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What I understand by a spontaneous reaction is that - A reaction whose G is negative and which occurs without the input of energy.

Gasoline reacts with oxygen only if a spark is provided. So how can it be a spontaneous reaction?

Do nonspontaneous reactions occur in nature? If so, are they always slow? If they are slow then what is the difference between a slow spontaneous reaction and a nonspontaneous reaction?

Are nonspontaneous reactions active energies more than that of spontaneous reactions?

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Spontaneous reactions simply mean that the products will be lower in free energy than the reactants ($\Delta G$ is negative). If $\Delta G$ is negative, then it is energetically favourable for the reaction to occur - in other words, there will be a release of energy due to the reaction.

The magnitude of $\Delta G$ (how large it is) does not say anything about the reaction speed. For example, the reaction between gasoline and oxygen in the atmosphere at room temperature will be extremely slow, despite a large, negative $\Delta G$. This is because this reaction has a large activation energy, $E_A$.

Hold on. When two reactants collide, they interact to form an extremely unstable structure called the transition state. The transition state is high in energy and is NOT energetically favourable, and thus the transition state quickly collapses again and becomes either the reactants or the products. You can see this in the following picture.

Transition state

Depending on the kinetic energy of the reactants, they might not reach the top of the curve, in which case they will fall back to being reactants. But if they collide fast enough and are positioned appropriately with respect to each other, the top of the curve is reached, and the products will form. The energy required to form the transition state is the activation energy.

With gasoline and oxygen, the activation energy is high. So even though a tremendous amount of energy is released during the reaction, it needs a spark to overcome its activation energy. From here on, the heat generated from the reaction supplies the activation energy.

If this reaction was not spontaneous, we might be able to force the reaction along, but the reaction would not continue on its own.

As you know, not all reactions run to completion. This is because Gibbs free energy is dependent on concentrations of reactants and products, so as the products accumulate, and the reactants are used, $\Delta G$ becomes closer to 0, eventually reaching equilibrium, where $\Delta G = 0$. Visualise this scenario on the above picture. If there is no difference in Gibbs free energy, the reactants will still get to the transition state and turn into products. But the energy required for products to reach the transition state and becomes reactants is equivalent, and thus the rate of the forward reaction is equal to the rate of the reverse reaction.

The difference between reactions that will reach an equilibrium and the ones that run to completion is the magnitude of the $\Delta G$ at standard conditions (denoted $\Delta G°$). This is a measure of how energetically favourable the reaction "inherently" is. If $\Delta G°$ is large and negative, the reaction will run to completion. If it is smaller, the reaction will establish an equilibrium at some point.

That's at least generally true for reactions, where all of the species stay in the same phase I think. If you envision a reaction where one of the products is a gas, that escapes the solution of reactants, then the products will be unable to reform into reactants.

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The word 'spontaneous' has different meanings in everyday life and this is unhelpful. I prefer to think of a spontaneous reaction as one that 'is allowed to occur', without any prediction of how fast the reaction occurs. If a reaction is not allowed to occur - i.e. it is non-spontaneous, it cannot occur whatever kinetic tactics (catalyst, higher reactant concentration) we try, except that if we heat up the system, the reaction might be allowed to occur at the higher temp because delta G changes.

My preference would be not to use the word 'spontaneous' at all for chemical reactions but to define a reaction that is allowed as one with a large equilibrium constant. The link between free energy change and K means that a big K means a large negative delta G. The reason for this approach is that equilibrium constants (K) are direct experimental evidence of reaction - a big K means that the reaction has virtually gone to completion.

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