The pKa values of \(-\mathrm{COOH},-\mathrm{SH}\) and \(-\mathrm{N} \mathrm{H}_{3}\) groups present in the amino acid cysteine are \(1.8,8.2\) and \(10.8\) respectively. The structure of cysteine at a \(\mathrm{pH}=5\) will be

Understanding the pKa values of various groups within amino acids is essential in predicting their behavior in different pH environments. The pKa value is a measure of acidity that indicates the pH at which half of the molecules of that group are dissociated (deprotonated). In the case of cysteine, the carboxyl group (-COOH) has a pKa of 1.8, the sulfhydryl group (-SH) has a pKa of 8.2, and the amino group (-NH3) has a pKa of 10.8.

When the pH is lower than the pKa, the group remains protonated, while if the pH is higher, it becomes deprotonated. This principle is key to predicting the ionization states of the functional groups in cysteine at any given pH. For instance, at pH 5, we can expect the carboxyl group to lose its proton and exist as -COO-, while the amino and sulfhydryl groups will stay protonated as -NH3+ and -SH respectively, due to their pKa values being higher than the pH.

Amino acids, the building blocks of proteins, possess characteristic functional groups that determine their structure and function within biological systems. A typical amino acid like cysteine contains a central carbon (alpha carbon), an amino group, a carboxyl group, a hydrogen atom, and a unique side chain - in this case, a sulfhydryl (-SH) group.

The behavior and ionization state of these functional groups are influenced by the surrounding pH, governed by their respective pKa values. These values facilitate the understanding of how amino acids will interact with their environment, particularly how they will be charged. The charge in turn affects how amino acids can bind with each other, thus influencing protein structure and functionality. It is the ionization of these functional groups that determines the overall charge of the amino acid at a given pH.

Acid-base equilibria are concerned with the balance between proton donors (acids) and proton acceptors (bases) in a system. It's a core concept in understanding how functional groups in amino acids, like those in cysteine, act in varying pH levels. The Henderson-Hasselbalch equation is a formula used to relate the pH, pKa, and ratio of deprotonated to protonated species.

It helps explain why certain groups are protonated or deprotonated at different pH values. As observed in the given exercise, the pH substantially influences the state of the functional groups. At any given moment, amino acids like cysteine exist in an equilibrium between their protonated and deprotonated forms, and the ratio between these forms is defined by the pH compared to their respective pKa values.