Re: Could you please explain 'Gibb's Free Energy Equation' in laymen's terms?

While I'm not aware of anything specifically called "Gibbs Free Energy 
Equation", I can try and explain what the Gibbs energy (sometimes called 
Gibbs Free Energy) is and how it relates to enthalpy and entropy.

The Gibbs energy is defined as G = H - TS, where H is the enthalpy, T is 
the temperature, and S is the entropy.  But what does it represent?  
Basically, it is a way to talk about the way energy and entropy work in 
competing directions  in determining the equilibrium state of a system.  
This is sometimes called "energy-entropy compensation".

All other things being equal, systems will tend toward the state of lowest 
energy (enthalpy is a measure of energy that includes the energy due to 
the system having a pressure, but you don't need to worry about that 
detail).  However, systems at temperatures above absolute zero do not just 
sit in the lowest-energy state, because the thermal energy causes the 
molecules to fly around and get into other states.  The entropy can be 
thought of as a measure of how many different states a system samples due 
to thermal energy.  At very low temperatures, the system doesn't get to 
sample all these other states, and the equilibrium is determined by the 
low-energy state.  At very high temperatures, the intermolecular energies 
are relatively unimportant compared to the thermal energy, and the system 
tends to go to a state of maximum entropy.  The competition between these 
two is represented by the Gibbs energy.  The condition for equilibrium at 
a fixed temperature and pressure is that the Gibbs energy is a minimum -- 
when T is small that means minimizing the enthalpy and when T is big it 
means maximizing the entropy.

As an example, consider some argon.  At very low temperatures, the 
intermolecular attractions cause the atoms to freeze into a regular 
crystal lattice, which is the lowest energy state.  This state has a low 
entropy since the atoms are stuck in one spot and can't sample other 
configurations.  At high temperatures, the entropy is more important and 
the argon atoms fly around in a gaseous state with higher entropy, even 
though it means they have more energy than if they were sitting in a nice 
regular lattice.

With regard to the second part of your question, one can't just talk about 
"higher temperature"; one has to talk about the other constraints on the 
system such as whether you are increasing the temperature while holding 
volume constant, or holding pressure constant, or what.  However, under 
most constraints, both the enthalpy and entropy will increase with 
increasing temperature.  The enthalpy because it is primarily a measure of 
the energy in a system, and raising the temperature generally means 
putting in energy.  The entropy because more thermal energy generally 
allows the system to sample a wider range of states.