University of Babylon /College Of Engineering
Electrochemical Engineering Dept.
Second Stage /Thermodynamics
Heat effect
Heat capacity :it is amount of heat required to change a unit mass by a unit
temperature .Heat capacity of ideal gas depends on the type of the gas
1. For mono atom gas γ =1.66 ( Helium , Aaron )
2. For diatomic gases γ = 1.4( Co,N2 ,air)
3. For more than tri-atomic gases γ 1.3 (NH3, CH4 )
The higher molecular weight of gases , the lower value of γ
æ ¶U ö
÷
CV ( heat capacity at constant volume ) = ç
è ¶T ø V
æ ¶H ö
÷
CP (heat capacity at constant pressure ) = ç
è ¶T ø P
The temperature dependence may be shown graphically but the value which get
from graph with less accurate
University of Babylon /College Of Engineering
Electrochemical Engineering Dept.
Second Stage /Thermodynamics
On the other hand temperature dependence usually given by an empirical equation;
the two simplest expressions of practical value are:
CP
C
= a + b T + g T 2 and , P = a + bT + cT - 2
R
R
Where α , β and , γ and a , b , and c are constants characterized of particular gas ,
by combine the above two equations :
CP
D
= A + BT + CT 2 + 2
( 7-1 )
R
T
Where C or D is zero , depending on the gas considered .Value of A,B,C and D are
CP
given in table 4.1,since
dimensionless , the unit of CP is governed by choice
R
of R unit .equation (7-1) used for all gases as well as ideal gas.
More accurate but more complex equations are found in literatures.
CP = CV + R
CV
C
= P -1
R
R
CV
CP
, is readily found from equation
R
R
CP
CV
, and
are determined by experiment, most often
Effect of temperature on
R
R
from spectroscopic data and knowledge of molecular structure .
CP
equation
How to use the
R
T 2
T 2
D ù
é
D H = ò C P dT = ò R ê A + BT + CT 2 + 2 údT
T û
T1
T1 ë
B
C
1
1
= R [ A (T 2 - T 1 ) + (T 22 - T 12 ) + (T 23 - T 13 ) - D ( - )
2
3
T 2 T1
Thus the temperature dependence of
EX:
The molar heat capacity of methane in the ideal gas state is given in table 4.1as
CP
= 1 .702 + 9 .081 ´ 10 -3 T - 2 .164 ´ 10 - 6 T 2
R
CP
Where T in Kelvin , develop an equation for
for temperature in ○C
R
University of Babylon /College Of Engineering
Electrochemical Engineering Dept.
Second Stage /Thermodynamics
Solution
TK = t ○C +273.15
CP
= 1 .702 + 9 .081 ´ 10 -3 (t + 273 .15 ) - 2 .164 ´ 10 - 6 (t + 273 .15 ) 2
R
CP
= 4 .021 + 7 .899 ´ 10 -3 t - 2 .164 ´ 10 - 6 t 2
R
EX:
Calculate the heat required to raise the temperature of 1 mol of methane from 260
to 600 ○C in flow process at constant pressure approximately at 1 bar .
Solution
T 2 = 873 .15
T 2
Q = DH =
òC
P
dT = R
T1
ò(
1 .702 + 9 .081 ´ 10 - 3 T - 2 .164 ´ 10 - 6 T 2 ) dT
T 1= 533 .15
Q = 2378 .8 R = 2378 .8 * 8 .314 = 19780 J
As a matter convenience , we define a mean heat capacity
T 2
Cp mean =
ò CpdT
T1
T 2 - T1
When equation (7-1) written by use mean heat capacity equation
Cp mean
c
D
= A + BT am + ( 4T am2 - T 1T 2 ) +
R
3
T 1T 2
( 7-2 )
Where Tam= (T1+T2) / 2 is the arithmetic mean temperature.
The general equation for all gases and ideal gas
EX:
Rework the last example by applying equation (7-2)
Tam = (533.15 +873.15) / 2 = 703.15
University of Babylon /College Of Engineering
Electrochemical Engineering Dept.
Second Stage /Thermodynamics
Cp mean
2 .164 ´ 10 -6
-3
[ 4 * ( 703 .15 ) 2
= 1 .702 + 9 .081 ´ 10 * 703 .15 3
R
- (533 .15 ) * (873 .15 )] = 6 .997
Q = ∆ H = 6.997*8.314*(873.15-533.15) = 19780 J
For calculation of T2 in case given T1 and Q , used try and error to find T2 by
T2 =
DH
+ T1
Cp mean
(7-3)
Assume value of T2 for calculation Cp mean by use equation (7-2) substitution of
resulting value into equation (7-3) provides a new value of T2 which reevaluate
Cpmean. Iteration continues to convergence on T2 value.
Heat capacities of solid by used table 4.2 , while heat capacities of liquids from table
4.3.
Heat capacity for mixture
Cp mean = å Cp i y i
= Cp mean ( a ) y ( a ) + Cp mean ( b ) y ( b ) + ....
Where : y mole fraction