Problem 151 - A Complete Resource Book in Chemistry for JEE Main

Question

At what concentration of \(\mathrm{CH}_{3} \mathrm{COOH}\) will the \(\left[\mathrm{H}^{+}\right]\) obtained will be same as that obtained from \(10^{-2} \mathrm{M}\) \(\mathrm{HCOOH},\left(\mathrm{Ka}\left(\mathrm{CH}_{3} \mathrm{COOH}\right)=10^{-5}, \mathrm{Ka}(\mathrm{HCOOH})=10^{-4}\right)\) (a) \(10 \mathrm{M}\) (b) \(5 \mathrm{M}\) (c) \(10^{-1} \mathrm{M}\) (d) \(6 \mathrm{M}\)

Step-by-Step Solution

Answer

(c) \(10^{-1} \mathrm{M}\)

Step 1: Determine the concentration of hydrogen ions from HCOOH

Given the acid constant \( \mathrm{Ka}(\mathrm{HCOOH}) = 10^{-4} \). Since the concentration of \( \mathrm{HCOOH} = 10^{-2} \mathrm{M} \), we can approximate the concentration of dissociated \( [\mathrm{H}^+] \) as \( \sqrt{\mathrm{Ka} \times [\mathrm{HCOOH}]} \). Substitute the values:\[[\mathrm{H}^+] = \sqrt{10^{-4} \times 10^{-2}} = \sqrt{10^{-6}} = 10^{-3} \mathrm{M}\]

Step 2: Equate [H+] of CH3COOH with [H+] of HCOOH

We know that the hydrogen ion concentration \([\mathrm{H}^+]\) from \( \mathrm{CH}_3\mathrm{COOH} \) must be \(10^{-3} \mathrm{M} \). Using the given \( \mathrm{Ka}(\mathrm{CH}_3\mathrm{COOH}) = 10^{-5} \), we write:\[[\mathrm{H}^+] = \sqrt{\mathrm{Ka}(\mathrm{CH}_3\mathrm{COOH}) \times [\mathrm{CH}_3\mathrm{COOH}]} = 10^{-3}\]

Step 3: Solve for the concentration of CH3COOH

Rearrange the equation from previous step:\[10^{-3} = \sqrt{10^{-5} \times x}\]Squaring both sides yields:\[10^{-6} = 10^{-5} \times x\]Solving for \(x\) (the concentration of \(\mathrm{CH}_3\mathrm{COOH}\)) gives:\[x = \frac{10^{-6}}{10^{-5}} = 10^{-1} \mathrm{M}\]

Step 4: Match the solution to the given choices

The concentration of \( \mathrm{CH}_3\mathrm{COOH} \) calculated is \( 10^{-1} \mathrm{M} \). Match it with the options given.It corresponds to option (c) \( 10^{-1} \mathrm{M} \).

Key Concepts

Acetic Acid

Acetic acid is a common organic acid that you might know more familiarly as vinegar. Its chemical formula is \( \mathrm{CH}_{3}\mathrm{COOH} \), signifying it as a weak acid. The character of acetic acid being weak means that it doesn't dissolve completely in water. Instead, it partially dissociates into hydrogen ions \( \left( \mathrm{H}^+ \right) \) and acetate ions \( \left( \mathrm{CH}_3\mathrm{COO}^- \right) \).
When discussing the dissociation of acetic acid, the important factor to know is its acid dissociation constant, \( \mathrm{Ka} \). This constant allows us to understand how strong or weak an acid is. For acetic acid, the \( \mathrm{Ka} \) value is \( 10^{-5} \).
Understanding this value tells us that acetic acid doesn't dissociate much in aqueous solution. This is important because in situations where hydrogen ion concentration from acetic acid is needed, knowing how much of the acid will dissociate is key to performing accurate calculations.

Formic Acid

Formic acid, structurally written as \( \mathrm{HCOOH} \), is the simplest carboxylic acid. It naturally occurs in the venom of ant stings. Compared to acetic acid, formic acid has a slightly higher acid dissociation constant, \( \mathrm{Ka} = 10^{-4} \), which makes it stronger than acetic acid, meaning it dissociates more readily to release hydrogen ions.
In aqueous solution, formic acid partially dissociates into hydrogen ions \( \left( \mathrm{H}^+ \right) \) and formate ions \( \left( \mathrm{HCOO}^- \right) \). This stronger tendency to release hydrogen ions compared to acetic acid is a critical point when comparing the behaviors of different acids in a solution.
The dissociation ability of formic acid significantly affects the concentration of hydrogen ions in a solution, which will come into play when determining how it compares to acetic acid in terms of producing the same hydrogen ion concentration at different molar concentrations.

Hydrogen Ion Concentration

Hydrogen ion concentration is central in understanding acid behavior in solutions. It is often expressed in terms of molarity \( \left( \mathrm{M} \right) \) as \( [\mathrm{H}^+] \). This concentration defines an acid's strength and ability to change the pH of a solution.
In the case of acetic and formic acids, the dissociation constants \( \mathrm{Ka} \) are utilized to determine \( [\mathrm{H}^+] \). To find \( [\mathrm{H}^+] \) in weak acids like acetic or formic acid, you use the relation \( [\mathrm{H}^+] = \sqrt{\mathrm{Ka} \times [\text{acid}]} \).
This formula gives an approximation of the hydrogen ion concentration derived from the acid in solution, crucial for understanding balancing solutions, calculating pH, and solving for unknown concentrations in scenarios like determining the acetic acid concentration with a known \( [\mathrm{H}^+] \).

Concentration Calculations

Calculating concentration involves several steps, especially when dealing with weak acids and their partial dissociation in water. The key is to relate the known and unknown factors using the acid dissociation constant \( \mathrm{Ka} \).
To calculate the hydrogen ion concentration from a weak acid, we use the square root formula \( [\mathrm{H}^+] = \sqrt{\mathrm{Ka} \times [\text{acid}]} \). This formula allows us to approximate the \( [\mathrm{H}^+] \) without knowing the exact extent of dissociation for weak acids.
Once we determine the hydrogen ion concentration for one acid (in this case formic acid), we can equate it to the other acid (acetic acid) and solve for the unknown concentration of the other acid in the solution.
This involves simple algebraic manipulation, squaring both sides of the equated formula to isolate the unknown variable and then solving for its value. Such calculations are fundamental in understanding chemical equilibria in acid-base chemistry.