6

TOPIC 2 - REAL GAS

Examples of Solved Problems:

1. For CO(g) at 298.15 K, B(T) = -8.6×10-6

m3mol

-1 and C(T) =1550×10

-12 m

6 mol

-2, by

using the virial equation:

+++= ...

)()(1

2

mm

mV

TC

V

TBRTPV calculate the pressure

(in atm) of 38.8 g of CO(g) in a 10 dm3 container at 298.15 K. Compare with the

ideal-gas result and find Z.

Solution:

Calculation part by part , using the suitable unit

[ ]

9988.0389.3

385.3

385.39988.0389.3

10974.210192.11389.3

....)()(

1

10974.2)10219.7(

101550)(

10192.110219.7

106.8)(

.389.3219.7

)15.298)(08206.0(

10219.73852.1

1010

,3852.1999.15011.12

8.38

53

2

5

2133

2612

2

3

133

136

13

113

;

13333

===

=×=

×+×−=

+++=

×=×

×=

×−=×

×−=

===

×=×

==

=+

==

−−

−

−−

−−

−

−−

−−

−

−−

−−−

ideal

real

mmm

real

m

m

m

idea

m

CO

P

PZ

atm

atm

V

TC

V

TB

V

RTP

molm

molm

V

TC

molm

molm

V

TB

atmmoldm

KmolKatmdm

V

RTP

molmmol

m

n

VV

molM

mn

R values

= 0.08206 L atm K-1

mol-1

= 0.08314 L bar K-1

mol-1

= 8.314 L kPa K-1

mol-1

= 8.314 Pa m3

K-1

mol-1

1 L

= 1 dm3

= 1×10-3

m3

= 1×103 cm

3

7

2. Calculate TV

P

∂

∂and

VT

P

∂

∂for a gas that has the following equation of state;

V

a

bV

RTP −

−=

Solution:

Lets know these symbols :

∂ : is symbol for partial differential

d : is symbol for total or exact differential or normal differential

δ : is symbol for differential for non state function (for work, w and heat, q ).

To do these type of questions, you must remember the concept of basic derivatives in

calculus.

Let see a simple examples :

i. y = x5

415 55 xxdx

dy== −

ii. 5

53

3 −== xx

y

6

615 1515)5(3

xxx

dx

dy −=−=−= −−−

26

26

15

5

)1(

215)1(215

)1(231

23.

−−

−=−−−=

−+=−

+=

−−

−−

xxxx

dx

dy

xxxx

yiii

The meaning of TV

P

∂

∂is the partial deferential of P respect to V at constant T.

For equation of state: V

a

bV

RTP −

−= = (RT)(V-b)

-1- a(V)

-1

TV

P

∂

∂=(-1)(RT)(V-b)

-1-1- a(-1)(V)

-1-1 =

22)( V

a

bV

RT+

−−

bV

R

bV

RT

T

P

V −=−

−=

∂

∂ −

011

8

Exercise 2a

1. In an industrial process, nitrogen is heated to 500 K at a constant volume of

1.000 m3. The gas enters the container at 300 K and 100 atm. The mass of the

gas is 92.4 kg. Use the van der Waals equation to determine the approximate

pressure of the gas at its working temperature of 500 K. For nitrogen; a =

1.352 dm6 atm mol

−2, b = 0.0387 dm

3 mol

-2. (140.48 atm)

2. A gas at 250 K and 15 atm has a molar volume 20 per cent smaller than that

calculated from the perfect gas law. Calculate the molar volume of the gas.

Calculate the compression factor under these conditions. Which are

dominating in the sample, the attractive or the repulsive forces? (0.80, the

attractive forces)

3. Suppose that 10.0 mol C2H6(g) is confined to 4.860 dm3 at 27°C. Predict the

pressure exerted by the ethane from the van der Waals equations of state.

Calculate the compression factor based on these calculations. For ethane, a =

5.507 dm6 atm mol

−2, b = 0.0651 dm

3 mol

−1. (35.16 atm, 0.694)

4. Cylinders of compressed gas are typically filled to a pressure of 200 bar.

For oxygen, what would be the molar volume at this pressure and 25°C

based on (a) the perfect gas equation, (b) the van der Waals equation. For

oxygen, a = 1.382 dm6 bar mol

−2, b = 3.19 × 10

−2 dm

3 mol

−1. ( A cubic

equation, can solved using a computer with math app.) ( 0.112 dm3 mol

−1)

5. At 300 K and 20 atm, the compression factor of a gas is 0.86. Calculate the

volume (L) occupied by 8.2 mmol of the gas under these conditions.(8.68).

6. The experimentally determined density of H2O at 1200 bar and 800 K is

537 g L–1

. Calculate Z and Vm from this information. (0.0334 L, 0.602)

7. The critical temperature and pressure of n-butane are 425.2 K and 3800

kPa, respectively. Calculate the critical volume of the gas? (0.35 L mol-1

)

8. What is the molar volume of N2(g) at 500 K and 600 bar according to the

virial equation? The virial coefficient B of N2(g) at 500 K is 0.0168 mol-1

.

(0.0861)

9. A gas following the hard-sphere potential without attraction obeys the

equation P(V-b) = RT. Derive TTP PVandPVTV )/()/(,)/( 22 ∂∂∂∂∂∂ .

(R/P, -RT/P2, 2RT/P

3)

10. At T=Tc, 0)/( =∂∂ =TcTmVP and .0)/( 22 =∂∂ =TcTmVP Use this information to

determine a and b in the van der Waals equation of state in terms of the

experimentally determined values Tc and Pc. (27R2Tc

2/64Pc, RTc/8Pc)

9

Exercise 2b

1. You want to calculate the molar volume of O2 at 398.15 K and 60 bar

using the van der Waals equation, but you do not want to solve a cubic

equation. Use the first two terms of equation

....1

1 +

−+== P

RT

ab

RTRT

PVZ m …………

The van der Waals constants of O2 are a = 0.138 Pa m6mol

-2 and b=

3.18×10-5

m3mol

-1. What is the molar volume in L mol

-1? (0.542 L mol

-1)

2. Calculate the density of CO2(g) at 375 K and 385 bar using the ideal gas

law, and the van der Waals equations of state. Solve the van der Waals

equation for Vm. Based on your result, does the attractive or repulsive

contribution to the interaction potential dominant under these conditions?

For CO2(g), a = 3.658 dm6 bar mol

−2, b = 0.0429 dm

3 mol

−1. (7738.6 g/L,

Z = 0.07>1, the attractive contribution is dominance).

3. The critical constants of methane are Pc = 45.99 bar,Vc = 98.60 cm3 mol

−1,

and Tc = 190.56 K. Calculate the van der Waals parameters of the gas.

(b = 0.0431 dm3mol

-1, r =1.94×10

-10 m, a = 2.303 dm

6 bar mol

-2).

4. At what temperature and pressure will H2 be in a corresponding state with

CH4 at 500.0 K and 2.00 bar pressure? Given Tc=33.2 K for H2, 190.6 K

for CH4; Pc=13.0 bar for H2, 46.0 bar for CH4. (0.5655 bar)

5. Assuming that methane is a perfectly spherical molecule, find the radius

(in cm) of one molecule using the value of b in the van der Waals equation.

For methane, b = 0.0428 cm3

mol-1

. (1.62×10-9

)

6. A certain gas obeys the van der Waals equation with a = 0.76 m6

Pa mol-2

.

It has a volume of 3.0×10-4

m3

mol-1

at 273.15 K and 3.0 MPa. From this

information, estimate the radius of the gas molecules (in nm) on the

assumption that they are spheres.(0.216 nm)

7. Gas A (having a molar mass of 26 g mol-1

) has a mass of 1.8 kg at - 43 oC

and pressure 78 bar. The gas occupies a volume of 12.3 L. Based on the

calculations explain whether it is easier or more difficult to compress this

gas than if it behaves ideally.( A is easier to compress than if it behaves

ideally)

10

Problems 2

1. A hypothetical gas has the equation of state P = (RT/Vm-b) - (a/TVm)

Where a and b are constants distinct from zero. Ascertain whether this

gas has a critical point. If it has, express Vmc and Tc, (the critical

constants) in term of a and b. (The gas has no critical point )

2. A non ideal-gas is represented by the equation of state:

++=

32

1

V

b

V

a

VRTP where a and b are constants. Calculate

TdV

P

∂and

∂

∂2

2

V

P. By requiring that this equation of state at the critical

point, show that 3

and2

c

c

VbVa =−= .

3. A hypothetical gas has the equation of state

2

mm V

aT

bV

RP −

−= where a

and b are constants distinct from zero. Ascertain whether this gas has a

critical point. If it has, express Vmc, and Tc (the critical constants) in term

of a and b. (3b, 27Rb/8a).

4. Derive the compression factor (Z) of 0.225 kg of N2 behaving as a van

der Waals gas, confined in 5.860 dm3 at 27

oC and find its value. Which

intermolecular forces are dominating in the sample? Given for N2 : a =

1.35×106 atm cm

6 mol

-2 ; b = 38.6 cm

3 mol

-1 . (0.97, attractive forces)

5. Compression factor, Z is a direct measure of the deviation of a real gas

from ideality.

a) Define Z in the form of mathematical expression.

b) Show that Z = o

m

m

V

V =

0P

P , where o denotes ideal state.

c) Derive an expression for the compression factor of a van der Waals

gas in terms of Vn, T, a and b.

d) Calculate the compression factor of 10.0 mol CO2 behaving as a van

der Waals gas and is confined in 4.860 dm3 at 27

oC. Which

intermolecular forces are dominating in the sample? [ Given for CO2 :

a = 3.610 atm dm6 mol

-2 ; b = 4.29 x 10

-2 dm

3 mol

-1]

Answer:

RT

PVm ,bV

V

m

m

−-

mRTV

a , 0.80, Z < 1 ; attractive force.

6. The critical temperatures of gases O2, H2 and CO2 are -118.2, -239.8 and

31 oC respectively. (a) Which of these gases will easily be liquefied?

(b) Which of these gases will liquefy when compressed isothermally at

100K? (Gas with the lowest Tc ,CO2 and O2 )

11

7. The following equations of state are occasionally used for approximate

calculation on gases; (gas A) PVm= RT(1+b/Vm), (gas B) P(Vm-b) = RT.

Assuming that there were gases that actually obeyed these equations of

state, would it be possible to liquefy either gas A or B? Would they have

a critical temperature? Explain your answer.

8. State two conditions whereby van der Waals equation is more

appropriate to apply then ideal gas law equation. Explain.

9. The critical constants of ethane are Pc = 48.20 atm, Vc = 98.7 cm3 mol

-1,

and

Tc = 190.6 K. Calculate the van der Waals parameters of the gas and

estimate the radius of the molecules.

( b = 32.9 cm3 mol

-1, a = 1.33 L

2 atm mol

-2, r = 0.24 nm)

10. A hypothetical gas is found to obey the following equation of state

P = bV

RT

m − -

mTV

a

Where a and b are constants not equal to zero. Show whether the critical

temperature exists. Would it be possible for this gas to liquefy?