Question

Two different proteins \(\mathrm{X}\) and \(\mathrm{Y}\) are dissolved in aqueous solution at \(37^{\circ} \mathrm{C}\). The proteins bind in a \(1: 1\) ratio to form XY. A solution that is initially \(1.00 \mathrm{mM}\) in each protein is allowed to reach equilibrium. At equilibrium, \(0.20 \mathrm{~m} M\) of free \(\mathrm{X}\) and \(0.20 \mathrm{~m} M\) of free \(\mathrm{Y}\) remain. What is \(K_{c}\) for the reaction?

Short answer

Based on the given information and the ICE table, the equilibrium concentrations of X, Y, and XY are 0.20 mM, 0.20 mM, and 0.80 mM, respectively. The equilibrium constant Kc for the reaction can be found using the expression \(K_c = \frac{[XY]}{[X][Y]}\). Substituting the equilibrium concentrations into this expression gives \(K_c = \frac{0.80}{0.20 \times 0.20} = 20\). Therefore, the equilibrium constant for the given reaction is 20.

Step-by-step solution

Write the balanced chemical equation

For the given reaction, we have: \[X + Y \rightleftharpoons XY\]

Set up an ICE table

Create an ICE table to keep track of the changes in the concentrations of X, Y, and XY during the reaction: | | X | Y | XY | |----|-----|-----|----| | I | 1.0 | 1.0 | 0 | | C | -x | -x | +x | | E | 1-x | 1-x | x |

Substitute given equilibrium concentrations

At equilibrium, the free concentrations of X and Y are given as 0.20 mM. We can substitute this value in the equilibrium row (E) of the ICE table: | | X | Y | XY | |----|----|----|----| | I | 1.0| 1.0| 0 | | C | -x | -x | +x | | E | 0.2| 0.2| x |

Determine the change in concentration and the concentration of XY at equilibrium

We can determine the change in concentration (x) and the concentration of XY at equilibrium by solving for x: \[1.0 - x = 0.20 \Rightarrow x = 1.0 - 0.20 = 0.80\] So the concentration of XY at equilibrium is 0.80 mM.

Write the expression for Kc and plug in the equilibrium concentrations

The expression for Kc is given by: \[K_c = \frac{[XY]}{[X][Y]}\] Now, we can substitute the equilibrium concentrations of each species into this expression: \[K_c = \frac{0.80}{0.20 \times 0.20} = \frac{0.80}{0.04} = 20\]

State the result

The equilibrium constant Kc for the given reaction is 20.

Key concepts

ICE Table

An ICE table is a powerful tool in chemistry to easily visualize and calculate the changes in concentrations of reactants and products during a chemical reaction. The name "ICE" stands for Initial, Change, and Equilibrium. These three stages help track the concentration of substances:

• **Initial (I):** Represents the starting concentrations before the reaction begins.
• **Change (C):** Shows how much the concentration of each species changes as the reaction proceeds towards equilibrium.
• **Equilibrium (E):** Displays the final concentrations once the reaction has reached equilibrium.

To use an ICE table, you need to write down the balanced chemical equation and then fill in the initial concentrations. During the reaction, changes are denoted by a variable, often "x", allowing you to calculate how substances convert from reactants into products. Finally, the equilibrium row informs you of the concentrations of each species once equilibrium has been achieved. By substituting these equilibrium values into the equilibrium constant equation, you can find the value of the equilibrium constant, which characterizes the reaction at the given temperature.

Protein Binding

Protein binding is a process where molecules such as proteins interact with one another, forming complexes. In the specified exercise, proteins X and Y bind to form a complex, XY. This interaction is crucial as it determines the biological function and behavior of proteins in biological systems.

When considering protein binding, it is important to understand that the ratio of binding is often specified—here, it is 1:1. This ratio is important for setting up your equilibrium equation and determining the changes in concentration over time. When analyzing such a reaction, knowing the initial amounts of each component is key to predicting the complex's behavior at equilibrium.

Protein complexes like XY in this scenario are central in biochemical processes and can influence various pathways in living organisms. Understanding how proteins bind and under what conditions can have implications in biochemical research and medicine.

Chemical Equilibrium

Chemical equilibrium is achieved in a reaction when the rate of the forward reaction is equal to the rate of the reverse reaction. At this point, the concentrations of reactants and products remain constant over time, although molecules continue to react.

For a reversible reaction like the one involving proteins X and Y forming XY, reaching equilibrium means that the amount of proteins converting to the complex is balanced by the amount of complex reverting to individual proteins. This state does not imply that the concentrations are equal, only that they are constant.

The concept of equilibrium is fundamental in understanding how reactions behave under different conditions. Knowing the equilibrium concentrations, you can calculate the equilibrium constant ( K_c ) which gives insights into how far the reaction has proceeded to favor products or reactants. In general, a large K_c suggests that the reaction heavily favors products, like in the given problem where K_c is 20.

Reaction Quotient

The reaction quotient (Q) is similar to the equilibrium constant ( K_c ) but can be calculated at any point during a reaction, not just at equilibrium. It is used to predict the direction in which a reaction will proceed to reach equilibrium.

• If Q < K_c , the reaction will proceed in the forward direction to produce more products.
• If Q > K_c , the reaction will proceed in the reverse direction to produce more reactants.
• If Q = K_c , the system is at equilibrium.

Calculating Q involves substituting the current concentrations of the reactants and products into the expression used for K_c . During experiments or simulations, you might frequently calculate Q to monitor reactions and understand how they shift towards equilibrium under varying conditions. In practical terms, the reaction quotient is an important tool for diagnosing and predicting the status of reactions in progress, especially in dynamic biological and chemical environments.