4. A consideration of the noble gas which exists in the monatomic state shows that there is no internal energy due to potential energy. The only energy is kinetic energy. The kinetic theory of matter (Chapter 58) derives the fundamental equation:

$PV=\frac{1}{3}Nm{u}^{-2}$

    for 1 mole of gas, and the ideal gas law for one mole of gas is:

$PV=RT$

    By combining these equations:

$RT=\frac{1}{3}Nm{u}^{-2}orNm{u}^{-2}=3RT$

    The total internal energy is U, and since the average kinetic energy of 1 mole of gas is 1/2Nmu⁻² then

$U=\frac{1}{3}Nm{u}^{-2}or2U=Nm{u}^{-2}$

    Since 3RT = Nmu⁻² then

$2U=3RTorU=\frac{3}{2}RT$

    Differentiating with respect to T gives:

$\frac{dU}{dT}=\frac{3R}{2}$

    The expression dU/dT is the rate of change of the internal energy with temperature, which is the energy change associated with 1 degree Kelvin. This is defined as the molar heat capacity of the gas at constant volume and symbolised Cv. Hence

$Cv=\frac{3}{2}R=\frac{3}{2}\times 8.314=12.465Jmo{l}^{-1}{K}^{-1}$

5. Let 1 mole of a monatomic gas be contained in a cylinder fitted with a weightless friction-free piston as shown in Figure 61.1, and let the pressure due to the gas above and below the piston be P and the volume of the gas be V. If the cylinder is heated, the kinetic energy of the gas increases and causes an increase in temperature. This increases the force acting on the piston and the piston will move to accommodate the increased volume of the gas. The heat energy given to the cylinder increases the internal energy of the gas and in addition does work in moving the piston.

    This can be represented by the equation

$q=\Delta U=w$

    the first law of thermodynamics, where q is the heat energy supplied, ΔU is the change in internal energy and w is the work done by the system. This means that the value of Cp, the molar heat capacity at constant pressure, is greater than Cv according to the equation:

$C_P=C_v+workdone$

    where the work done=pressure × increase in volume.

    For a pressure P the work done in changing the volume from V to V’ is

$P\left(V’-V\right)orPV’-PV$

    For 1 mole of a monatomic gas

$PV=RTandPV’=RT’$

    The work done can be expressed as

$RT’-RT=R\left(T’-T\right)$

    The temperature change for the molar heat capacity is 1 K hence

$T’-T=1$

    This means that for a monatomic gas

$C_p=C_v+workdone=C_v+R$

    Cv has been shown to be 12.465 Jmol⁻¹ K⁻¹ and since R=8.314 Jmol⁻¹ K⁻¹ then Cp= 20.779 Jmol⁻¹ K⁻¹. For a diatomic gas it can be shown that Cv= 20.8 and Cp= 29.1 Jmol⁻¹ K⁻¹ and for a triatomic gas Cv= 25.0 and cp= 33.3 Jmol⁻¹ K⁻¹.

    Hence at constant pressure, the heat energy absorbed is a combination of the internal energy of the system and the work done by the system. This is represented by the equation:

$\Delta H=\Delta U+P\Delta V$

    where ΔH is the enthalpy change of the system.

6. A consideration of the internal energy of a chemical system involving several polyatomic molecules is difficult. In order to obtain information about energy changes that occur, the measurements are made at constant pressure and hence are called the enthalpy changes, ΔH.

7. When a chemical reaction releases heat energy to its surroundings it is called an exothermic reaction and is represented by the equation:

$\text{Reactans=Products}\Delta H\text{is}\text{negative}$

8. There are many different types of enthalpy changes found in chemistry and they are usually stated under the standard conditions of 101.3 kPa and 298 K, and given the symbol ΔH.

9. The standard enthalpy of combustion ΔH_c is the enthalpy change when one mole of a substance (the relative molecular mass, in grams) is completely burned in oxygen, under standard conditions. For example:

$C\left(s\right)+O_2\left(g\right)=C{O}_2\left(g\right),\Delta H_c^{\theta }=-393.4kJmo{l}^{-1}$

    The standard enthalpy of formation, ΔH_f the enthalpy change when one mole of a substance is prepared from its elements, under standard conditions. For example

$H_2\left(g\right)+S\left(s\right)=H_{2}S\left(g\right),\Delta H_f^{\theta }=-20.6kJmo{l}^{-1}$

    The standard enthalpy of solution, ΔH_s is the enthalpy change when one mole of a substance is dissolved in a solvent to give a solution of defined concentration, under standard conditions. For example,

$HCl\left(g\right)+aq=HCl\left(aq\right),\Delta H_s^{\theta }=-72.38kJmo{l}^{-1}$

    Since the most common solvent used in chemistry is water, the standard enthalpy of hydration ΔH_h is of importance. The standard enthalpy of neutralisation is the enthalpy change when one mole of aqueous hydrogen ions are neutralised by a base in dilute solution. For strong acids and strong bases, this value is – 57.33 kJ. This is because, independent of the strong acid or base, the neutralisationreaction is:

$H_3^{+}O\left(aq\right)+O{H}^{-}\left(aq\right)=2H_{2}O\left(I\right)$

    (For weak acids and bases, the enthalpy of neutralisation is lower because some of the heat energy produced is required by the reaction to complete the ionisation of the acids and bases).

    The standard ehtalpy of atomisation is the enthalpy change which occurs when one mole of an element in its standard state at 298 K and 101.325 kPa is converted into free atoms.

    The standard enthalpy of ionisation is the enthalpy change which occurs when one mole of gaseous atoms are converted into one mole of gaseous ions accompanied by a loss of one mole of electrons, measured under standard conditions. It should be noted that each element has a different number of ionisation energies dependent upon the number of electrons in the element. Usually the formation of stable metallic ions are associated with a loss of 1, 2, 3 or 4 electrons.

    The standard enthalpy of lattice energy is the enthalpy change which occurs when one mole of an ionic solid is formed from its constituent gaseous ions measured under standard conditions.

    The standard enthalpy of electron affinity is the enthalpy change which occurs when one mole of gaseous atoms are converted into one mole of gaseous negative ions by accepting one mole of electrons, measured under standard conditions. It should be noted that some non-metals can have more than one electron added: for example, oxygen has two electron affinity enthalpy values.

    For covalent molecules the energy required to separate one mole of covalent bonds between two atoms in a diatomic molecule is called the bond dissociation enthalpy, or bond dissociation energy. Some typical examples of values are given in Table 61.1, where B.D._x−y symbolises those terms.

    In covalent molecules containing more than two atoms the bond dissociation energies are found for all of the bonds and an average value is derived. For example, in methane, CH₄ there are four carbon to hydrogen bonds. The enthalpy required to break the four bonds is 1666 kJmol⁻¹, and the average value is thus 416.5 kJmol⁻¹.

10. The law of Hess states that ‘the total change in enthalpy in a chemical reaction is independent of the number of stages used to complete the reaction’. Considerable use is made of this law in the calculation of enthalpy values that are difficult to measure directly.

11. The Born-Haber cycle is an application of Hess’s law used to relate together all of the enthalpy changes involved in the formation of an ionic solid. An example of such a Born-Haber cycle is shown in Figure 61.2 for calcium sulphide. In this diagram the combined 1st and 2nd electron affinities have been omitted. To find the value, Hess’s law is applied to the system and stated as:

$\Delta H_1+\Delta H_2+\Delta H_3+\Delta H_4+\Delta H_5=\Delta H_6$

    Substituting the known values of enthalpy changes into the equation gives:

$+176.6+590+1100+238.1+\Delta H_4\left(X\right)-3084=-482.4$

    which gives a value for the enthalpy of electron affinity of

$\Delta H_4=+496.9kJmo{l}^{-1}$

12. For covalently bonded molecules similar enthalpy cycles as above, can be constructed to obtain information about covalent reactions. The enthalpy cycle shown in Figure 61.3 shows the interrelationships between methane and its elements. For example, the equation for the formation of methane can be written:

${\text{C(s)+2H}}_{\text{2}}\text{(g) }\stackrel{\Delta H_1}{\to }{\text{ CH}}_{\text{4}}\text{(g)}$

    where by definition, ΔH₁ is the enthalpy of formation of methane. Before covalent bonds are formed by atoms of carbon and hydrogen the carbon must be atomised and the hydrogen molecules dissociated into atoms. The enthalpies involved are ΔH₂ and ΔH₃ both of which are endothermic. Hence

$C(s)\stackrel{\Delta H_2}{\to }C(g)and2H_2(g)\stackrel{\Delta H_3}{\to }4H(g)$

    ΔH₂= the enthalpy of atomisation of carbon = ΔH_a carbon

    ΔH₃ = 2 × the bond dissociation enthalpy of hydrogen = 2 x B.D._{H - H}

    The next step involves the formation of four carbon to hydrogen single covalent bonds. Bond formation is an exothermic process and ΔH₄ in the equation:

$\text{C(g)+4H(g) }\stackrel{\Delta H_4}{\to }{\text{ CH}}_{\text{4}}\text{(g)}$

    is 4 × the average bond dissociation energy of a C—H bond,

$i.e.\Delta H_4=4\times E_{C-H}$

    By applying Hess’s law of constant heat summation:

$\begin{array}{l}\Delta H_1=\Delta H_2+\Delta H_3+\Delta H_4\\ or\Delta H_f=\Delta H_{a}carbon+2\times B.D{.}_{H-H}+4\times E_{C-H}\end{array}$

    Hence by knowing the required enthalpy values, the enthalpy of formation can be calculated.

13. Another useful application of Hess’s law is the determination of the enthalpy of covalent reactions. When any reaction takes place chemical bonds must be first broken and then new bonds formed. The bond breaking is an endothermic process and bond forming an exothermic process. By considering the energy associated with covalent bonds in all of the molecules, the difference between reactants and products will be the enthalpy of the reaction. For example, the hydrogenation of ethene takes place according to the equation:

$\begin{array}{l}\Delta H_1=\Delta H_2+\Delta H_3+\Delta H_4\\ or\Delta H_f=\Delta H_{a}carbon+2\times B.D{.}_{H-H}+4\times E_{C-H}\end{array}$

    The mean bond energies of the bond to be broken and formed can be presented in tabular form as follows:

14. The enthalpy changes which are measured are determined using calorimetry. A calorimeter is a container, which is insulated against heat loss, and contains a liquid of known specific heat capacity, c (usually water) in which the chemical reaction takes place. The rise in temperature of a known mass of liquid, m kg, from t°₁ C to t°₂ C, is associated with an energy change ΔH given by the expression:

$\Delta H=mc\left(t_2-t_1\right)joules$

    (See Chapter 40, para 4.)