Picturing Entropy: Some Examples

The first law of thermodynamics considers interpretations of the law of conservation of energy and how it applies to the system (the reaction under consideration). The second law, however, is interested in the measurement of randomness in a system, the entropy. Before we get down to the nitty-gritty on what that actually means, let's examine some ways in which entropy influences the world around us.

Example 1: Cup of Tea

Say you add an ice cube to your cup of tea. Aside from the wonderful crackling noise the ice cube makes as it melting, what is really going on? First of the enthalpy of the reaction is increasing as the ice cube requires heat to melt. And then, the entropy is also increasing.

Why? Well, in a solid the molecules have less degrees of rotation, as they solely vibrate in a dense structure, but when they melt to become liquid water, the molecules can move around and slide past one another, which is why if you tip over your tea it will spill all over the floor and will make the mess that it does. These further degrees of freedom increase the amount of randomness in the system. And hence, liquids have greater entropy than solids.

Now, let's consider the steam coming off of your tea cup. As you may have anticipated, gases of a substance have even higher entropy than liquids, because the molecules have greater degrees of freedom than liquids.

So in summary: $$entropy_{solid} < entropy_{liquid} < < entropy_{gas}$$
More degrees of freedom = more entropy

Example 2: Denaturing DNA

 Denaturing DNA = breaking DNA from its classic double helix shape into two separate strands.

 Often in the laboratory denaturation of DNA is done in a test tube by increasing the temperature so that the hydrogen bonds that connect the bases (A,T,G,C) break. This is an endothermic process as it requires input of energy, high temperature and results in something with more entropy (2 strands instead of 1).

Denaturing DNA is also a reversible process, so if you decrease the temperature, the strands will re-anneal to the double-stranded helix structure. This can be investigated more clearly by checking out the Gibb's Free Energy equation $G=H-TS$, which I will discuss in a later post.

 Basically, the take home message is that for any reaction, there is a dialogue between the favourable enthalpy and favourable entropy conditions, which will determine whether the reaction is spontaneous or not.

In example 1 with the tea cup, the ice melting and the steam rising, was all spontaneous at room temperature. However, in example 2, this is not as clear. The reaction is temperature dependent, as temperature is the factor that mediates the relationship between the enthalpy and entropy of the reaction.

 But now you are probably wondering: what are the favourable enthalpy and favourable entropy conditions for a reaction to occur spontaneously? Fair enough.

 For a reaction to occur spontaneously, the it prefers to have negative enthalpy, which is an exothermic process (releases heat into the environment).

 "Okay," you say, "But what if I have a giant stick of dynamite in my hands? That releases a lot of heat because it explodes. So why isn't this reaction spontaneous? Clearly it is an exothermic reaction."

Good point. Recall a potential energy diagram for an exothermic process. Some reactions are required to overcome their activation energy for the reactants to turn into the products. Here is a quick 30 second video to trigger your memory if you have forgotten this concept. Activation Energy Definition So in the case of dynamite, it will not spontaneously combust because it requires a certain input of energy - the spark to ignite it - to blow up and overcome its activation energy.

 So, we've covered the spontaneous conditions for enthalpy, but what about entropy? The system is favours states of greater disorder, so the entropy increases positively. A way representing this formally is defining our system (because we can take a system to be as small or as big as we like, as we are the ones who define it) as the universe. Since the universe is an isolated system, we know that $$\Delta{S}_{universe}\geq{0}$$

 Now, this explains why certain reactions with respect to a smaller system can occur at all. As long as the $\Delta{S}_{universe}$ remains positive, which it will because of the massive size of the system, a small negative blip where $\Delta{S}_{system}\leq{0}$, will simply not matter as it is insignificant in the grand scheme of things. We represent this by $$\Delta{S}_{universe}+\Delta{S}_{system}\geq{0}$$ or $$\Delta{S}_{system}+\Delta{S}_{surroundings}\geq{0}$$

 Also, note that a reversible process is defined by $$\Delta{S}_{system}=-\Delta{S}_{surroundings}$$

 All right, this leads us to example 3.

 Example 3: An Endothermic Reaction

 $$\text{N}_2\text{O}_4\longrightarrow 2\text{NO}_2$$

 So what is happening here? Dinitrogen dioxide on the left hand side of the equation is reacting to become the product nitrogen monoxide. The key thing to notice is that there is only 1 mole of reactant to 2 moles of products, so even though this reaction absorbs energy and is endothermic, and hence should be unfavourable, the positive increase in entropy (more products than reactants, therefore more disorder) favours the reaction to proceed. In this case, the drive of the entropy overcomes the enthalpy of the reaction.

In the next post I will discuss a statistical representation of entropy, so stay tuned! :)