Question

A student designs an ammeter (a device that measures electrical current) that is based on the electrolysis of water into hydrogen and oxygen gases. When electrical current of unknown magnitude is run through the device for \(2.00 \mathrm{~min}, 12.3 \mathrm{~mL}\) of water-saturated \(\mathrm{H}_{2}(\mathrm{~g})\) is collected. The temperature of the system is \(25.5^{\circ} \mathrm{C}\), and the atmospheric pressure is 768 torr. What is the magnitude of the current in amperes?

Short answer

The magnitude of the current in amperes is 8.17 A, which was calculated using the Ideal Gas Law and Faraday's Law of Electrolysis. First, we converted the volume of hydrogen gas to moles, then calculated the moles of electrons passed through the system based on the moles of hydrogen gas. Finally, we used Faraday's Law to find the current in amperes.

Step-by-step solution

Convert the volume of hydrogen gas to moles

First, we need to convert the volume of hydrogen gas collected to moles. We will use the Ideal Gas Law for this purpose, which is given as follows: \(PV = nRT\) Where: \(P\) = Pressure \(V\) = Volume \(n\) = Moles of gas \(R\) = Ideal Gas Constant \((=0.0821\, L\,atm \,mol^{-1} K^{-1})\) \(T\) = Temperature in Kelvin First, we must convert the temperature from Celsius to Kelvin: \(T_K = T_C + 273.15 = 25.5 + 273.15 = 298.65\,K\) Since the hydrogen gas is water-saturated, we need to correct the pressure of hydrogen gas by subtracting the vapor pressure of water at the given temperature. At 25.5°C, the vapor pressure of water is approximately 23.8 torr, so the partial pressure of hydrogen is: \(P_{H_2} = P_{atm} - P_{water} = 768 - 23.8 = 744.2\, torr\) Now, we should convert the pressure to atm by dividing it by 760: \(P_{H_2(atm)} = \frac{744.2}{760} = 0.979\,atm\) Next, we will convert the volume of hydrogen gas from mL to L: \(V_{H_2(L)} = \frac{12.3}{1000} = 0.0123\,L\) Now, we can use the Ideal Gas Law to find the moles of hydrogen gas: \(n_{H_2} = \frac{PV}{RT} = \frac{(0.979\, atm)(0.0123\, L)}{(0.0821 L\, atm \,mol^{-1} K^{-1})(298.65\, K)} = 5.10 \times 10^{-4}\, mol\)

Calculate the moles of electrons passed through the system

The balanced chemical equation for the electrolysis of water is: \(2H_2O(l) \rightarrow 2H_2(g) + O_2(g)\) For every 2 moles of hydrogen gas produced, 4 moles of electrons are consumed. Therefore, the moles of electrons passed through the system can be calculated as follows: \(n_{e^-} = 2 \times n_{H_2} = 2 \times 5.10 \times 10^{-4} = 1.02 \times 10^{-3}\, mol\) of electrons

Calculate the magnitude of current in amperes using Faraday's Law of Electrolysis

Now, we can calculate the current using Faraday's Law of Electrolysis: \(I = \frac{n_{e^-}F}{t}\) Where: \(I\) = Current in amperes \(n_{e^-}\) = Moles of electrons \(F\) = Faraday Constant \((=96485\, C\, mol^{-1})\) \(t\) = Time in seconds First, we must convert the time from minutes to seconds: \(t_s = 2.00\, min \times 60\, s\, min^{-1} = 120\, s\) Now we can calculate the current: \(I = \frac{(1.02 \times 10^{-3}\, mol)(96485\, C\, mol^{-1})}{120\, s} = 8.17\, A\) The magnitude of the current in amperes is 8.17 A.

Key concepts

Ideal Gas Law

Understanding the behavior of gases under various conditions is crucial for many scientific analyses, and the Ideal Gas Law provides a simple relation between pressure, volume, temperature, and the number of moles of a gas. It is expressed as \(PV = nRT\), where \(P\) is the pressure, \(V\) is the volume, \(n\) is the moles of the gas, \(R\) is the gas constant, and \(T\) is the temperature in Kelvin.

When working with gases as products of chemical reactions like in the electrolysis of water, we can collect the gas and measure its volume. To find other properties such as the number of moles, we use the Ideal Gas Law. We must be cautious with conditions: temperatures should be in Kelvin, pressure may need adjustment for non-ideal conditions (e.g., subtracting vapor pressure of water), and volumes must be in liters. This calculation is a critical step in solving the exercise as it provides a bridge between the physical properties of the gases involved and the chemical reaction taking place.

Faraday's Law of Electrolysis

Faraday's Law of Electrolysis is a quantum of electrochemistry that connects electrical charge and chemical reactions. It tells us that the amount of chemical change during electrolysis is proportional to the quantity of electricity that passes through the electrolyte. In equations, Faraday's first law can be written as \(m = (Q \times M) / (n \times F)\), where \(m\) is the mass of the substance altered at an electrode, \(Q\) is the total electric charge passed through the substance, \(M\) is the molar mass of the substance, \(n\) is the valence number of ions of the substance, and \(F\) is Faraday's constant, which is approximately \(96485\text{ C mol}^{-1}\).

In the context of the solved exercise, Faraday's Law is used to convert moles of electrons (calculated from the stoichiometry of the reaction) to a flow of electrical current over time, which gives us the magnitude of the current. This principle is foundational in designing devices that measure electrical current through electrolysis, as well as in many industrial processes where electrochemical reactions are controlled.

Electrical Current Measurement

In the realm of electricity, current measurement is a fundamental concept involving the determination of the flow of electric charge. It's quantified in amperes (A), and the device typically used to measure this flow is known as an ammeter. The setup described in the exercise situates the ammeter in series with the electrolysis apparatus to ensure that all electric charge flows through the device.

The use of electrolysis to measure the current capitalizes on Faraday's Law, directly correlating the amount of electricity used in the process with the amount of gas produced. By precisely controlling the conditions and carefully measuring the volume of gas, students can infer the amount of current that passed through the system. This application beautifully illustrates the direct relationship between chemical changes and electrical quantities, helping students grasp the more abstract concept of electrical flow through tangible chemical products.

Stoichiometry

Stoichiometry is the area of chemistry that involves the calculation of reactants and products in chemical reactions. In a balanced chemical equation, the stoichiometric coefficients represent the relative amounts of substances involved in the reaction.

Chemical Equation Interpretation

For the electrolysis of water in our exercise, the equation \(2H_2O (l) \rightarrow 2H_2 (g) + O_2 (g)\) tells us that for every 2 moles of water, we get 2 moles of hydrogen gas and 1 mole of oxygen gas.

Electron Involvement

In electrolysis, the stoichiometry also extends to the electrons during oxidation and reduction. For example, each mole of hydrogen gas involves 2 moles of electrons. This relationship allows us to calculate the number of moles of electrons necessary to produce a known quantity of hydrogen, which is crucial in determining the passage of electric current over time per Faraday's Law.

Connecting Concepts

Thus, stoichiometry is invaluable because it connects quantitative aspects of chemical formulas and reactions with physical measurements and observations such as the volume of gas produced or consumed.