## Enthalpy Dependence on Temperature

Let AH be the enthalpy change owing to a chemical reaction.

By definition, AH is equal to the sum of the enthalpies of the products of a reaction minus the sum of the enthalpies of all reactants. Thus, AH = enthalpy of products — enthalpy of reactants. When there is a single product, then the enthalpy change of the reaction can be expressed per kilomole of product by dividing AH by the number of kilomoles of product created. An equivalent definition holds for the free energies. In mathematical shorthand,

where, nPi = number of kilomoles of the ith product, nRi = number of kilomoles of the ith reactant, Hp = enthalpy of the ith product, and Hr = enthalpy of the ith reactant.

For the water synthesis reaction,

where HH2O is the enthalpy of water, and so on.

Of course, at RTP, for water vapor, HH2O = —241.8 MJ/kilomole, and HH2 and HO2 are both zero by choice. 1 Hence, AH = —241.8 MJ/kilomole.

If the value of the enthalpy, H0 of a given substance (whether product or reactant), is known at a given temperature, T0, then at a different temperature, T, the enthalpy is

J To where cp is the specific heat at constant pressure.

^Remember that, by convention, the enthalpy of all elements in their natural state under normal conditions on Earth is taken as zero.

cp is somewhat temperature dependent, and its value for each substance can be found either from tables or from mathematical regressions derived from such tables. For rough estimates, cp, can be taken as constant. In this latter case,

Replacing the various enthalpies in Equation 9.49 by their values as expressed in Equation 9.51, pTp. ____('Tr.

For a simple estimate of the changes in AH, one can use the constant cp formula:

AH = AH0 + nPiCpPi (Tp - ToP. ) nRiCpRi (T^ - T0Ri ).

Example 1

The standard enthalpy of formation of water vapor is -241.8 MJ/kmoles. What is the enthalpy of formation when both reactants and product are at 500 K?

We need to know the specific heats at constant pressure of H2, O2, and H2O(g). To obtain an accurate answer, one must use Equation 9.52 together with tabulated values of the specific heat as a function of temperature. An approximate answer can be obtained from Equation 9.52a using constant values of the specific heats. We shall do the latter.

We saw in Chapter 2 that, if one can guess the number of degrees of freedom, v, of a molecule, one can estimate the specific heat by using the formula

The advantage of this procedure is that it is easier to remember v than cp. For diatomic gases, v can be taken as 5, yielding a cp = 29.1 kJ kmole_1K_1, and for water vapor, v can be taken as 7, yielding cp = 37.4 kJ kmole^K-1.

(Continued)

Using Equation 9.52a (in this problem, all AT are the same: 500 K —298 K),

—243.1 MJ per kmole of water vapor.

Since the enthalpy changes only little with temperature, these approximate results are close to the correct value of —243.7 MJ/kmole obtained through the use of Equation 9.52.

## Solar Stirling Engine Basics Explained

The solar Stirling engine is progressively becoming a viable alternative to solar panels for its higher efficiency. Stirling engines might be the best way to harvest the power provided by the sun. This is an easy-to-understand explanation of how Stirling engines work, the different types, and why they are more efficient than steam engines.

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