Question

When \(0.05\) moles of the following acid are dissolved in \(1000 \mathrm{ml}\) of \(\mathrm{H}_{2} \mathrm{O}\), the \(\left[\mathrm{H}^{+}\right.\) will be greatest in (a) \(\mathrm{HNO}_{2} ; \mathrm{p} K_{\mathrm{a}}=3.0\) (b) \(\mathrm{HCOOH} ; \mathrm{p} K_{\mathrm{a}}=3.75\) (c) \(\mathrm{HCN} ; \mathrm{p} K_{\mathrm{a}}=9.4\) (d) \(\mathrm{CH}_{3} \mathrm{COOH} ; \mathrm{p} K_{\mathrm{a}}=4.75\)

Short answer

\(\mathrm{HNO}_{2}\) with a pKa of 3.0 will have the greatest \[\mathrm{H}^{+}\] concentration when 0.05 moles are dissolved in 1000 ml of \(\mathrm{H}_{2}\mathrm{O}\).

Step-by-step solution

Understanding the Acid Dissociation Constant (Ka)

The acid dissociation constant, Ka, measures the strength of an acid in solution. It is the equilibrium constant for the chemical reaction where an acid donates a proton to water. Stronger acids have larger Ka values, resulting in greater \[\mathrm{H}^{+}\] concentration. The pKa is the negative logarithm of the Ka value and, therefore, a smaller pKa value indicates a stronger acid.

Relating pKa to \[\mathrm{H}^{+}\] Concentration

Since pKa is inversely related to the strength of the acid, the smaller the pKa, the stronger the acid, and the greater the concentration of \[\mathrm{H}^{+}\] when the acid is dissociated in water. To find the acid that produces the highest concentration of \[\mathrm{H}^{+}\], we need to find the acid with the smallest pKa value.

Comparing pKa Values

We compare the pKa values of each acid to determine which one is smallest:\(\mathrm{HNO}_{2}\); pKa = 3.0\(\mathrm{HCOOH}\); pKa = 3.75\(\mathrm{HCN}\); pKa = 9.4\(\mathrm{CH}_{3}\mathrm{COOH}\); pKa = 4.75 Out of all the given pKa values, \(\mathrm{HNO}_{2}\) has the smallest pKa value.

Determining the Acid with Greatest \[\mathrm{H}^{+}\] Concentration

Since \(\mathrm{HNO}_{2}\) has the smallest pKa value among the given choices, it will produce the greatest concentration of \[\mathrm{H}^{+}\] when dissolved in water.

Key concepts

pKa Value

Understanding the concept of the pKa value is essential for anyone studying chemistry, especially when it comes to analyzing the strength of acids. The pKa value is a numerical scale used to measure the acidity of a solution. It is related to the acid dissociation constant (Ka) by the negative logarithm: the pKa is the negative base-10 logarithm of the Ka value, represented as \( pKa = -\log(Ka) \). Essentially, the lower the pKa value, the stronger the acid.

Therefore, when we look at acids with different pKa values, such as \( HNO_2 \) with a pKa of 3.0, \( HCOOH \) with a pKa of 3.75, and so on, we're comparing their relative strengths. A lower pKa value indicates not only a stronger acid but also implies that this acid will donate protons (hydrogen ions) more readily when in aqueance. This property is crucial in predicting the behavior of acids in different chemical reactions and biological systems.

Strength of Acids

The strength of an acid can be quantified in several ways, but one of the most commonly used methods is through its acid dissociation constant (Ka). Strong acids have a higher tendency to donate their protons to water, which leads to more free hydrogen ions \( \left[ H^+ \right] \) in the solution. On the Ka scale, stronger acids have larger numerical values while weaker acids have smaller Ka values.

There's a direct relationship between the strength of an acid and its pKa value: stronger acids have a lower pKa. This is why examining the pKa value is a quick way to assess an acid's strength. Strong acids like hydrochloric acid (HCl) have very low pKa values, while weak acids like acetic acid (CH3COOH) have higher pKa values.

Hydrogen Ion Concentration

The concentration of hydrogen ions [H^+] in a solution determines the solution's acidity. It's a critical factor in the acid-base chemistry that influences various chemical equilibria and reactions. When acid is added to water, it dissociates to a certain extent and releases hydrogen ions, which affects the pH level of the solution.

The relationship between the hydrogen ion concentration and pH is logarithmic, making the pH scale a convenient way to express the acidity. Mathematically, pH is defined as the negative logarithm of the hydrogen ion concentration: \( pH = -\log \left[ H^+ \right] \). When comparing different acids like HNO2, HCOOH, HCN, and CH3COOH, the one with the smallest pKa value typically releases more hydrogen ions in water, leading to a lower pH and a more acidic solution.

Chemical Equilibrium

Chemical equilibrium refers to the state in which the rate of the forward reaction equals the rate of the reverse reaction, resulting in no net change in the concentration of reactants and products over time. This concept is not only central to acid-base reactions but to all reversible chemical processes.

For the reaction where an acid (HA) donates a proton to water (H2O) forming its conjugate base (A-) and a hydronium ion (H3O+), the equilibrium expression can be written as: \( Ka = \frac{[A^-][H3O^+]}{[HA]} \). Here, Ka is the acid dissociation constant, and a large Ka value indicates that the equilibrium favors the products, meaning the acid is strong and readily gives up its proton. Equilibrium is dynamic - although the concentrations of the reactants and products remain constant at equilibrium, the molecules continue to react with one another.