10.1

Heat and Heat Capacity

Heat (like work) is not a property of the system and is only transfered between the system and the surroundings, as a result of temperature differences in the system and surroundings.

A lower case q will be the symbol that we will use to describe heat.

Sign convention for heat:

• If the surroundings are at a higher temperature than the system, then heat will flow from the surroundings to the system:

With respect to the system heat transfer into the system is positive and the process in the system is said to be endothermic.

• If the surroundings are at a lower temperature than the system, then heat will flow from the system to the surroundings:

With respect to the system heat transfer out of the system is negative and the process in the system is said to be exothermic.

systemsurroundings

q < 0 (an exothermic process)

Tsys > Tsurr

systemsurroundings

q > 0 (an endothermic process)

Tsys < Tsurr

10.2

The heat capacity, C, of a substance is the amount of heat required to raise the temperature of a given amount of that substance by one degree:

C dq/dT

• Heat capacities reported for a gram of substance are known as specific heat capacities (in general values reported per gram are often refered to as specific values).

Since the definition of the calorie is the amount of heat necessary to raise one gram of liquid water 1 oC (actually from 14.5 oC to 15.5 oC, since heat capacities, as we shall see, are functions of temperature), this is also the specific heat capacity of liquid water:

cH2O (l) = 1.00 cal / gram oC

Note that a lower case c will be used to designate specific heat capacities.

A British thermal unit or Btu is the amount of heat required to raise one lbm of liquid water 1 oF. How many calories are there in one Btu?

In June I paid $23.18 for 300 kWh of electricity and $21.93 for 3.90 Dkt of gas. 1 Dkt (Dekatherm) equals 10+6 Btus. What is the ratio of the cost of electricity to the cost of gas? Should you purchase a gas or electric hot water heater?

What is the molar heat capacity of liquid water in Joules / mole K?

10.3

• Heat capacities are different for different phases of the same substance:

cH2O (s) = 0.502 cal / gram oC

cH2O (l) = 0.998 cal / gram oC

cH2O (g) = 0.439 cal / gram oC

• Heat capacities are different for different substances:

cH2O (l) = 0.998 cal / gram oC

cHg (l) = 0.033 cal / gram oC

Are the above values what you would have expected? Why are cherries a significant agricultural crop in the area immediately surrounding Flathead Lake?

• Heat capacities are functions of the conditions under which they were measured, i.e., they are path dependent.

The difference in the heat capacity measured at constant pressure, Cp, (what you generally find tabulated) and the heat capacity measured at constant volume, Cv, is significant for gases:

Cp, H2O (g) = 7.91 cal / mole oC

Cv, H2O (g) = 6.05 cal / mole oC

The difference in the Cp and Cv given above for water vapor is approximately equal to the value of what constant?

and small or insignificant for condensed phases:

Cp, H2O (l) = 18.0 cal / mole oC

Cv, H2O (l) = 18.6 cal / mole oC

10.4

• Heat capacities vary with temperature. This temperature dependence is often given by functions of the general form:

Cp = a + b T + c T2 + d T-2

The a, b, c, and d in this equation are constants or fitting parameters that were obtained by least squares fitting heat capacities measured at different temperatures to an equation like that shown above (typically either the constant c or the constant d is set equal to zero). Values of the constants for O2 (g) taken from a table in the Chemical Rubber Company Handbook of Chemistry and Physics are:

a 10+3 b 10+6 c 10-5 d

Cp (cal/mole K) 8.27 0.258 ----- - 1.877

The heading 10+3 b in the 2nd column literally means that what is tabulated is 1000 times the constant b, so that the value of b is actually 0.258 x 10 -3. The temperature dependence of the heat capacity of O2 (g) is then:

Cp = 8.27 + 0.258x10-3 T - 1.877x10+5 T-2

At 100.0 oC the heat capacity of O2 (g) would be:

Cp = 8.27 + 0.258x10-3 (373.2 K) - 1.877x10+5 (373.2 K)-2

= 8.27 + 0.0963 - 1.348 = 7.02 cal / mole K

A different tabulation gives:

a 10+3 b 10+6 c 10-5 d

Cp (cal/mole K) 6.095 3.253 - 1.017 -----

What is the heat capacity of O2 (g) at 100.0 oC according to this data? Why are the two heat capacities not the exactly the same?

10.5

Temperature Dependence of Heat Capacity

Bonus Problem

This bonus problem is worth 10 points and is due, if you decide to pursue it, at 5:00 P.M. one week from the day that it is assigned and will only be graded on the answer.

Least squares fit heat capacity versus temperature data for some alkane hydrocarbon (other than methane) that can be found in a table of Thermodynamic Properties of the Alkane Hydrocarbons in the Chemical Rubber Company Handbook of Chemistry and Physics (note some newer editions of the handbook do not have these tables) to a heat capacity function of the form:

Cp = a + b T + d T-2

to obtain best values for the constants a, b, and d. Most spreadsheets and math applications (like MathCad and Mapel) have least squares fitting routines built into them (it is up to you to figure out how to use them). As a test case the heat capacity function for methane is:

Cp = 6.25 + 10.5x10-3 T - 0.318x10+5 T-2

Hand in a function for your compound like that given for methane, above.

If you wish (I might even give additional bonus points for demonstrating that you have done this correctly) you can write your own least squares routine (a good discussion of least squares can be found in Shoemaker, D. P., et. al., Experiments in Physical Chemistry, McGraw-Hill, San Francisco, (1981), Chapter 20) and then solve the resulting set of simultaneous normal equations using matrix algebra and functions for inverting and multiplying matrices found in most spreadsheets.

10.6

How much heat is required to heat a 100.0 gram block of silver from 25.0 oC to 100.0 oC at constant pressure? The constant pressure specific heat capacity of silver is:

a 10+3 b 10+6 c 10-5 d

Cp (cal/mole K) 5.09 1.02 ----- 0.36

We can calculate the heat transferred to the silver, the system, by integrating over dq and using the definition of heat capacity:

q = dq = n Cp, Ag (s) dT

= 298.2 K

373.2 K

n [5.09 + 0.00102 T + 0.36x10+5 T-2] dT

= n [5.09 (373.2K - 298.2K)

+ (0.00102/2) [(373.2K)2 - (298.2K)2]

- (0.36x10+5) [1/(373.2K) - 1/(298.2K)]]

= n [382 + 25.7 + 24]

= (100.0 g) (1.000 mol/107.9 g) (432 cal/mol) = + 400 cal

Are T22 - T1

2 and (T2 - T1)2 equal? Are 1/T2 - 1/ T1 and 1/(T2 - T1) equal?

What was the heat transfer in the surrounding, qsurr?

How much heat would be required to heat 50.0 grams of silver between the same two temperatures? Comment on the difference between heat and temperature.