Question

A solution is prepared by bubbling \(15.0 \mathrm{~L}\) of \(\mathrm{HCl}(g)\) at \(25^{\circ} \mathrm{C}(298 \mathrm{~K})\) and \(101,325 \mathrm{~Pa}\) into \(250.0 \mathrm{~mL}\) of water. (a) Assuming all the HCl dissolves in the water, how many moles of \(\mathrm{HCl}\) are in solution? (b) What is the \(\mathrm{pH}\) of the solution?

Short answer

(a) 0.614 moles; (b) pH = -0.39.

Step-by-step solution

Calculate Moles of HCl Using Ideal Gas Law

To find the moles of HCl, use the ideal gas law formula: \( PV = nRT \). Here, \( P = 101,325 \text{ Pa} \), \( V = 15.0 \text{ L} \), \( R = 8.314 \text{ J mol}^{-1} \text{ K}^{-1} \) (ideal gas constant), and \( T = 298 \text{ K} \). First, convert \( V \) from liters to cubic meters: \( 1 \text{ L} = 0.001 \text{ m}^3 \), so \( 15.0 \text{ L} = 0.015 \text{ m}^3 \). Next, plug in the values: \[ n = \frac{PV}{RT} = \frac{101,325 \times 0.015}{8.314 \times 298} \approx 0.614 \text{ moles} \].

Determine the Concentration of HCl in Solution

Since all HCl gas dissolves in the water, the moles of HCl in the gas are equal to the moles dissolved. The volume of the solution is \(250.0 \text{ mL} = 0.250 \text{ L}\). Calculate the molarity \(M\) of the solution using \[ M = \frac{n}{V} = \frac{0.614}{0.250} = 2.456 \text{ M} \].

Calculate the pH of the Solution

HCl is a strong acid and completely dissociates in water to form \( H^+ \) and \( Cl^- \) ions. Thus, the concentration of \( H^+ \) ions is the same as the molarity of HCl: \( [H^+] = 2.456 \text{ M} \). Calculate the pH using the formula \[ \text{pH} = -\log[H^+] = -\log(2.456) \approx -0.39 \].

Key concepts

Moles of HCl

Understanding how to calculate the moles of a gas like hydrogen chloride (HCl) requires a grasp of the Ideal Gas Law. The Ideal Gas Law is an equation of state that describes the behavior of an ideal gas in terms of pressure (P), volume (V), temperature (T), and moles (n). The equation is expressed as \(PV = nRT\), where \(R\) is the ideal gas constant.

In this problem, we need to find the moles of HCl gas that are bubbled into the water. By using the Ideal Gas Law, we rearrange the formula to solve for \(n\), the number of moles:

• Given: \(P = 101,325\) Pa, \(V = 15.0\) L (converted to cubic meters as \(0.015\) m³), \(T = 298\) K, and \(R = 8.314\) J/(mol⋅K).
• Apply the formula: \(n = \frac{PV}{RT}\). Substituting the known values, we find approximately \(0.614\) moles of HCl in the solution.

This result tells us the amount of HCl gas that has been dissolved in the water which is crucial for further calculations.

Molarity

Molarity is a way to express the concentration of a solute in a solution, defined as the number of moles of solute per liter of solution. It is an important concept in chemistry because it helps describe the strength and reactivity of a solution.

In this exercise, after determining the moles of HCl gas, we calculate its molarity in the prepared solution. The process involves:

• The total volume of the solution is given as \(250.0\) mL, which must be converted to liters, resulting in \(0.250\) L.
• Using the moles of HCl calculated previously, which is approximately \(0.614\) moles, the molarity \(M\) is calculated using the formula: \(M = \frac{n}{V}\).
• Substituting the values gives a molarity of \(2.456\) M.

This molarity indicates how concentrated the HCl solution is, which will affect its chemical behavior, such as in subsequent pH calculations.

pH Calculation

The pH of a solution measures its acidity or alkalinity, calculated as the negative logarithm of the hydrogen ion concentration. For strong acids like HCl, which fully dissociates, the concentration of hydrogen ions, \([H^+]\), is equal to the molarity of the solution.

To find the pH of our HCl solution, we use the formula:\[ \text{pH} = -\log[H^+] \]

• With a molarity of \(2.456\) M, the \([H^+]\) is also \(2.456\) M, since each HCl molecule provides one \(H^+\).
• Substituting the hydrogen ion concentration into the formula gives us the pH: \( \text{pH} = -\log(2.456) \).
• Upon calculation, the result is approximately \(-0.39\).

In this particular scenario, the negative pH value suggests an extremely high concentration of hydrogen ions, characteristic of a very strong acidic solution. This calculation is essential for understanding the solution's reactivity and potential applications or hazards.