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Новосибирский государственный технический университет

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General Physics part 1 / PhysPraktikum / Woan G. The Cambridge Handbook of Physics Formulas (CUP, 2000)(ISBN 0521573491)(230s)_PRef_.pdf

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110	Thermodynamics

5.3 Gas laws

Ideal gas

Joule’s law	U = U(T )				(5.55)

Boyle’s law	pV \|T = constant				(5.56)

Equation of state	pV = nRT				(5.57)
(Ideal gas law)	pV = nRT				(5.57)
(Ideal gas law)

	pV γ = constant				(5.58)
Adiabatic	T V (γ−1) = constant				(5.59)
Adiabatic	T γp(1−γ) = constant
equations	T γp(1−γ) = constant				(5.60)
	1			(p2V2 − p1V1)
	∆W =				(5.61)
	∆W =		γ − 1		(5.61)
Internal energy	nRT				(5.62)
	U =
	U =	γ − 1
Reversible
isothermal	∆Q = nRT ln(V2/V1)				(5.63)
expansion

Joule expansiona	∆S = nR ln(V2/V1)				(5.64)

Uinternal energy

Ttemperature

ppressure

Vvolume

nnumber of moles

Rmolar gas constant

γratio of heat capacities (Cp/CV )

∆W work done on system

∆Q heat supplied to system

1,2 initial and ﬁnal states

∆S change in entropy of the system

aSince ∆Q = 0 for a Joule expansion, ∆S is due entirely to irreversibility. Because entropy is a function of state it has the same value as for the reversible isothermal expansion, where ∆S = ∆Q/T .

Virial expansion

					B2(T )			p	pressure
	pV = RT		1 +		B2(T )			V	volume
	pV = RT		1 +					V	volume
Virial expansion					V		(5.65)	R	molar gas constant
				B3(T )				T	temperature
			+			+ · · ·		T	temperature
			+	V 2		+ · · ·		Bi	virial coe cients
Boyle	B2	(TB) = 0					(5.66)	TB	Boyle temperature
temperature	B2	(TB) = 0					(5.66)	TB	Boyle temperature
temperature

5.3 Gas laws	111

Van der Waals gas

				p	pressure
		a		Vm	molar volume
Equation of state		a		R	molar gas constant

	p+ Vm2 (Vm − b) = RT		(5.67)
	p+ Vm2 (Vm − b) = RT		(5.67)	T	temperature
				a,b	van der Waals’ constants

	Tc = 8a/(27Rb)		(5.68)	Tc	critical temperature
	pc = a/(27b2)			Tc	critical temperature
Critical point	pc = a/(27b2)		(5.69)	pc	critical pressure
	Vmc = 3b		(5.70)	Vmc	critical molar volume
				pr	= p/pc
Reduced equation	3			pr	= p/pc
Reduced equation	3		(5.71)	Tr	= T /Tc
of state	pr + Vr2 (3Vr − 1) = 8Tr		(5.71)	Tr	= T /Tc
				Vr	= Vm/Vmc

Dieterici gas

pressure

−a

molar volume

Equation of state

p =

exp

(5.72)

molar gas constant

Vm − b

RT Vm

temperature

a ,b Dieterici’s constants

Tc = a /(4Rb )

(5.73)

critical temperature

Critical point

pc = a /(4b 2e2)

(5.74)

critical pressure

Vmc = 2b

(5.75)

Vmc

critical molar volume

= 2.71828...

Reduced equation

= p/pc

pr =

(5.76)

= T /Tc

of state

2Vr − 1 exp 2

− VrTr

= Vm/Vmc

		Van der Waals gas					2
	1.4						2
	1.4	1.1	Tr = 1.2				1.8
		1.1	Tr = 1.2				1.8
	1.2	1.0					1.6
	1	1.0					1.4
							1.4
							1.2
	0.8						1.2
r	0.8					r	1
p		0.9				p	1
	0.6	0.9					0.8
	0.6						0.8
	0.4						0.6
	0.4						0.4
	0.2	0.8					0.4
	0.2	0.8					0.2
							0.2
	0	1	2		4	5	0
	0	1	2	3	4	5
			Vr

Dieterici gas

		Tr = 1.2
	1.1
	1.0
	0.9
	0.8
0	1	2	3	4	5
		Vr

112 Thermodynamics

5.4 Kinetic theory Monatomic gas

pressure

number density = N/V

Pressure

p = 3 nm c

(5.77)

particle mass

mean squared particle

velocity

Equation of

volume

Boltzmann constant

state of an ideal

pV = NkT

(5.78)

number of particles

gas

temperature

Internal energy

internal energy

U = 2 NkT

= 2 m c

(5.79)

(5.80)

CV =

heat capacity, constant V

Heat capacities

Cp = CV + Nk =

(5.81)

heat capacity, constant p

ratio of heat capacities

γ =

(5.82)

Entropy

entropy

(Sackur–

mkT

3/2

5/2

¯h

= (Planck constant)/(2π)

Tetrode a

S = Nkln 2π¯h2

(5.83)

= 2.71828...

equation)

For the uncondensed gas. The factor

mkT

3/2

is the quantum concentration of the particles, n

. Their thermal de

2π¯h2

1/3

Broglie wavelength, λT , approximately equals nQ−

Maxwell–Boltzmann distributiona

probability density

3/2

particle mass

Particle speed

pr(c) dc =

exp

−mc

4πc2 dc

distribution

2πkT

2kT

Boltzmann constant

(5.84)

temperature

particle speed

Particle energy

pr(E) dE =

2E1/2

exp

−

dE (5.85)

particle

kinetic

distribution

π1/2(kT )3/2

energy (= mc /2)

c =

1/2

Mean speed

(5.86)

mean speed

πm

crms =

1/2

rms

root mean squared

rms speed

(5.87)

speed

Most probable

cˆ =

1/2

speed

(5.88)

cˆ

most probable speed

aProbability density functions normalised so that %0∞ pr(x) dx = 1.

5.4 Kinetic theory	113

Transport properties

Mean free patha

l =

√

πd2n

(5.89)

Survival

pr(x) = exp(−x/l)

(5.90)

equationb

Flux through a

J =

n c

(5.91)

planec

Self-di usion

J = −D n

(5.92)

(Fick’s law of

where D

l c

di usion)d

(5.93)

H = −λ T

(5.94)

Thermal

1 ∂T

conductivityd

T =

∂t

(5.95)

for monatomic gas

ρl c cV

(5.96)

Viscosityd

η 2 ρl c

(5.97)

Brownian

kT t

motion (of a

(5.98)

= 3πηa

sphere)

Free molecular

1/2

πm

ﬂow (Knudsen

−

T 1/2

ﬂow)e

(5.99)

lmean free path

dmolecular diameter

nparticle number density

pr probability

xlinear distance

Jmolecular ﬂux

c mean molecular speed

Ddi usion coe cient

Hheat ﬂux per unit area

λthermal conductivity

Ttemperature

ρdensity

cV	speciﬁc heat capacity, V	5
	constant
η	dynamic viscosity
η	dynamic viscosity

xdisplacement of sphere in x direction after time t

kBoltzmann constant

ttime interval

asphere radius

dM mass ﬂow rate dt

Rp pipe radius

Lpipe length

mparticle mass

ppressure

aFor a perfect gas of hard, spherical particles with a Maxwell–Boltzmann speed distribution. bProbability of travelling distance x without a collision.

cFrom the side where the number density is n, assuming an isotropic velocity distribution. Also known as “collision number.”

dSimplistic kinetic theory yields numerical coe cients of 1/3 for D, λ and η.

eThrough a pipe from end 1 to end 2, assuming Rp l (i.e., at very low pressure).

Gas equipartition

Classical

Eq = 1 kT

(5.100)

energy per quadratic degree of

freedom

equipartition

Boltzmann constant

temperature

heat capacity, V constant

CV =

fNk =

fnR

(5.101)

heat capacity, p constant

Ideal gas heat

number of molecules

= Nk 1 +

(5.102)

number of degrees of freedom

capacities

number of moles

(5.103)

molar gas constant

γ =

= 1 +

ratio of heat capacities

aSystem in thermal equilibrium at temperature T .

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