Under conditions of constant temperature and pressure, the equilibrium state of a system is determined by the Gibbs energy (G). Gibbs energy combines both the enthalpy (H) and the entropy (S) of the system, and it is defined as G = H - TS, where T is the temperature and S is the entropy.

To understand why Gibbs energy is at a minimum at equilibrium, let's consider the other options:

a) Internal energy (E or U):
Internal energy is the sum of the kinetic and potential energies of the particles in a system. Under constant temperature and pressure, changes in internal energy do not affect the equilibrium state. Therefore, internal energy is not at a minimum at equilibrium.

b) Enthalpy (H):
Enthalpy is a measure of the heat energy absorbed or released during a process at constant pressure. While enthalpy changes can affect the equilibrium state, it is not the most relevant factor. Enthalpy minimization alone does not guarantee the system will be at equilibrium.

c) Gibbs energy (G):
Gibbs energy is a thermodynamic potential that takes into account both the enthalpy and entropy of a system. It provides a measure of the spontaneity of a process. At equilibrium, the system is in its most stable state, which corresponds to the minimum value of Gibbs energy. This is known as the principle of minimum Gibbs energy.

d) Helmholtz energy (A):
Helmholtz energy is another thermodynamic potential that combines internal energy (E) and entropy (S). However, it is not as commonly used as Gibbs energy. Under constant temperature and volume, Helmholtz energy is minimized at equilibrium. In the given conditions of constant temperature and pressure, Helmholtz energy is not at a minimum at equilibrium.

In summary, under constant temperature and pressure, the quantity at a minimum at equilibrium is the Gibbs energy (G). The principle of minimum Gibbs energy is a fundamental concept in thermodynamics and provides insight into the stability and spontaneity of chemical reactions and processes.