THERMOCHEMISTRY
4.10 DIFFERENTIAL SCANNING CALORIMETRY
When moderate amounts of heat are supplied to a solution of simple salts in water, we expect a smooth heating curve between, say 290 and 320 K, like the lower curve in Fig. 4.3. The heat capacity of water is nearly constant over this temperature range,
DIFFERENTIAL SCANNING CALORIMETRY 67 and it will be little affected by small amounts of dissolved salts. Electrical circuitry exists that permits one to supply heat to a dilute solution in an adiabatic (insulated) calorimeter in very small pulses which may be regarded as infinitesimals dq. We nor- mally carry out the experiment at constant pressure, so the definition of heat capacity at constant pressure Cp =dqp/d T is satisfied. The gradual temperature rise over many small pulses can be followed by means of a thermistor circuit or its equivalent.
If, instead of a dilute solution of simple salts, the calorimeter contains a solute that is capable of undergoing a thermal reaction, which is a reaction brought about by heat, the heating curve is more complicated. Thermal reactions are important in many areas, especially in biochemistry. Proteins undergo heat denaturation. Heat denaturation involves unfolding of the native protein and requires breaking of some or many of the bonds holding it in its native structure. Heat denaturation may be quite specific as to the temperature at which it occurs, and it may bring about subtle changes in the protein, like changes in physiological activity, or it may bring about gross changes in the form of the protein as in the cooking of an egg.
Because heat denaturation involves breaking of internal bonds in the protein, it requires an enthalpy input at constant pressure. The reaction is endoenthalpic. A dilute solution of salt and protein takes more heat to bring about a small temperature change than would the solution without the protein. The difference is observed only at or near the temperature of denaturation. Thus we have a normal temperature rise until denaturation begins, after which the heat capacity of the solution is abnormally large until we achieve complete thermal denaturation whereupon the temperature rise drops back to the normal baseline of a salt solution. Plotting Cp as a function of T, we see a peak at the denaturation temperature. This is the upper line in Fig. 4.3. It is a simple matter to interface a computer to the scanning calorimeter output and to integrate under the experimental curve:
denH= Tf
Ti
Cpd T
Cp
T
FIGURE 4.3 Schematic diagram of the thermal denaturation of a water-soluble protein. The straight line is the baseline of salt solution without protein. The peak is due to endoenthalpic denaturation of the protein.
The heat capacity curve of the simple salt solution (the baseline) is subtracted from the experimental result. There may be multiple peaks if there is more than one protein in the test solution or if the protein is capable of unfolding in sequential steps.
PROBLEMS AND EXAMPLE
Example 4.1 Oxygen Bomb Calorimetry
Exactly 0.5000 g of benzoic acid C6H5COOH were burned under oxygen. The combustion produced a temperature rise of 1.236 K. The same calorimetric setup was used to burn 0.3000 g of naphthalene (C10H8) and the resulting temperature rise was 1.128 K. The heat of combustion of benzoic acid is qV =cU298=–3227 kJ mol−1 (exothermic). What is the heat of combustion qV =cU298of naphthalene?
Solution 4.1 The corresponding molar masses are: benzoic acid, 122.12 g mol−1; and naphthalene, 128.19 g mol−1. The temperature rise for each combustion was: ben- zoic acid, 1.236/0.5000=2.472 K g−1; and naphthalene, 1.128/0.3000=3.760 K g−1. Multiplying by the molar masses in each case, one obtains 301.9 K mol−1for benzoic acid and 482.0 K mol−1for naphthalene. This gives us the ratio
301.9
482.0 = −3227 x
x= −3227482.0
301.9 = −5152 kJ mol−1 qV =cU298= −5152 kJ mol−1
Notice that the units cancel on the left; thus x has the units of kJ mol−1, not K mol−1. The handbook value iscU= −5156 kJ mol−1.
Problem 4.1
A resistor of precisely 1 ohm is immersed in a liter (1 dm3) of water in a perfectly insulated container. Suppose that precisely 1 ampere flows through the resistor for precisely 1 second. What is the temperature rise of the water?
Problem 4.2
Exactly one gram of a solid substance is burned in a bomb calorimeter. The bomb absorbs as much heat as 300 g of water would absorb. (Its water equivalent is 300 g.) The bomb was immersed in 1700 g of water in an insulated can. During combustion of the sample, the temperature went from 24.0◦C to 26.35◦C. What is the heat of combustion per gram of the sample? What is the molar enthalpy of combustion if the molar mass of the substance is 60.0 g mol−1 and 2 mol of gas are formed in excess of the O2burned?
PROBLEMS AND EXAMPLE 69 Problem 4.3
The enthalpy of formation of liquid acetic acid CH3COOH(l) is fH◦=
−484.5 kJ mol−1. What iscH ? Problem 4.4
The enthalpy of combustion of solidα-d-glucose, C6H12O6(s) is−2808 kJ mol−1. What is its enthalpy of formation?
Problem 4.5
EstimatecH298(n-octane(g)) of n-octane by the hydrogen-atom counting method for alkanes.
Problem 4.6
FindfH298(2,4-dimethylpentane(g)) by the hydrogen atom counting method. What is the enthalpy of isomerization of n-heptane(g) to 2,4-dimethylpentane(g) according to this method? Compare your answer with the experimental result of −14.6 ± 1.7 kJ mol−1.
Problem 4.7
The input file for a GaussianC quantum mechanical calculation will be discussed in later chapters. Briefly, it consists of a few lines of instructions to the computer concerning memory requirements, the number of processors to be used, and the Gaussian procedure to be used, followed by an approximate geometry of the molecule.
In simple cases, the input geometry can be merely a guess based on what we learned in general chemistry. The machine specifications will vary from one installation to another. Our input file for the water molecule is
%mem=1800Mw
%nproc=1
# g3 water
0 1
H -1.012237 0.210253 0.097259 O -0.260862 0.786229 0.119544 H 0.489699 0.209212 0.142294
Adapt this input file for your system and run the water molecule. What is the optimized geometry in the form of Cartesian coordinates (like the input file)? What is the O H bond length? What is the H O H bond angle? What energy is given for water?
Problem 4.8
Plot the heat capacity of ethylene from the following data set:
Temperature (K) Heat capacity, Cp(J K−1mol−1)
300.0000 43.1000
400.0000 53.0000
500.0000 62.5000
600.0000 70.7000
700.0000 77.7000
800.0000 83.9000
900.0000 89.2000
1000.0000 93.9000
Problem 4.9
Suppose that the heat capacities Cp for N2, H2, and NH3 in the standard state are constant with temperature change (they aren’t) at 29.1, 28.8, and 35.6 J K−1mol−1. Suppose further thatrH◦for the reaction
N2+3H2→2NH3
is−92.2 kJ mol−1(of N2consumed) at 298 K. What isCpfor the reaction? What isrH◦at 398 K? What is the (hypothetical)rH◦at 0 K?