dp=0 then {[Q/m]/dT}p = cp = cv + R (11.4)

  23. cp = cv + R; cp – cv = R • Divide by cv, & let k = cp/cv; k - 1 = R/cv, or cv =R/(k-1) (11.6b) Multiply by cp/cv • cp = kR/(k-1) (11.6a) • usually k =  in most books

  24. IDEAL GAS h = u + pv dh = du + d(pv) dh = cvdT + RdT dh = (cv + R)dT dh = cpdT (11.3)

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