Question 14.15: A 1.000-gram sample of octane, C8H18(l), is burned in a calo......
A 1.000-gram sample of octane, C_{8}H_{18}(l), is burned in a calorimeter like that shown in Figure 14.18, and the observed temperature increase is 1.679 K. The total heat capacity of the calorimeter is c_{V,cal}= 28.46\ kJ·K^{–1} (where we write c_{V,cal} to emphasize that the value of the heat capacity of the calorimeter is that at constant volume in this case).
(a) Calculate the energy of combustion per gram and per mole of C_{8}H_{18}(l).
(b) Using the molar enthalpy of formation values listed in Table 14.3, show that ΔH_{rxn} ≈ ΔU_{rxn} for this reaction at 25°C.
| TABLE 14.3 standard enthalpies of formation, ΔH^{\circ}_{f}, for various substances a 25°C | |||||
| Substance | Formula | ΔH^{\circ}_{f}/kJ\cdot mol^{-1} | Substance | Formula | ΔH^{\circ}_{f}/kJ\cdot mol^{-1} |
| aluminum oxide | Al_{2}O_{3}(s) | -1675.7 | hydrogen fluoride | HF(g) | -273.3 |
| ammonia | NH_{3}(g) | -45.9 | hydrogen iodide | HI(g) | +26.5 |
| benzene | C_{6}H_{6}(l) | +49.1 | hydrogen peroxide | H_{2}O_{2}(l) | -187.8 |
| benzoic acid | C_{6}H_{5}COOH(s) | -385.2 | iodine vapor | I_{2}(g) | +62.4 |
| bromine vapor | Br_{2}(g) | +30.9 | magnesium carbonate | MgCO_{3}(s) | -1095.8 |
| butane | C_{4}H_{10}(g) | -125.7 | magnesium oxide | MgO(s) | -601.6 |
| calcium carbonate | CaCO_{3}(s) | -1207.6 | magnesium sulfide | MgS(s) | -346.0 |
| carbon (diamond) | C(s) | +1.897 | methane | CH_{4}(g) | -74.6 |
| carbon (graphite) | C(s) | 0 | methanol (methyl alcohol) | CH_{3}OH(l) CH_{3}OH(g) |
-239.2 -201.0 |
| carbon (buckminster fullerene) | C_{60}(s) | +2327.0 | methyl chloride | CH_{3}Cl(g) | -81.9 |
| carbon dioxide | CO_{2}(g) | -393.5 | nitrogen dioxide | NO_{2}(g) | +33.2 |
| carbon monoxide | CO(g) | -110.5 | nitrogen oxide | NO(g) | +91.3 |
| carbon tetrachloride | CCl_{4}(l) CCl_{4}(g) |
-128.2 -95.7 |
dinitrogen tetroxide | N_{2}O_{4}(g) N_{2}O_{4}(l) |
+11.1 -19.5 |
| chromium (III) oxide | Cr_{2}O_{3}(s) | -1139.7 | octane | C_{8}H_{18}(l) | -250.1 |
| cyclohexane | C_{6}H_{12}(l) | -156.4 | pentane | C_{5}H_{12}(l) | -173.5 |
| ethane | C_{2}H_{6}(g) | -84.0 | propane | C_{3}H_{8}(g) | -103.8 |
| ethanol (ethyl alcohol) | CH_{3}CH_{2}OH(l) | -277.6 | sodium carbonate | Na_{2}CO_{3}(s) | -1130.7 |
| ethene (ethylene) | C_{2}H_{4}(g) | +52.4 | sodium oxide | Na_{2}O(s) | -414.2 |
| ethyne (acetylene) | C_{2}H_{2}(g) | +227.4 | sucrose | C_{12}H_{22}O_{11}(s) | -2226.1 |
| freon-12 (dichloro difluoromethane) | CF_{2}Cl_{2}(g) | -477.4 | sulfur dioxide | SO_{2}(g) | -296.8 |
| glucose | C_{6}H_{12}O_{6}(s) | -1273.3 | sulfur trioxide | SO_{3}(g) | -395.7 |
| hexane | C_{6}H_{14}(l) | -198.7 | tin(IV) oxide | SnO_{2}(s) | -577.6 |
| hydrazine | N_{2}H_{4}(l) N_{2}H_{4}(g) |
+50.6 +95.4 |
water | H_{2}O(l) H_{2}O(g) |
-285.8 -241.8 |
| hydrogen bromide | HBr(g) | -36.3 | |||
| hydrogen chloride | HCl(g) | -92.3 | |||
| Data from CRC Handbook of Chemistry and Physics, 86th Ed., Ed. David R. Lide, CRC Press, 2005–2006. (More thermodynamic data are given in Appendix D.) | |||||
(a) The equation for the combustion reaction is
2\ C_{8}H_{18}(l) + 25\ O_{2}(g) → 16\ CO_{2}(g) + 18\ H_{2}O(l)The energy of combustion is given by
\Delta U = –c_{V,cal}\Delta Twhich is analogous to Equation 14.36 under conditions of constant volume.
Using this equation, we find that the value of ΔU for 1.000 grams of C_{8}H_{18}(l) is
\Delta H = –c_{P,cal}\Delta T (14.36)
\Delta U_{rxn}\ [per\ 1.000\ g\ C_{8}H_{18}(l)] = –(28.46\ kJ·K^{–1})(1.679\ K·g^{–1})= –47.78\ kJ·g^{–1}because the observed ΔT is associated with the combustion of 1.000 grams of octane. The value of \Delta U_{rxn} per mole of octane (molar\ mass = 114.22\ g·mol^{–1}) is
\Delta U_{rxn}\ [per\ mol\ C_{8}H_{18}(l)] = (-47.78\ kJ·g^{–1})(114.22\ g·mol^{–1})= –5457\ kJ·mol^{–1}or in terms of the overall reaction equation we write,
C_{8}H_{18}(l) + \frac{25}{2} O_{2}(g) → 8\ CO_{2}(g) + 9\ H_{2}O(l) \quad \Delta U_{rxn} = –5457\ kJ·mol^{–1}Here we balanced the chemical equation using fractional coefficients rather than whole number coefficients because we are interested in the standard enthalpy of combustion per mole of octane.
(b) Using the values listed in Table 14.3 and applying Equation 14.24, we find that
ΔH^{\circ}_{rxn} =ΔH^{\circ}_{f}\ [products] − ΔH^{\circ}_{f}\ [reactants] (14.24)
\Delta H_{rxn} = \{8\ ΔH^{\circ}_{f} [CO_{2}(g)] + 9\ ΔH^{\circ}_{f} [H_{2}O(l)]\} – \{ΔH^{\circ}_{f} [C_{8}H_{18}(l)] + \frac{25}{2} ΔH^{\circ}_{f} [O_{2}(g)]\}\\= \{8(–393.5\ kJ·mol^{–1}) + 9(–285.8\ kJ·mol^{–1})\} – \{(–250.1\ kJ·mol^{–1})\}\\= –5470.1\ kJ·mol^{–1}Hence, \Delta H_{rxn} and \Delta U_{rxn} differ by two-tenths of a percent, showing that ΔH_{rxn} ≈ ΔU_{rxn} is a good approximation in this case.