10.1 Stoichiometry and extent of reaction

Extent of reaction is given the symbol ξ (Greek xi). A balanced stoichiometric chemical reaction is written with stoichiometric coefficients a, b, and so on, which are all positive numbers

The IUPAC defines stoichiometric numbers ν_A, ν_B, and so on, as signed quantities, which are negative for reactants and positive for products. Stoichiometric coefficients are the absolute values of the stoichiometric numbers. In other words, a = |v_A| = −v_A but z = |v_z|v_z. The extent of reaction has units of moles and is defined by

(10.1)

This is a positive number that is the same for all reactants and products. n_i is the amount of substance (moles) present at a given time, while n_i,0 is the amount of substance present initially. The stoichiometric equation must be specified to define the extent of reaction. From Eq. (10.1) the extent of reaction can be used to calculate the amount of substance present at any given time,

(10.2)

By differentiation, it yields the change in the amount of substance

(10.3)

10.2 Standard enthalpy change

The standard enthalpy change, ΔH°, is obtained when the reactants and products of a given reaction are present in their standard states at a specified temperature. The standard state is the pure form in equilibrium at 1 bar. Generally, the temperature – if not specified – is taken to be 298.15 K (25 °C). Most elements in the Periodic Table are solids in their standard state at 298 K. The most stable form of the solid is taken as the standard state. For example, graphite is the standard state of carbon, and not diamond or graphene or carbon nanotubes or amorphous carbon. Several other elements have multiple allotropic forms with the standard states as follows: Sb (metallic), As(gray), P(white), Se(gray), S(rhombic) and Sn(white). You should be familiar with the exceptions. The noble gases are all monatomic gases in their standard states. H₂, N₂, O₂, F₂ and Cl₂ are all diatomic gases in the standard state. Br₂ is a diatomic liquid and I₂ is a diatomic solid, but At is an atomic solid. Hg is a monatomic liquid. Ga becomes a liquid just above room temperature at 302.91 K.

Enthalpy changes can be measured for a variety of processes. The process is specified by a subscript after the delta, for example Δ_rH for the enthalpy change that occurs during a reaction. The subscript m after the H specifies the value as a molar enthalpy. The m is often dropped in the literature, so always check the units to determine whether the enthalpy change is given as an absolute value in kJ or molar value in kJ mol⁻¹. The subscripts used to denote a number of processes are listed in Table 10.1. Recommended superscripts appear in Table 10.2.

Reactions are classed as either exothermic or endothermic based on the value of the standard enthalpy change for the reaction, Δ_rH°. Δ_rH° < 0 defines an exothermic reaction, Δ_rH° > 0 defines an endothermic reaction, and Δ_rH° = 0 defines a thermoneutral reaction. An increase in temperature tends to favor product formation for endothermic reactions but disfavors products for exothermic reactions.

From the First Law of Thermodynamics and the definition of work at constant pressure for a system in which only pV work is available, the relationship between changes in internal energy and enthalpy is

(10.4)

Therefore, at constant pressure where Δp = 0,

(10.5)

In other words, internal energy changes and enthalpy changes are virtually identical for any process run at atmospheric pressure that does not involve a change in the volume. If gases are not involved in the process, the change in volume is unlikely to be significant. However, under extremes of pressure – for instance in a geochemical setting or in a diamond anvil cell – the p ΔV term may be significant even if only solids are involved.

10.2.1 Example

By what amount will the change in internal energy differ from the change in enthalpy for the reaction of 0.0500 mol of MgCO₃ with excess dilute HCl(aq) at standard temperature and pressure (STP) for which p = 1 bar and T = 273 K?

Start with a balanced chemical reaction

Accordingly, 0.05 mol MgCO₃ produces 0.05 mol CO₂. The production of a gas leads to a volume change, and the work performed by the system is accomplished by expanding this gas against the pressure of the surroundings. Use the ideal gas law to calculate the volume change and then the work. At constant p and T, the work is given by

where Δn_gas is the change in gas moles = 0.0500 mol.

The work is done by the system on the surroundings and is, therefore, negative. The difference between the internal energy change and the enthalpy change on a molar basis is

10.2.3 Temperature dependence of enthalpy of reaction

In Section 9.6 we have shown that the temperature dependence of the enthalpy is given by

(10.6)

Since Eq. (10.6) is true for each individual substance in the reaction, we can perform the appropriate sums and differences of the enthalpies of all of the reactants and products to calculate Δ_rH for the reaction. From this, we can show that the change in enthalpy of reaction with temperature follows Kirchhoff’s law

(10.7)

where

(10.8)

For all reactions over small ranges of temperature, and for some reactions over extended ranges of temperature change, Δ_rC°_{p, m} is constant and can be pulled from the integral. The enthalpy of reaction at T₂ is then given by

(10.9)

10.3 Calorimetry

The standard method for determining the enthalpy change during a reaction is calorimetry – the measurement of temperature rise after a known amount of material has reacted. The temperature rise is related to the heat evolved in the process, q, and the heat capacity, C, by the fundamental equation of calorimetry,

(10.10)

A calorimeter must correspond as close as possible to a closed system. Neither heat nor matter should be exchanged with the surroundings. This is not that difficult to approximate in practice if the reaction can be run in a thermos flask and there is some way to initiate the reaction so that it is clear what the temperature is before the reaction and after the reaction. Initiation of the reaction may occur after mixing of reagents, which requires some sort of mechanical intervention inside of the thermos flask. This may require dropping one container into another, breaking a barrier between containers, or the injection of fluids.

One of the most common types of calorimeter is a constant volume bomb calorimeter. This is shown in Fig. 10.2. The bomb is a stainless steel tube with a removable cap that can be sealed. A solid or liquid sample is placed in a cup near an igniter. The igniter is a wire through which current is passed to heat it sufficiently to initiate combustion. The igniter is consumed in the reaction. The bomb is sealed after the sample is loaded into the cup and a fresh wire has been coiled to form the igniter. The bomb is pressurized with excess O₂ to a pressure of approximate 3 MPa. The bomb is then placed in a known quantity of water inside of a thermos flask. A thermometer is placed in the water, which is stirred vigorously. The water provides good thermal contact with the stainless steel bomb. The mass of the sample used is matched to the mass of the water and the bomb so that the temperature rise that accompanies the reaction is no more than a few degrees, usually about 2–3 K. In principle, the measurement should be made at close to constant temperature so that the temperature dependence of the heat capacities of materials involved need not be considered. On the other hand, if the temperature rise is too small, uncertainties in the measurement are increased. Hence, the temperature rise of ΔT ≈ 2–3 K represents a good compromise.

The bomb, its contents, and the water-filled thermos with temperature measurement are then placed inside of a thermally isolated jacket. Highly accurate work will correct for the energy provided by mixing and the igniter, as well as accounting for thermal drift. The system is allowed to equilibrate thermally, after which the temperature is recorded for several minutes before the ignition and several minutes after ignition, as shown in Fig. 10.3. Extrapolation is then used to determine the temperature change, ΔT. In many practical applications benzoic acid is used as a standard for calibration.

10.3.1 Example

A bomb calorimeter is built with a specific heat capacity of 1.58 kJ K⁻¹. Calculate the molar internal energy change if the combustion of 120 mg of naphthalene, C₁₀H₈(s), leads to a temperature rise of 3.06 K.

Because of the small temperature change, we treat the heat capacity as constant, pull it from the integral in Eq. (10.10) and arrive at Eq. (10.11),

(10.11)

The heat q_V generated by the reaction at constant volume is given by the product of the amount of substance combusted and the internal energy change,

(10.12)

Note how the subscript c is used to denote this internal energy change specifically corresponds to the change during combustion. With a molar mass of 128.17 g mol⁻¹, 120 mg corresponds to 0.936 mmol of naphthalene. By the First Law of Thermodynamics, the heat released by combustion − q_V must be equal and opposite to the energy transferred to the calorimeter q,

(10.13)

Equating these expressions for heat evolved by the reaction and heat absorbed by the calorimeter – that is, assuming a perfect calorimeter with no contribution from the igniter – we solve the resulting equation for internal energy change

10.3.2 Example

Calculate Δ_cH°_m at 298.15 K for the benzoic acid calorimetry experiment in Example 7.9.2.3. Recall that 0.4089 g of C₆H₅COOH were combusted at T_i = 20.17 °C with T_f = 22.22 °C corresponding to an internal energy change of Δ_cU°_m = −3226 kJ mol⁻¹.

Start with a balanced reaction

This is written as a standard combustion reaction. A standard combustion reaction corresponds to the reaction of 1 mol of the named reactant with however many moles of gaseous O₂ it takes to oxidize it completely to products. The reaction temperature is 298.15 K unless otherwise stated. If the reactant is a hydrocarbon (or contains only H, C and O), the products are CO₂(g) and H₂O(l). In principle, the same equation applies to a compound containing another element if that element ends in its standard state; for example, N if the product is N₂(g). In general, however, the exact products containing the other elements must be known in order to calculate the enthalpy of combustion.

To calculate the value of Δ_cH°_m, recall that Δ_cH = Δ_cU + Δ(pV). When all reactants and products are in condensed phases at near atmospheric pressure, Δ(pV) is negligible in comparison with Δ_cH and Δ_cU. When gases are involved, Δ(pV) cannot be ignored. Assuming ideal gas behavior, we have

(10.14)

(10.15)

Δn_gas is the change in the amount of substance of gas during the course of one full stoichiometric equivalent of reaction. In other words, Δn_gas is the difference between the stoichiometric numbers of the gaseous products minus the stoichiometric numbers of the gaseous reactants, Δn_gas = ν_gas,products – ν_{gas,reactants}. T refers to the final temperature. For the reaction as written above Δn_gas = 7 – 7.5 = −0.5. Hence

and the enthalpy of combustion is Δ_cH°_m = −3227 kJ mol⁻¹ at the final temperature of 293.32 K. To correct this value to the reference temperature of 298.15 K, we apply Eq. (10.9)

with T₂ = 298.15 K, T₁ = 293.32 K, ΔT = T₂ − T₁ and

The temperature correction amounts to only −33 J mol⁻¹ and is inconsequential in this case. What does the mol⁻¹ refer to? It refers to one full stoichiometric equivalent of reaction as written. Energy in the amount of 3227 kJ is evolved per mole of C₆H₅COOH combusted but per 7.5 mol of O₂ consumed. In other words, per mole for the balanced reaction as written. The enthalpies of combustion of several compounds are listed in Table 10.3.

Compound	Formula	M / g mol−1	ΔcH°m / kJ mol−1	Specific heat of combustion / kJ g−1
Graphite	C	12.011	393.5	32.76
Hydrogen	H2	2.0159	285.8	141.77
Ammonia	NH3	17.031	382.8	22.48
Methane	CH4	16.043	890.8	55.53
Propane	C3H8	44.096	2219.3	50.33
Octane	C8H18	114.229	5470	47.89
Naphthalene	C10H8	128.171	5156.3	40.23
Biphenyl	C12H10	154.207	6251	40.54
Methanol	CH4O	32.042	726	22.66
Ethanol	C2H6O	46.068	1366.8	29.67
1-Naphthol	C10H8O	144.17	4957	34.38
Benzophenone	C13H10O	182.217	6510	35.73
Benzoic acid	C7H6O2	122.122	3228.2	26.43
Glucose	C6H12O6	180.155	2803	15.56
Lactose	C12H22O11	342.296	5630	16.45

10.4 Phase transitions

When heat is added to a substance, its temperature increases in accord with Eq. (10.11). This is shown in Fig. 10.4. As long as no phase transitions or reactions occur, the temperature increases in a simple manner determined by the heat transfer rate and the heat capacity. At a phase transition temperature, further addition of heat does not increase the temperature. Instead, the system enters an isothermal region in which the added heat is used to facilitate the phase transition rather than increasing the temperature. Once all of the substance has been converted to the new phase, further heat transfer leads to increases in temperature again. The new phase will have a different heat capacity.

The example of water is shown in Fig. 10.4. starting in the liquid phase. The liquid generally has the highest heat capacity of any phase of a material so the response of temperature to heat transfer is the smallest for this phase – that is, the slope of the temperature versus time curve is the shallowest. However, upon heating, the pattern of increasing temperature, followed by an isothermal region, followed by further increasing temperature is the same for any first-order phase transition. For first-order phase transitions (the most common kind, as discussed and defined further in Chapter 13) there is always a discontinuous change in the heat capacity from one phase to the next. There also will always be an isothermal region because such phase transitions are accompanied by an enthalpy of transition that has to be supplied to make the transition occur (or removed upon cooling).

Phase changes are accompanied by changes in the orientations and distances between molecules (or atoms). The existence of intermolecular interactions means that phase changes will also be accompanied by enthalpy changes. The more drastic the change in molecular orientations and intermolecular distance, the larger the enthalpy changes. The melting of water is endothermic by 6.01 kJ mol⁻¹ at 273 K, whereas vaporization of water at 298 K is endothermic by 43.98 kJ mol⁻¹. The vaporization of Au is endothermic by 324 kJ mol⁻¹. Therefore, both the strength of intermolecular interactions and the extent of change of relation of molecules to one another in the two phases determine the magnitude of the enthalpy change during the phase transition.

The enthalpy change for a phase transition is denoted Δ_trsH. The heat evolved at constant pressure is equal to the enthalpy change. The standard temperature of a phase transition is the temperature at which the phase transition occurs at equilibrium when the pressure is equal to the standard state value of p° = 1 bar. In condensed phases it is of little consequence whether the standard state pressure is taken as 1 bar or 1 atm. The difference is more measureable for gases. Therefore, one will encounter the terms normal transition temperature (corresponding to p = 1 atm) in addition to the standard transition temperature (p = 1 bar).

The heat evolved at the standard pressure while a phase transition occurs at its standard transition temperature is equal to the standard enthalpy of the phase transition. In molar units this is

(10.16)

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