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Magazine Chapter 6: Problem 3
The dissociation constant for salicylic acid, C.H.(OH)COOH, is \(1.0 \times 10^{-3}\) Calculate the percent dissociation of a \(1.0 \times 10^{-3} \mathrm{M}\) solution. There is one dicsociable proton. (Sce also Excel Problem 25 below.
Short Answer
Expert verified
The percent dissociation of the solution is 100%.
Step by step solution
01
Identify the Reaction
Salicylic acid can be represented as HA, and its dissociation in water can be written as: \[ \text{HA} \rightleftharpoons \text{H}^+ + \text{A}^- \] The dissociation constant \( K_a \) for this reaction is given as \( 1.0 \times 10^{-3} \).
02
Write the Ka Expression
For the dissociation of HA, the expression for \( K_a \) is given by: \[ K_a = \frac{[\text{H}^+][\text{A}^-]}{[\text{HA}]} \]
03
Set Up the ICE Table
The initial concentration of HA is \( 1.0 \times 10^{-3} \) M. Let the change in concentration of \( [\text{H}^+] \) and \( [\text{A}^-] \) be \( x \). The ICE (Initial, Change, Equilibrium) table will be:\\[ \begin{array}{c|c|c|c} & \text{HA} & \text{H}^+ & \text{A}^- \ \hline \text{Initial} & 1.0 \times 10^{-3} & 0 & 0 \ \text{Change} & -x & +x & +x \ \text{Equilibrium} & 1.0 \times 10^{-3} - x & x & x \end{array} \]
04
Substitute into Ka Expression
Using the equilibrium concentrations into the \( K_a \) expression gives us: \[ 1.0 \times 10^{-3} = \frac{x^2}{1.0 \times 10^{-3} - x} \]
05
Simplify the Equation
Assume \( x \ll 1.0 \times 10^{-3} \), then simplify the equation to \[ 1.0 \times 10^{-3} = \frac{x^2}{1.0 \times 10^{-3}} \] which leads to \( x^2 = (1.0 \times 10^{-3})^2 \).
06
Solve for x
Solving the simplified equation \( x^2 = 1.0 \times 10^{-6} \), we find \( x = \sqrt{1.0 \times 10^{-6}} = 1.0 \times 10^{-3/2} = 1.0 \times 10^{-3} \).
07
Calculate Percent Dissociation
Percent dissociation is calculated using the formula: \[ \text{Percent Dissociation} = \left( \frac{x}{\text{Initial concentration of HA}} \right) \times 100 \] Substitute \( x = 1.0 \times 10^{-3} \) and the initial concentration \( 1.0 \times 10^{-3} \): \[ \frac{1.0 \times 10^{-3}}{1.0 \times 10^{-3}} \times 100 = 100\% \]
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Dissociation Constant
The dissociation constant, often represented as \( K_a \), is a crucial parameter in acid-base chemistry. It reflects the ability of an acid to donate protons (\( H^+ \)) to a solution. For an acid like salicylic acid, the dissociation constant is given by the expression:\[ K_a = \frac{[\text{H}^+][\text{A}^-]}{[\text{HA}]} \].
In this formula:
In this formula:
- \([\text{H}^+]\) represents the concentration of hydrogen ions.
- \([\text{A}^-]\) is the concentration of the acid's conjugate base.
- \([\text{HA}]\) is the concentration of the undissociated acid.
ICE Table
An ICE table, which stands for Initial, Change, Equilibrium, is a useful tool in chemistry to organize our thoughts when dealing with equilibrium reactions. It helps track the changes in concentration of species during a chemical reaction. Consider the dissociation of salicylic acid (HA) in water, forming \( H^+ \) and \( A^- \):
- **Initial** - Before the dissociation begins, the concentration of HA is \( 1.0 \times 10^{-3} \) M, while \([\text{H}^+]\) and \([\text{A}^-]\) are zero.
- **Change** - As the reaction proceeds, the concentration of HA decreases by \( x \), and \([\text{H}^+]\) and \([\text{A}^-]\) both increase by \( x \).
- **Equilibrium** - At equilibrium, \([\text{HA}] = 1.0 \times 10^{-3} - x\), and \([\text{H}^+]=[\text{A}^-]=x\).
Percent Dissociation
Percent dissociation is a measurement of the extent to which an acid dissociates in a solution. It shows how much of the acid's molecules donate their protons. The formula to calculate percent dissociation is:\[ \text{Percent Dissociation} = \left( \frac{x}{\text{Initial concentration of HA}} \right) \times 100 \].
In the context of salicylic acid, we find \( x = 1.0 \times 10^{-3} \).
By substituting this value into the percent dissociation formula, we get:
In the context of salicylic acid, we find \( x = 1.0 \times 10^{-3} \).
By substituting this value into the percent dissociation formula, we get:
- \( \frac{1.0 \times 10^{-3}}{1.0 \times 10^{-3}} \times 100 = 100\% \).
Acid-Base Equilibrium
Acid-base equilibrium pertains to the balance achieved in a solution between the dissociated ions and the undissociated molecules. This balance is dictated by the dissociation constant \( K_a \) and the initial concentrations. Equilibrium helps us predict how an acid behaves once dissolved.In the case of salicylic acid:
- The equilibrium formula is framed as \( [\text{HA}] \rightleftharpoons [\text{H}^+] + [\text{A}^-] \).
- A change in temperature, concentration, or presence of other chemicals could shift this balance according to Le Chatelier's principle, affecting the dissociation extent.
