Question

Calculate the pH of a \(0.25 \mathrm{M}\) solution of the salt ethyl-amine hydrochloride, \(\mathrm{C}_{2} \mathrm{H}_{5} \mathrm{NH}_{3} \mathrm{CI}\). The dissociation constant for the base, ethylamine \(\left(\mathrm{C}_{2} \mathrm{H}_{5} \mathrm{NH}_{2}\right)\) is \(\mathrm{K}_{\mathrm{b}}=5.6 \times 10^{-4}\).

Short answer

The pH of a \(0.25 \mathrm{M}\) solution of ethyl-amine hydrochloride, \(\mathrm{C}_{2} \mathrm{H}_{5} \mathrm{NH}_{3} \mathrm{CI}\), is approximately 1.93.

Step-by-step solution

Write the dissociation equation for the salt

We know that the salt ethyl-amine hydrochloride, \(\mathrm{C}_{2} \mathrm{H}_{5} \mathrm{NH}_{3} \mathrm{CI}\), dissociates in water to give ethylamine \(\left(\mathrm{C}_{2} \mathrm{H}_{5} \mathrm{NH}_{2}\right)\) and chloride ions \(\mathrm{Cl}^{-}\). Write the dissociation equation as follows: \[ \mathrm{C}_{2}\mathrm{H}_{5}\mathrm{NH}_{3}\mathrm{Cl} \rightleftharpoons \mathrm{C}_{2} \mathrm{H}_{5} \mathrm{NH}_{2} + \mathrm{H}^{+} \]

Set up the ICE table

Create an Initial, Change, and Equilibrium (ICE) table to track the changes in the concentrations of the species involved in the reaction: | \(\mathrm{C}_{2}\mathrm{H}_{5}\mathrm{NH}_{3}\mathrm{Cl}\) | \(\mathrm{C}_{2} \mathrm{H}_{5} \mathrm{NH}_{2}\) | \(\mathrm{H}^{+}\) --------------|-----------------|----------------|--------------- Initial (M) | 0.25 | 0 | 0 Change (M) | -x | +x | +x Equilibrium (M)| \(0.25-x\) | x | x Now, we will use the given \(\mathrm{K}_{\mathrm{b}}\) value to relate the concentrations of these species at equilibrium.

Write the expression for \(\mathrm{K}_{\mathrm{b}}\) and solve for x

The expression for the \(\mathrm{K}_{\mathrm{b}}\) for ethylamine is: \[ \mathrm{K}_{\mathrm{b}}=\frac{[\mathrm{C}_{2}\mathrm{H}_{5}\mathrm{NH}_{2}][\mathrm{H}^{+}]}{[\mathrm{C}_{2}\mathrm{H}_{5}\mathrm{NH}_{3}\mathrm{Cl}]} \] Substitute the equilibrium values from the ICE table into the \(\mathrm{K}_{\mathrm{b}}\) expression: \[ 5.6 \times 10^{-4}=\frac{x\cdot x}{0.25-x} \] Assuming that x is very small compared to \(0.25\), we can approximate \(0.25-x \approx 0.25\). Thus, we have: \[ 5.6 \times 10^{-4}=\frac{x^2}{0.25} \] Solve for x, which represents the concentration of \(\mathrm{H}^{+}\) at equilibrium: \[ x=\sqrt{5.6 \times 10^{-4}\times 0.25} \] And calculate its value: \[ x\approx 0.01183 \] Since x is small compared to \(0.25\), our approximation is valid.

Calculate pH from \([\mathrm{H}^{+}]\)

Now we have the concentration of hydrogen ions, \([\mathrm{H}^{+}]\), at equilibrium. To find the pH of the solution, use the formula: \[ \mathrm{pH}=-\log_{10} [\mathrm{H}^{+}] \] Plug the obtained value of x into the formula: \[ \mathrm{pH}=-\log_{10} (0.01183) \] Calculate the pH of the solution: \[ \mathrm{pH} \approx 1.93 \] So, the pH of a \(0.25 \mathrm{M}\) solution of ethyl-amine hydrochloride is approximately 1.93.

Key concepts

ethylamine

Ethylamine, also known by its chemical formula \((\mathrm{C}_{2}\mathrm{H}_{5}\mathrm{NH}_{2})\), is a simple amine that plays a crucial role in chemistry when dissolved in water. As a weak base, ethylamine partially dissociates in water to produce hydroxide ions (\(\mathrm{OH}^-\)) and its conjugate acid, ethylammonium ion \((\mathrm{C}_{2}\mathrm{H}_{5}\mathrm{NH}_{3}^+)\). This chemical process is central to the calculation of the pH of solutions containing ethylamine or its derivatives.

Understanding ethylamine's behavior in water is essential for chemical equilibrium calculations. It highlights the nature of weak bases compared to strong bases, which dissociate completely. This partial dissociation influences how we calculate the equilibrium concentrations and further leads to determining the pH of the solution.

dissociation constant

The dissociation constant, represented as \(\mathrm{K}_b\) for bases, like ethylamine, provides insight into the extent of dissociation in an aqueous solution. This constant is crucial in determining the pH of the solution when dealing with weak bases.

For ethylamine, the dissociation constant is given as \(5.6 \times 10^{-4}\). This value indicates the equilibrium position of the reaction, where less than 1% of ethylamine molecules dissociate in solution.

Recognizing the small value of \(\mathrm{K}_b\) helps us understand why the change in initial concentration upon reaching equilibrium (denoted as \(x\) in calculations) is often negligible, simplifying our calculations. Such simplifications are critical in solving equations for equilibrium concentrations, essential for determining pH.

ICE table

An ICE table, which stands for Initial, Change, Equilibrium, is an organized way to track concentrations of chemical species in a reaction. When calculating pH for weak acids or bases, the ICE table becomes a vital tool in managing these concentrations.

Consider the reaction of ethylamine dissociation:

• Initial concentrations depict the conditions before any reaction occurs. For our example, the ethylamine hydrochloride's initial concentration is given.
• Changes signify the amount that each species gains or loses as the reaction approaches equilibrium.
• Equilibrium concentrations reflect the actual amounts of species present once the reaction stabilizes.
  

Using the ICE table helps streamline the calculation of changes in concentrations and applying these values in the dissociation constant expression aids in solving for unknowns, like the concentration of hydrogen ions \([\mathrm{H}^+]\), needed for pH calculation.

equilibrium concentration

Equilibrium concentration refers to the concentrations of reactants and products once a reversible reaction achieves a state of balance. In our context, it involves ethylamine reaching an equilibrium state in water.

Achieving equilibrium doesn't mean that reactions stopped; instead, the rates of the forward and reverse reactions are equal, leading to stable concentrations of products and reactants.

These concentrations are fundamental for using the dissociation constant formula to solve for unknowns like \(x\), the change in concentration shared among products due to partial dissociation.

Understanding how to determine equilibrium concentrations ensures accurate pH computations, as equilibrium concentrations directly feed into the expression for calculating \([\mathrm{H}^+]\) concentrations, thus leading to the determination of the solution's pH.