Question

For alanine, \(K_{\mathrm{al}}=5.1 \times 10^{-3}, K_{\mathrm{a} 2}=1.8 \times 10^{-10}\), Calculate the ratios \([\mathrm{Z}] /\left[\mathrm{C}^{+}\right]\) and \([\mathrm{Z}] /\left[\mathrm{A}^{-}\right]\) at \(\mathrm{pH}\) (a) \(2.00\). (b) \(6.00\). (c) \(10.50\).

Short answer

Question: Calculate the ratios \(\frac{[\mathrm{ZH}]}{[\mathrm{Z^+}]}\) and \(\frac{[\mathrm{ZH}]} {[\mathrm{Z}^-]}\) for alanine at pH values 2.00, 6.00, and 10.50. Answer: At pH = 2.00: \(\frac{[\mathrm{ZH}]}{[\mathrm{Z^+}]} = 5.1 \times 10^{-1}\) and \(\frac{[\mathrm{ZH}]} {[\mathrm{Z}^-]} = 1.8 \times 10^{-8}\) At pH = 6.00: \(\frac{[\mathrm{ZH}]}{[\mathrm{Z^+}]} = 5.1 \times 10^3\) and \(\frac{[\mathrm{ZH}]} {[\mathrm{Z}^-]} = 1.8 \times 10^{-4}\) At pH = 10.50: \(\frac{[\mathrm{ZH}]}{[\mathrm{Z^+}]} = 5.1 \times 10^7.5\) and \(\frac{[\mathrm{ZH}]} {[\mathrm{Z}^-]} = 1.8\sqrt{10}\)

Step-by-step solution

1. Identify the acidic groups and write the equilibrium expressions

The dissociation of alanine can be represented by two equations because it has two acidic groups: (a) \(H^{+} + ZH = Z^{+}\) (b) \(H^{+} + Z^{-} = ZH\) For the above equations, we can write equilibrium expressions using the given dissociation constants: (a) \(K_{al}=\frac{[\mathrm{Z}^+][H^+]}{[\mathrm{ZH}]}\) (b) \(K_{a 2}=\frac{[\mathrm{ZH}][H^+]}{[\mathrm{Z}^-]}\) We need to find the ratios \([\mathrm{Z}] /\left[\mathrm{C}^{+}\right]\) and \([\mathrm{Z}] /\left[\mathrm{A}^{-}\right]\), which are equal to \(\frac{[\mathrm{ZH}]}{[\mathrm{Z^+}]}\) and \(\frac{[\mathrm{ZH}]} {[\mathrm{Z}^-]}\), respectively.

2. Convert pH to \([H^+]\)

We need to find the \([H^+]\) for given pH values before substituting them into the dissociation equations: (a) pH = 2.00, \([H^+] = 10^{-2}\) (b) pH = 6.00, \([H^+] = 10^{-6}\) (c) pH = 10.50, \([H^+] = 10^{-10.5}\)

3. Calculate the ratios for each pH value

Now, we need to substitute the \([H^+]\) values and the dissociation constants in the equilibrium constants and solve for the ratios. (a) pH = 2.00: \(\frac{[\mathrm{ZH}]}{[\mathrm{Z^+}]}=\frac{K_{al}}{[H^+]}\ =\frac{5.1 \times 10^{-3}}{10^{-2}} = 5.1 \times 10^{-1}\) \(\frac{[\mathrm{ZH}]} {[\mathrm{Z}^-]}=\frac{K_{a 2}}{[H^+]}\ =\frac{1.8 \times 10^{-10}}{10^{-2}}=1.8 \times 10^{-8}\) (b) pH = 6.00: \(\frac{[\mathrm{ZH}]}{[\mathrm{Z^+}]}=\frac{K_{al}}{[H^+]}\ =\frac{5.1 \times 10^{-3}}{10^{-6}} = 5.1 \times 10^3\) \(\frac{[\mathrm{ZH}]} {[\mathrm{Z}^-]}=\frac{K_{a 2}}{[H^+]}\ =\frac{1.8 \times 10^{-10}}{10^{-6}}=1.8 \times 10^{-4}\) (c) pH = 10.50: \(\frac{[\mathrm{ZH}]}{[\mathrm{Z^+}]}=\frac{K_{al}}{[H^+]}\ =\frac{5.1 \times 10^{-3}}{10^{-10.5}} = 5.1 \times 10^7.5\) \(\frac{[\mathrm{ZH}]} {[\mathrm{Z}^-]}=\frac{K_{a 2}}{[H^+]}\ =\frac{1.8 \times 10^{-10}}{10^{-10.5}}=1.8 \times10^{0.5}=1.8\sqrt{10}\)

Key concepts

pH Calculation

Understanding the concept of pH calculation is crucial for many areas of chemistry, particularly when studying acid-base reactions. The pH scale measures the acidity or basicity of a solution and is calculated based on the concentration of hydrogen ions (H^+) present. The formula for pH is given by -log[H^+]. For instance, a solution with a high concentration of hydrogen ions is more acidic and has a lower pH value, whereas a solution with a low concentration of hydrogen ions is more basic, with a higher pH value. This inverse relationship plays a vital role in determining the nature of the chemical substances in a reaction.

When solving exercises involving pH, it's important to convert the given pH value into the hydrogen ion concentration by reversing the pH formula, which is achieved by using the mathematical operation 10^{-ph}. For example, if the pH is 6.00, the concentration of hydrogen ions is 10^{-6}. This conversion is a fundamental step in pH-related calculations and forms the bedrock upon which further analysis of an acid-base equilibrium is built.

Dissociation Constants

In the world of dissociation constants, we delve into the quantitative aspect of acid-base reactions. The dissociation constant (K_a) is a measure of the extent to which a compound (acid or base) separates into its ions in a solution. It reflects the strength of an acid or base; the larger the dissociation constant, the stronger the acid or base. When dealing with amino acids like alanine, it's imperative to note that they can have more than one dissociation constant because they contain multiple ionizable groups.

The values of these constants are integral to the task of calculating ratios of different ionic species at various pH levels, as seen in our original problem. Students should recognize that each constant relates to a specific dissociation step, allowing them to solve complex equilibria by breaking them down into simpler, more manageable parts. Understanding how to apply these constants to acid-base equilibrium equations is essential to mastering pH calculations.

Chemical Equilibrium

The concept of chemical equilibrium is a fascinating area, representing a state in which the rates of the forward and reverse reactions are equal, and the concentrations of the reactants and products remain constant over time. This doesn't mean the reactions have stopped, but rather they occur at the same pace in both directions.

In acid-base reactions, the equilibrium point can be described using the dissociation constants, denoting the ratio of the concentrations of the products to those of the reactants, at equilibrium. The equilibrium expressions are central to solving problems like the one illustrated, where one must manipulate these expressions to find the desired ratios of species within a solution. A solid grasp of equilibrium principles is not only helpful but necessary for students delving into any in-depth study of reactions, as it allows for the prediction and control of reaction behaviors.

Amino Acids

The structures of amino acids are at the heart of biochemistry. Amino acids, the building blocks of proteins, contain both an amine group and a carboxylic acid group and can act as both acids and bases. This dual-functionality stems from their ability to donate and accept protons (H^+), which causes them to behave differently depending on the pH of their environment.

Each amino acid has its own unique set of dissociation constants which correlate to the ionizable side chains present in its structure. When students analyze problems involving amino acids and pH, they should consider the amino acid's isoelectric point, the specific pH at which the amino acid has no net charge. Exercises, like the one we have with alanine, often involve calculating the ratios of different forms of amino acids at different pH values, which entails a comprehensive understanding of how an amino acid's structure dictates its behavior in various pH environments.