Reactions : Rate vs. Extent

The reaction of nitrogen with hydrogen also illustrates several very important features of chemical reactions. The first is their rate. Just as we talk about the speed at which we drive and bemoan the length of time to get to the movies when traffic is high, we use essentially the same language to describe the speed of a reaction. Some reactions are very fast--the products are produced in less than a millisecond. Other reactions, such as the reaction of nitrogen with hydrogen, are quite slow. At room temperature, this reaction does not produce any appreciable amount of ammonia over a period of several days. Indeed, the German chemist Fritz Haber discovered the right catalyst and conditions that would produce ammonia within a reasonable amount of time (this discovery indirectly gave the Germans a significant military advantage because ammonia can be converted to nitric acid, which in turn is used to prepare explosives such as trinitrotoluene). Other reactions, such as the conversion of diamond to graphite, take thousands or millions of years, even at elevated temperatures.

The extent of the reaction of nitrogen with hydrogen at room temperature is fairly high. What we mean by this is that if the reaction were fast, or if we wait long enough, most of the nitrogen will be converted to ammonia. An example of a reaction that has a low extent but a fast rate is the process used to prepare HCl in the laboratory:

H₂SO₄ + 2 NaCl → 2 HCl + Na₂SO₄

This reaction is very fast and is over in less than a second. However, very little HCl is produced. The reaction is similar to a fellow in a Ferrari who sets out to travel from Philadephia to New York, but only travels 10 miles, at 100 miles per hour, before he is stopped by a policeman for speeding.

The extent of a reaction can be predicted by using thermodynamics. The two main laws of themodynamics, called the first and second laws, tells us that energy is conserved and that disorder increases. Both of these laws are relatively easy to relate to. We have all experienced the transfer of heat from a flame to a pan of water and the first law of thermodynamics simply tells us that energy of the flame is used to heat the metal of the pan, heat the water, and even heat some of the surrounding air. The total energy before the pan was heated is the same as the energy after the pan was heated; it has just been transferred to other parts of the surroundings.

The second law of thermodynamics--that disorder increases--is also part of everyone's experience. Our rooms and homes become more and more cluttered unless we work hard to keep things tidy. A gas confined to a vial rapidly permeates a room and even a whole building when the vial is opened. These are both movements toward greater disorder. A special term is used to describe disorder-- entropy (S).

Now consider our pan of boiling water. The energy of the steam produced is greater than that of the liquid water; after all, heat is required to produce the steam. We will now use a fancy term for energy -- enthalpy (H) --and say that in heating the liquid water to produce steam, the enthalpy of the water has increased. Because steam consists of water molecules undergoing random motion in the gaseous state, we can also say that the steam is more disordered than liquid water and that therefore the entropy of the water has increased.

Enthalpy and entropy can be combined to give a new thermodynamic function called the free energy (G). The mathematical relationship is

G = H - TS

or, since only changes in enthalpy and entropy can be measured, the change in free energy (ΔG) can be expressed as

ΔG = ΔH - TΔS

At this point you are asking, "Okay, but what does ΔG tell us about our water?" If the temperature is maintained constant and all the thermodynamic variables are measured at so-called standard conditions (one atmosphere pressure and 1.0 molar concentrations), ΔG° is related to the extent of the process. If ΔG° is negative, the process is favored, and the more negative its value the greater the extent of the reaction. If ΔG° is positive, the process is not favored. Thus, if we heat the water to 50 °C, keep the temperature constant from this point on, and determine ΔG°, we will find that it is positive and that the conversion of liquid water to gaseous water is not favored. However, if we heat the water to 100 °C, then ΔG° will be negative (actually when the temperature is exactly at the boiling point of water, ΔG° will be zero), and the conversion to steam will be favorable. It should now be obvious that ΔG° can be very useful because it is directly related to the extent of the reaction.

Actually, the extent of a reaction is usually measured by the equilibrium constant for a reaction. This constant, K, is a measure of the amount of products formed in a reaction relative to the amount of reactants. Consider the reaction of N₂ with H₂ to produce ammonia.

N₂ + 3 H₂ → 2 NH₃

We already know that this reaction is slow at room temperature. Let us suppose that we mix together 1.0 mole of N₂ and 3.0 mole of H₂ in a one liter flask. The following table presents some hypothetical data on the amount of ammonia present in the flask as a function of time.

Table 5. Amount of ammonia present as a function of time.

time, h	moles of ammonia
1	1 x 10-5
10	1 x 10-3
20	1 x 10-2
50	0.30
100	0.70
200	0.80
300	0.80

It is clear from the table that the ammonia is formed slowly, but after 200 hours there does not seem to be a change in the amount formed. At this point, the reaction has reached equilibrium. This equilibrium state is a dynamic one: ammonia continues to be formed, but it also reverts back to N₂ and H₂ as quickly as it is formed. To put it another way, the reaction never stops, but at equilbrium the rate at which products are formed is equal to the rate at which reactants are formed from products.

In the experiment above, the equilibrium constant can be calculated as:

N₂ + 3 H₂ → 2 NH₃

Notice that the equilibrium expression contains the concentrations (moles per liter) of each substance raised to a power that is the same as the coefficient in front of that substance in the equation.

Finally, we can now relate ΔG° to the equilibrium constant.

ΔG° = -RT lnK

This equation contains R, the ideal gas constant, T, the absolute temperature, and the natural logarithim of the equilibrium constant. It is easy to see that when ΔG° is negative, lnK will be greater than one and that negative free energy changes lead to high extents.

Problem Seventeen

Calculate InK if ΔG° is -100 kJ/mol, R is 0.008 kJ/K-mol and T is 298 K.

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