Entropy $S$

  • entropy is a measure of order (or disorder) in a system
  • a very regular and ordered system will have a low entropy
  • a very regular system also has low energy
  • changes that result in positive $\Delta{S}$:
    • increase in the number of moles
    • increase in temperature
    • increase in volume
    • phase changes from solid to liquid to gas
    • forming of more complicated molecules

Laws of thermodynamics

Zeroth law od thermodynamics

Of two thermodynamic systems are each in thermal equilibrium with a third system, then they are in thermal equilibrium with each other.

First law of thermodynamics

No energy can be created or destroyed.

Second law of thermodynamics - Clausius

Heat does not spontaneously pass from a colder to a hotter body.

Second law of thermodynamics - Entropy

The entropy of isolated systems left spontaneous evolution cannot decrease, as they always arrive at a state of thermodynamic equilibrium, where the entropy is highest.If all processes in the system are reversible, its entropy is constant.

Third law of thermodynamics

The entropy of any perfectly ordered, crystalline substance at a temperature of absolute zero is zero.

Absolute entropy and Boltzmann formula

  • absolute entropy of a system is determined by the number of microstates (the configuration of all atoms in the system)
  • zero entropy corresponds to a perfectly ordered crystalline system with only one microstate
  • for known values of standard entropy, the standard entropy change for any process can be calculated as the difference between the sums of entropies of products and reactants

$$\Delta{S^0} = \sum{nS^0_p} - \sum{mS^0_r}$$

  • where $n$ and $m$ are stoichiometric coefficients, $p$ stands for products and $r$ stands for reactants

  • Boltzmann formula approximates the absolute entropy of a system

$$S=k_B\ln{P}$$

  • where:
    • $k_B$ is the Boltzmann constant
    • $P$ is the number of microscopic states

Implications of the third law of thermodynamics

  1. The sign of entropy of any substance at temperatures above absolute zero is positive.
  2. A fixed reference point was found, which allows us to measure the absolute entropy of any substance, at any temperature.
    • absolute entropy is this determined by the heat increase required to bring the substance from 0 K to the desired temperature

Gibbs free energy $\Delta{G}$

  • knowing the change in enthalpy is not enough to determine whether a process is or is not spontaneous
  • if the change in enthalpy is negative, the reaction is more likely to happen spontaneously
    • an exothermic reaction which results in a more disordered system (creating more fragments) is more propable than an exothermic reaction which results in the creation of a less disordered system (energy needs to be added to keep the system ordered)
    • if heat is not removed from the system during the reaction, the reaction slows down and can even stop
  • combining enthalpy, entropy a temperature together results in the change in Gibbs free energy

$$\Delta{G} = \Delta{H} - T\Delta{S}$$

  • the lower the change in Gibbs energy the more spontaneous reaction
    • spontaneous reaction is described by a negative change in Gibbs energy (exothermic processes)
  • an endothermic process and a positive change in entropy, the reaction will be spntaneous, though less favorable than an exothermic reaction
  • a process with a positive change in enthalpy and a negatve change in entropy will not occur spontaneously

Calculation

$$\Delta{G^0} = \sum{nG^0_p} - \sum{mG^0_r}$$

  • where $n$ and $m$ are stoichiometric coefficients, $p$ stands for products and $r$ stands for reactants

$$\Delta{G} = \Delta{G^0} + RT\ln{q}$$

  • if $\Delta{G}$ is zero, then:

$$\Delta{G^0} = -RT\ln{K_{eq}}$$

Kinetic control

  • it is not the case that exothermic reactions (or indeed with any thermodynamically favorable process) proceed faster than thermodynamically less favorable processes
  • the rate of the rection is not determined by reaction thermodynamics but reaction kinetics
  • the speed of a reaction is determined by the activation energy and not by thermodynamic favorability
  • controling activation energy is called kinetic control