Physical Chemistry Third Edition

2.2 Heat 53

Solution

a. c^1 .000 cal K−^1 g−^1 ^4 .184JK−^1 g−^1

q(3.20 mol)(18.015 g mol−^1 )(4.184JK−^1 g−^1 )(70.00 K)1690 J 1 .69 kJ

b. qtotal^0 qAl+qwater

CAl(0.215 cal K−^1 g−^1 )(25.00 g) 5 .38 cal K−^1

Cwater(1.000 cal K−^1 g−^1 )(100.00 g) 100 .0 cal K−^1

(5.38 cal K−^1 )(T− 363 .15 K)+(100.00 cal K−^1 )(T− 293 .15 K) 0

(5.38 cal K−^1 )(T−1954 cal)+(100.00 cal K−^1 )(T−29315 cal) 0

T

31269 cal

105 .36 cal K−^1

 296 .8K

tC 23. 6 ◦C

Exercise 2.5

TheBritish thermal unit(Btu) was originally defined as the amount of heat required to

raise the temperature of 1 pound of water by 1 degree Fahrenheit (◦F). It equals 252.0 cal.

Express the amount of heat in part a of the previous example in Btu.

The heat capacities of gases are often more strongly temperature dependent than

those of liquids. If the heat capacity depends on temperature, it can be represented by

a polynomial or other function.

EXAMPLE2.10

The molar heat capacity of water vapor at a constant pressure of 1.000 atm is represented by

CP, m 30 .54JK−^1 mol−^1 +(0.01029 J K−^2 mol−^1 )T

whereTis the Kelvin temperature. Find the amount of heat required to raise the temperature

of 2.000 mol of water vapor from 100.0◦C to 500.0◦C.

Solution

q(2.000 mol)

∫ 773 .15 K

373 .15 K

[

30 .54 J K−^1 mol−^1 +

(

0 .01029 J K−^2 mol−^1

)

T

]

dT

(2.000 mol)

{

(30.54JK−^1 mol−^1 )(400 K)

+(0.01029 J K−^2 mol−^1 )

1

2

[

(773.15 K)^2 −(373.15 K)^2

]}

(2.000 mol)(12216 J mol−^1 +2359 J mol−^1 ) 2. 915 × 104 J