(a) Is the standard free-energy change, \(\Delta G^{\circ}\), always larger than \(\Delta G ?\) (b) For any process that occurs at constant temperature and pressure, what is the significance of \(\Delta G=0 ?\) (c) For a certain process, \(\Delta G\) is large and negative. Does this mean that the process necessarily has a low activation barrier?

It is not true that the standard free-energy change (ΔG°) is always larger than the Gibbs free-energy change (ΔG). The relationship between ΔG and ΔG° is given by the Gibbs-Helmholtz equation: \[ \Delta G = \Delta G^{\circ} + RT \ln{Q} \] Where R is the gas constant, T is the temperature in Kelvin, and Q is the reaction quotient. As the reaction progresses, Q deviates from 1 and the value of ΔG changes. Depending on the reaction's progress, temperature, and Q's value, ΔG might be larger or smaller than ΔG°.

When ΔG = 0 for a process occurring at constant temperature and pressure, it implies that the process is in equilibrium. Neither the forward nor the reverse process is favored, and the concentrations of reactants and products remain constant. At equilibrium, the reaction quotient (Q) is equal to the equilibrium constant (K): \[ \Delta G = 0 = \Delta G^{\circ} + RT \ln{K} \] This signifies that in a reversible process, the forward and reverse reactions occur at the same rate, maintaining a constant equilibrium between reactants and products.

A large negative ΔG means that the reaction is thermodynamically favorable and will proceed spontaneously in the forward direction. However, it does not necessarily imply that the reaction has a low activation barrier. The activation barrier is related to the reaction kinetics, which determines the reaction rate. This activation barrier is represented by the activation energy (Ea) in the Arrhenius equation: \[ k = Ae^{\frac{-Ea}{RT}} \] Even if a reaction is thermodynamically favorable (large negative ΔG), it might have a high activation energy that requires external input (e.g., a catalyst) to lower the barrier and increase the reaction rate. So, a large negative ΔG does not provide information about the height of the activation barrier.