Thermochemical equations and Heat of reaction

Thermochemistry

It is well known fact that energy changes do take place in almost all the chemical reactions. Quite often, the energy change accompanying a chemical reaction is more significant than reaction itself. For example, liberation of heat energy during the burning of coal is more significant to a common man than the production of carbon dioxide.

The branch of chemistry which deals with the energy changes associated with chemical reactions is called thermochemistry.

Thermochemical Equations

We know that the chemical equation is a brief representation of a chemical change in terms of symbols or formulae of reactants and products. When the heat change accompanying the chemical reaction is also included in the chemical equation, it is known as thermochemical equation. While, writing a thermochemical equation, it is known as thermochemical equation. While writing a thermochemical equation, the heat evolved in case of exothermic reactions or the heat absorbed in case of endothermic reactions is indicated on the product side of the balanced chemical equation. For example,

H₂(g) + 1\2 O₂ (g) \(\longrightarrow\) H₂O(l), \(\Delta\)H = -68.4 kcals

C(graphite) + 2S(rhombic) \(\longrightarrow\) CS₂(l); \(\Delta\)H = +22.0 kcals

The most stable physical state of reacting species and the products is also indicated in brackets. Thus, a thermochemical equation gives complete information about the material change and the associated heat change. It is clear that the first reaction is exothermic and the second one endothermic.

Heat of Reaction

Heat of a reaction is defined as the quantity of heat evolved or absorbed when evolved or absorbed when molar quantities of substances react in amounts represented by chemical equation at a given temperature.

Now. If the reaction is carried out at constant temperature and constant volume, heat evolved or absorbed is equal to the change in internal energy. Thus, the heat of reaction at constant volume at a certain temperature is defined as the change in internal energy (\(\Delta\)E ) of the system when number of moles of substances react in amounts represented by chemical equation. If E_R and E_p represents the internal energy of reactants and products respectively, then heat of reaction at constant volume is given by

$$\Delta E = E_P – E_R = q_v $$

In the similar manner, the heat of reaction at constant pressure at a certain temperature is defined as the change in enthalpy (\(\Delta\)H) of the system when number of moles of substance react in amounts represented by chemical equation. Thus, if H_R and H_P represents the enthalpy of reactants and products respectively, then heat of reaction at constant pressure is given by

\(\Delta\)H = H_p – H_R = q_p

Relation between heat of reaction at constant volume(\(\Delta\)E) and

at a constant pressure (\(\Delta\)H) :

The quantities \(\Delta\)H and \(\Delta\)E are related to each other by the relation :

$$\Delta H =\Delta E + P.\Delta V$$

Where \(\Delta\)V is the change in volume that takes place in a given reaction. In most of the cases, the heat of reaction is measured at constant pressure, however, in some cases it is measured at constant pressure, however, in some cases it is measured at constant volume, e.g., measurement of heat of combustion in a bomb caloriment in a bomb calorimeter. The above relationship can be used if necessary for the conversion of \(\Delta H\) into \(\Delta\)E and vice-versa.

The above relationship can be simplified further. Assuming the gas laws to be valid, we have the general gas equation

PV = nRT

Where V is the volume occupied by ‘n’ moles of gas.

Let n₁ and n₂ represent the number of moles of gaseous reactants and gaseous products respectively. Then, the change in number of moles \(\Delta\) is given by

$$\Delta n = (n_2 – n_1)$$

Thus, the corresponding change in volume ‘\(\Delta\)V’ will be given by

\(\Delta\)V = V/n * \(\Delta\)n

Or, P. \(\Delta\)V = PV/n* \(\Delta\)n

Or, P.\(\Delta\)V = RT * \(\Delta\)n

Hence, \(\Delta\)H = \(\Delta\)E +\(\Delta\)nRT

In this equation, \(\Delta\)n is the difference between the number of moles of the gaseous products and the gaseous reactants. For example,

In the dissociation of phosphorus pentachloride, PCl₅

PCl₅(g) \(\longrightarrow\) PCl₃(g) + Cl₂(g)

1 mole 1 mole 1 mole

\(\Delta\)n = Number of moles of gaseous products – Number of moles of gaseous reactants

Or, \(\Delta\)n = 2-1 = 1

Therefore, \(\Delta\)H = \(\Delta\)E + RT

In the reaction involving combination of NO and O_2,

2NO(g) + O₂(g) \(\longrightarrow\) 2NO₂(g)

2moles 1 moles 2moles

\(\Delta\)n = 2-3 = -1

Therefore, \(\Delta\)H = \(\Delta\)E – RT

For reactions involving only solids and liquids, the P.\(\Delta\)V term will be negligible, therefore, \(\Delta\)H = \(\Delta\)E, i.e., the heat of reaction at constant pressure is virtually the same as the heat of reaction at constant volume. When the reaction mixture contains gaseous components in addition to solid and liquid, then for calculating \(\Delta\)n, only the number of moles of gaseous products and reactants are considered.

Variation of heat of reaction with temperature ( The Kirchoff’s equation)

Heat of a chemical reaction depends on temperature due to variation of specific heat, KIrchoff derived an expression with help of which the dependence of the heat of reaction on temperature can be expressed. Kirchoff’s equation can be derived as follows.

Heat of a reaction at constant volume is given by the relation

\(\Delta\)E = E_P – E_R …………………………………… (1)

Where E_R and E_P are internal energies of reactants and products.

Differentaiting equation (1) with respect to ‘T’ at constant ‘V’, we have

( d\(\Delta\)E/dT)_v = (dE_P/dT)_V – (dE_R/dT)_v …………………………(2)

But, (dE/Dt)_v = C_v (Heat capacity at constant volume)

Therefore, equation (2) can be written as

(dE\dT)_v = (C_v)_p – (C_V)_R = \(\DeltaC_V ………………………….(3)

Where (C_v)_R and (C_v)_p are the heat capacities at constant volume of the reactants and products respectively. From equation (3), we see that change in heat of reaction at constant volume per degree change in temperature is equal to the difference in heat capacities at constant volume of products and reactants.

For determining the heat of reaction at particular temperature T₂, when it is known at a known temperature T_1, the above equation (3) is integrated within the limits T₁ and T₂.

Thus,

(\(\Delta\)E)_T2 – (\(\Delta\)E)_T1 = \(\int\) \(\Delta\)C_v dT

Or, (\(\Delta\)E)_T2 – (\(\Delta\)E)_T1 = \(\DeltaC_v(T₂- T₁) ………………………. (4)

Where (\(\Delta\)E)_T2 and (\(\Delta\)E)_T1 are the heat of reaction at temperatures T₂ and T₁ respectively.

Similarly, heat of reaction at constant pressure, \(\Delta\)H, is given by equation

\(\Delta\)H = H_p – H_R …………………………………….. (5)

Where H_P and H_R are the heat contents of reactants and products respectively.

Differentiating equation (5) with respect to ‘T’ at constant P, we have

[d(\(\Delta\)H/dT]_p = (dH_p/dT)_p – (dH_R\dT)_p ………………………………………….(6)

But , we know that

[dH/dT]_p = C_p (heat capacity at constant pressure)

Thus, equation (6) can be written as

[d(\(\Delta\)H\dT]_p = (C_p)_P – (C_P)_R = \(\Delta\)C_P …………………………………(7)

Where (C_p)_p and (C_p)_R are heat capacities of products and reactants at constant pressure respectively. From equation (7), we see that the change in heat of reaction at constant pressure per degree change in temperature is equal to the difference in heat capacities at constant pressure of products and reactants.

Integrating equation (7) between temperatures T₁ and T_2, we get

(\(\Delta\)H)_T2 – (\(\Delta\)H)_T1 = \(\int (\Delta\)C_p.dT

Or, (\(\Delta\)H)_T2 – (\(\Delta\)H)_T1 = \(\Delta\)C_p (T₂ – T₁) ……………………………(8)

Equations (3), (4), (7) and (8) are known as Kirchoff’s equations and are used to calculate the heat of reaction at a particular temperature, if it is known at some other temperature.

Reference :

Maron, S.H. and Prutton C.F.,. Principles of Physical Chemistry. Oxford and IBH publication Company, 1992.

Moore, Walter J.,. Physical Chemistry. New Delhi: Orient Langman Ltd, 1999.

Puri, B.R., Sharma, L.R., and Pathania M.S.,. Principles of Physical Chemistry. Jalandhar: Vishal Publishling Co., , 2008.