Chap 1. The properties of gases Homework

Homework

Chap 1. The properties of gases

Problems: 1A.1 1A.8 1A.10 1C.1 1C.2

1C.3 1C.4 1C.9 1C.12 1C.16

Chap 1. The properties of gases

• Gas: the simplest state of matter

– A collection of molecules (or atoms) in continuous random motion – Average speeds increases as T is raised

– The molecules of a gas are widely separated (negligible intermolecular forces)

• The perfect gas: an idealized version of a gas

– Obey the perfect gas law: pV = nRT

• Real gas: do not obey the perfect gas law, (high p or low T) – Van der Waals eqn

– Virial equation

1장 수업목표 1: 기체의 성질

• Pressure: Standard pressure: 1 bar = 10

⁵

Pa

cf. 1 atm = 101325 Pa =760 Torr = 760 mmHg

• Temperature: the direction of the flow of energy

• Perfect gas law:

• Partial pressure:

o m

p nRT V

V RT

V n p



 

J J

p  x p

A B C

p  p  p    p

1A.1 Variables of states

The physical state of a sample of a substance (its physical condition):

defined by its physical properties (V, p, T, n)

(a) pressure: p = F/A (1 Pa = N/m ² = kgm ⁻¹ s ⁻² )

Standard pressure: 1 bar = 10 ⁵ Pa

1A.1 Variables of states

)

( p ^o

1A.1 Variables of states

A ^(area) h ^(height)

V=Ah ^(volume) Barometer: measures the atmosphere

F = mg = ρAhg Δp = F/A = ρgh

Example 1A.1

Mechanical equilibrium

1A.1 The variables of states

(b) Temperature (thermometer)

the property that indicates the direction of the flow of energy

Celsius scale ( Ө/ºC ): the length of a column of a liquid

perfect gas temperature scale: the pressure of the perfect gas

= thermodynamic temperature scale (T/K): T/K = Ө/ºC + 273.15

1A.2 Equation of state

Boyle’s law: pV = constant (at cont. T) Charles’s law: V = constant • T (at const p)

Avogadro’s principle: V = constant • n (at const. p, T)

The perfect gas law: pV = nRT

• The perfect gas (or ideal gas) law:

A real gas obeys in the limit of p→ 0

1A.2 Equations of states

V p  nRT

m

V V

 n

2 3 3

1 22.414 1 273.15

101325 22.414 10

1 273.15

pV atm l

R nT mol K

Nm m

mol K

 

  



 

 

 

^{3 3 3 3}

1 l  10 cm  1 dm  10

^

m

• Standard ambient temperature and pressure (SATP) T=298.15 K, p = 1 bar

V _m ^o = 24.789 l/mol

• Standard temperature and pressure (STP) T= 0 ºC, p = 1 atm

V _m ^o = 22.414 l/mol Molar volume:

(R=N

_A

k)

o m

V RT

 p

1A.2 (b) Mixtures of gases

J

J total A B C J

total

x n n n n n n

 n       

• Partial pressure: p _J = x _J p

• Mole fraction: x _J

• Dalton’s law:

부분압 (partial pressure): 혼합기체에서 특정 한 기체의 압력

 

A B C 1

A B C A B C

x x x

p p p x x x p p

    

         

A B C

p  p  p    p

1A.2 (b) Mixtures of gases

1장 수업목표 2: 기체운동론

• Maxwell-Boltzmann distribution of speed

• Root mean square (rms)speed:

• Mean speed:

• Collision frequency:

• Mean free path:

 

3 2

2 2 2

4 2

Mv

M

RT

f v v e

 RT



 



  

  ₁ ¹

2 2 3RT 2

c v

M

 

   

 

1

8 2 mean

v RT

 M

 

    

1

rel rel

rel

z v N v p

kT

v kT

z N p

 

  

 

  

1B. The kinetic model

1. The gas consists of molecules of mass m in ceaseless random motion.

2. The size of the molecules is negligible, in the sense that their diameters are much smaller than the average distance travelled between collisions.

3. The molecules interact only through brief, infrequent, and elastic collisions (탄성충돌).

In the kinetic theory of gases it is assumed that KE of the molecules is the only contribution to E of the gas.

1B.1 The model

1B.1(a) Pressure and molecular speeds-1a

1 2

pV  3 nMc

p of a perfect gas according to the kinetic model

 

a molecule

2

2

2

Momentum change of a molecule after collision:

2

Number of colliding molecule in : 1 2

= 2 =

2

x

A x

A x x

total x

x

x

P mv

t Av t nN

V nN Av t nMAv t

P mv

V V

P nMAv

F t V

F nMv

p A V

 

 

     

 



  



 

2 2

2 2 2 2 2

3 where

x

x y z

nM v nMc

V V

c v v v v

 

   

2 1 2

M mN

A

c v





2

2

3

3

pV nMc nR

c RT

M

T



 



반은 오른쪽 나 머지 반은 왼쪽.

반만 벽과 충돌

1B.1(b) The Maxwell-Boltzmann distribution of speed

 

 

 

2 2

2

2

1

2 2

direction 1

2 Probability having ,

2

x

x

x x

x x

mv E

kT

m k k

v T x

T x

f v m

v KE mv

v v

f v e e

kT e





 

 



 

 

 













: Maxwell-Boltzmann velocity distribution

 

3

2

2 2 2

4 2

Mv

M RT

f v v e

 RT



  

  

 

Maxwell-Boltzmann distribution of speeds

molar mass, gas constant,

A A

M mN R kN





 

2

2

2

1

2 1

2

x

ax

x x

mv kT

x

e dx

a f v dv

Ne dv N kT

m a m

kT





 







 





 



  

 

 







 

3 2

2 2 2

4 2

Mv

M

RT

f v v e

 RT



 



    

2 2 2

2

~ ~

~ ,

si

x x x

y y y

z z z

x y z

x y z

v v dv v v dv v v dv

v v v v

dv dv dv v







  



    ^{ }

2 2 2

2

2

3

2 2 2 2

3 2

2 2 2

0 0

3

2 2

n

2 2 sin

4 2

x x y y z z

x y z

x y z

x x y y z z

x y z

v dv v dv v dv

x x y y z z

v v v

mv mv mv

v dv v dv v dv

kT kT kT

x y z

v v v

v dv mv

kT v

mv k

dvd d f v dv f v dv f v dv

m e e e dv dv dv kT

m e v dvd d

kT

m e

kT

 

  



  



 

  

     

 



 

    

 

    

 

    

  

  

  

 

 

3 2

2 2 2

2

4

mv

v dv v dv

T

v v

m

kT

f v v e

v dv f v dv

 



 

 

    





 

molar mass, gas constant,

M mN

A

R kN





 

1 2

2 2

direction

2

mvx

kT x

x

f v m e

v

 kT

 



  

 



1B.1(b) The Maxwell-Boltzmann distribution of

speed

 

3 2

2 2 2

4 2

Mv

M

RT

f v v e

 RT



 



  

 

1B.1(b) The Maxwell-Boltzmann distribution of

speed

1B.1(c) Mean values

0  

n n

f v

v   ^ v dv

 

3

2

2 2 2

4 2

Mv

M RT

f v v e

 RT



  

  

 

   

3 2

2 2 2

4 0

2

Mv

M

RT

f v v e v

 RT



 



     

 

2

2

2

2

2

0

0

2 3 2

0

3 0 2

4 5 2

0

1 2

1 2

4 1 2 3

8

ax

ax

ax

ax

ax

e dx

a xe dx

a

x e dx a

x e dx

a

x e dx a







 

 

  

 

  





















1B.1(c) Mean values

 

 

0

2 2 2

0

v mean vf v dv

c v v f v dv







 





Root mean square (RMS) speed

Mean speed

Most probable speed

1

1 2

2 2

1 1

2 2

1 1

2 2

3

8 8

3

2 2

3

mean rms

mp rms

c v RT

M

v RT v

M

v RT v

M

 

 

     

   

         

   

         

   

3 2

2 2 2

4 0

2

Mv

M

RT

f v v e v

 RT



 



       

1B.1(c) Mean values

Ex 1B.1 v

_mean

of N

₂

molecules in air at 298 K.

 

1 1

1

3 1

8 8 8.314 298

28.02 10 475

mean

RT JK mol K

v ms

M kg mol

 

 



 

 

  

 

1 2

1 2

2 2 8

8 where

rel mean

A B

A B

v v RT

M

kT m m

m m



 

 

   

 

 

      

Mean relative speed

 

1 1

1

3 1

8 8 8.314 298

2 2 728

28.02 10

RT JK mol K

c ms

M kg mol

 

 



 

 

  

 

1B.1(c) Mean values

Bi 1B.2 v

_rel

of N

₂

molecules in air at 298 K.

1

rel rel

rel

z v N v p

kT

v kT

z N p

 

  

 

  

 

 

freeze the position of all the molecules except one number density,

/

A

rel rel

N p

V kT pV nRT nN kT kT

z v t t N v

V  

  

   

  

       

1B.2 (a) The collision frequency, z (b) The mean free path, 

For 1 atm of N

₂

molecules at 298 K.

   

18 2 1 2

23 1

9 1

1

9 1

0.43 10 728 101325

1.381 10 298

7.7 10

728 95 (~ 1000 times of ) 7.7 10

rel

rel

z v p kT

m ms Nm

JK K

s

v ms

nm d

z s





  

 









  

  

 

  



2

collision crosssection,

   d

1장 수업목표 3: 실제기체

• Compression factor:

(압축인자)

• Van der Waals equation:

• Virial equation:

(라틴어의 “힘”)

m m

o m

V pV Z  V  RT

2

m V m

a b

V

p RT 

 

2 2

( ) ( )

1

1 '( ) '( )

m

m m

pV B T C T

RT Z V V

B T p C T p

     

    

1C.1 Deviations from perfect behavior

Real gases interact with one another ( high p low T )

1C.1 Deviations from perfect behavior

Consequences of molecular interactions

Real gases interact with one another ( high p low T )

CO ₂

1C.1(a) The compression factor, Z

^o

m m

V V RT

V V

n n p

  

m m

o m

V pV Z  V  RT

ZRT pV _m 

Even a real gas Z ≈ 1 at very low p

• Z > 1 repulsive F dominates

• Z < 1 attractive F dominates

Taylor expansion

Exponential function:

http://en.wikipedia.org/wiki/Taylor_expansion

( ) f x 



Geometric series:

^{2 3}

0

1 1 for 1

1

n n

x x x x x

x





       

 

 

^{2 3}

ln 1 2 3

x x x x

     

Natural log:

Taylor expansion

1C.1(b) Virial coefficients

1 for ideal gas pV

m

RT   Z

  ^{p 1 ' p '}

²

Z  Z   a  b p  

 

m

^{1 1 1}

₂

m m

Z Z V

V V

a b

     

2

2

1 ' '

1

m

m m

pV B p C p RT

B C V V

    

    

• Virial equation of state

, ,

' ' '( ) ( )

, ,

'(

) ) (

B C B T C T B C B T C T



Virial coefficients depend on Temperature. 

1C.1(b) Virial coefficients

2

'( ) ( ) ( 2

) 1

'( )

m 1

m m

B T C T B

pV p p T

R

C T

T        V  V  

2

1 ( ) ( )

2

' ) '( )

1 (

m

m m

B T C T

B T C T

pV Z p p

RT    V  V       

 

'( ) 2 '( ) ( ) 2 ( )

1

_{m m}

dZ B T C T p dp

dZ C T

d V B T V

   

   

At Boyle temperature T _B B(T) = 0

'( ( )

) B

B T

T T

 R

1C.1(b) Virial coefficients

At T _B the gas behaves

perfectly over a wider range

of conditions than at other

temperatures.

For large V

_m

and high T, the real-gas isotherms do not differ greatly from perfect-gas isotherms

1C.1(c) Critical constants

Critical constants

^{(임계상수 )}

(T _c , p _c , V _c )

liquids phase of substances does not form above T

_c

Supercritical fluids ( 초임계유체 ): the single phase at T > T

_c

and much denser than typical of gases

Vapor pressure

(A, B, C) (C, D, E) (E-F)

CO ₂

1C.1(c) Critical constants

• First assume “hard sphere” molecules (repulsive force) becomes

1C.2(a) The van der Waals equation

Only two parameters, derived from molecular concepts

RT

pV

_m

 p ( V

_m

 b )  RT

b V p RT

m





2

m m

RT a p  V b  V



• Now put in attraction

So becomes

1.C.2(a) The van der Waals equation

Ex 1C.1 van der Waals V _m of CO ₂ at 500 K and 100 atm.

 

^{2 2}

 

3 2

0

m m m m

m m m

V b V p RTV V b a

RT a ab

V b V V

p p p

   

   

        

   

6 2

2 3 1

3.592 4.267 10

a dm atm mol

b dm mol



 

 

   

 

3 1

2 3 1 2

3 3 1 3

0.453

3.61 10

1.55 10

b RT dm mol

p

a dm mol

p

ab dm mol

p



 

 

 

 

 

   

3 2 2 3

3 1

3 1

0.453 3.61 10 1.55 10 0

0.366 0.410

m m m

m o m

V V V

V dm mol

V dm mol

 





     





공식보다는 계산기나 컴퓨터 사용하여 푼다.

1C.2(b) The features of the equation

6 0 )

( 2

2 0 )

(

3 4 2

2

3 2



 





 

 

m m m

m m m

V a b

V RT dV

p d

V a b

V RT dV

dp

3 , V

m



,

3( ) 2

3

m m

m c

V b V

V b

 

 

2 2

3 3

2 2

2 2 8

( ) 4

27 27

8 27

2 9 27

m m

c

c

a a a

RT V b b

V b b

T a

Rb

RT a a

p b b b

   

 

  

… ①

… ②

① 6 0

) (

3

2



4







m m

m

V

a b

V V

RT … ③

② + ③, results in ₍ ^{2 3} _{) 0} )

(

²

 





_{m m}

m

b V b V

V RT

from ①,

8 3 8

3 27 27

²

,

 

 a

b b b a RT

V Z p

c c m c c

Critical compression factor

1C.2(b) The features of the equation

m m m

r r r

c c c

V p T

V p T

V  p  T 

1C.2(c) The principle of corresponding states

Reduced variables (dimensionless)