Question

\(\mathrm{KMnO}_{4}\) is added to a \(350 \mathrm{ml}\) solution of an approximately \(3 \%\) \(\mathrm{H}_{2} \mathrm{O}_{2}\) solution. It is found that \(3.94 \mathrm{~cm}^{3}\) of \(\mathrm{O}_{2}\) gas is released. It is collected over \(\mathrm{H}_{2} \mathrm{O}\) at a pressure (barometric) of \(0.941\) atm and \(28^{\circ} \mathrm{C}\). Assuming that the pressure due to water vapor is \(0.0373 \mathrm{~atm}\), what is the actual concentration of the \(\mathrm{H}_{2} \mathrm{O}_{2}\) solution?

Short answer

The actual concentration of the \(H_2O_2\) solution is approximately 3.06% after using the Ideal Gas Law, stoichiometry, and given data to calculate the moles of oxygen gas and hydrogen peroxide.

Step-by-step solution

Calculate the partial pressure of oxygen gas

To find the partial pressure of oxygen gas, subtract the pressure due to water vapor from the barometric pressure: \(P_{O_2} = P_{barometric} - P_{water}\) Substitute the given values to find the partial pressure of oxygen gas: \(P_{O_2} = 0.941 \, \text{atm } - 0.0373 \, \text{atm}\)

Calculate the number of moles of oxygen gas

Using the Ideal Gas Law: \(PV = nRT\) Solve for n (number of moles): \(n_{O_2} = \dfrac{P_{O_2}V_{O_2}}{RT}\) Where \(P_{O_2}\) is the partial pressure of oxygen gas, \(V_{O_2}\) is the volume of oxygen gas, \(R\) is the universal gas constant, and \(T\) is the temperature in Kelvin. First, convert the temperature in Celsius to Kelvin: \(T = 28^{\circ}C + 273.15 = 301.15 K\) Next, convert the volume of oxygen gas from cm³ to L: \(V_{O_2} = 3.94 \, \text{cm}^{3} \times \dfrac{1 \, \text{L}}{1000 \, \text{cm}^{3}} = 0.00394 \, \text{L}\) Now, plug the given values into the equation: \(n_{O_2} = \dfrac{(0.9037 \, \text{atm})(0.00394 \, \text{L})}{(0.0821 \, \text{L atm/mol K})(301.15 \, \text{K})}\)

Determine the number of moles of hydrogen peroxide

The balanced equation for the reaction of hydrogen peroxide with potassium permanganate is: \(\ce{2KMnO4 + 5H2O2 -> 2MnO_2 + 2KOH + 4H2O + O2}\) From this balanced equation, we see that 5 moles of \(H_2O_2\) produce 1 mole of \(O_2\). Thus, we can use stoichiometry to determine the number of moles of hydrogen peroxide: \(n_{H_2O_2} = \dfrac{5}{1}n_{O_2}\)

Calculate the concentration of hydrogen peroxide

To find the concentration of hydrogen peroxide in the solution, divide the number of moles of hydrogen peroxide by the volume of the solution in liters: \(\text{Concentration of }\ce{H2O2} = \dfrac{n_{H_2O_2}}{V_{solution}}\) First, convert the volume of the solution from mL to L: \(V_{solution} = 350 \, \mathrm{ml} \times \dfrac{1 \, \text{L}}{1000 \, \mathrm{ml}} = 0.350 \, \mathrm{L}\) Next, plug the numbers into the formula: \(\text{Concentration of }\ce{H2O2} = \dfrac{n_{H_2O_2}}{0.350 \, \mathrm{L}}\) Finally, multiply the concentration by 100 to express it as a percentage.

Key concepts

Ideal Gas Law

The Ideal Gas Law is a fundamental equation in chemistry that relates pressure, volume, temperature, and moles of a gas using the formula: \ \(PV = nRT\). Here, \(P\) stands for pressure, \(V\) is volume, \(n\) represents the number of moles, \(R\) is the universal gas constant (approximately \(0.0821 \, \text{L atm/mol K}\)), and \(T\) is temperature measured in Kelvin.
It is important to ensure all units are compatible before using the Ideal Gas Law. For temperature, always convert Celsius to Kelvin by adding \(273.15\). For volume, if given in mL or cm³, convert it to liters since the gas constant \(R\) uses liters.
Using this law helps us calculate the number of moles, a crucial step in understanding the amount of a substance involved in chemical reactions. Always check units and convert where necessary to avoid errors.

Stoichiometry

Stoichiometry refers to the calculation of reactants and products in chemical reactions. It involves using a balanced chemical equation to convert between moles of one substance to moles of another.
For example, in the reaction \(\ce{2KMnO4 + 5H2O2 -> 2MnO2 + 2KOH + 4H2O + O2}\), for every mole of \(\ce{O2}\) produced, 5 moles of \(\ce{H2O2}\) are consumed.
Steps in stoichiometry typically include:

• Writing a balanced chemical equation.
• Determining moles for known substances.
• Using mole ratios from the equation to find moles of unknown substances.

Stoichiometry provides precision in chemical calculations, ensuring proper proportions and amounts are used in reactions.

Solution Concentration

Solution concentration indicates the amount of solute dissolved in a given solution. It's commonly expressed in molarity (\(M\)), which is moles per liter, though it can also be expressed as a percentage.
To find molarity, divide the number of moles of solute by the volume of the solution in liters: \(\text{Molarity } = \frac{\text{moles of solute}}{\text{liters of solution}}\).
For percentage concentration, calculate the mass of solute per 100 grams of solution or multiply molarity by 100 for a simpler percentage form.
Concentration is crucial for predicting reaction yields and is fundamental in laboratory work and industrial applications, influencing product formation rates and saturation points.

Partial Pressure

Partial pressure is the pressure exerted by a single gas in a mixture of gases. It is a component of the total pressure, and each gas in a mixture contributes to it.
You can calculate partial pressure using Dalton's Law: \(P_{\text{total}} = P_{1} + P_{2} + \ldots + P_{n}\), meaning the total pressure is the sum of each individual gas's partial pressure.
In this exercise, the partial pressure is calculated by subtracting the vapor pressure of water from the barometric pressure to get \(P_{\ce{O2}}\). This distinguishes the contribution of \(\ce{O2}\) gas itself in the situation of collection over water.
Understanding partial pressure is important for processes like gas collection and determining behaviors of gases in mixed environments.