Problem 119 - Chemistry: The Central Science in SI Units, Global Edition

Question

A student designs an ammeter (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 90 min, \(32.5 \mathrm{~mL}\) of water-saturated \(\mathrm{H}_{2}(g)\) is collected. The temperature of the system is \(20^{\circ} \mathrm{C},\) and the atmospheric pressure is \(101.3 \mathrm{kPa}\). What is the magnitude of the average current in amperes?

Step-by-Step Solution

Answer

To find the magnitude of the average current, we first determine the moles of hydrogen gas produced using the ideal gas law: \(n=\frac{PV}{RT}\). Then, we find the number of electrons transferred during the electrolysis process using the balanced chemical equation. The total charge transferred is obtained by multiplying the number of electrons transferred by the Faraday constant. Finally, the average current is calculated by dividing the total charge transferred by the time duration (in seconds). Using the given information, we can find the average current through the device in amperes.

Step 1: Find the moles of hydrogen gas produced

To determine the moles of hydrogen produced, we'll use the ideal gas law: \[PV=nRT\] where P is the pressure, V is the volume, n is the number of moles, R is the gas constant, and T is the temperature in Kelvin. We have \(P = 101.3\ \mathrm{kPa}\), \(V = 32.5\ \mathrm{mL} = 0.0325\ \mathrm{L}\), and \(T = 20^\circ \mathrm{C} + 273.15 = 293.15\ \mathrm{K}\). Also, the gas constant is \(R = 8.314\ \mathrm{J\cdot K^{-1}\cdot mol^{-1}}\). Now, we can solve for n: \[n=\frac{PV}{RT}\]

Step 2: Convert pressure to required unit

The pressure is given in kPa, but we need it in the unit of atm for the ideal gas law. We can use the conversion factor \(1\ \mathrm{atm} = 101.325\ \mathrm{kPa}\) to get the pressure in atm: \[P_\mathrm{atm} = \frac{101.3\ \mathrm{kPa}}{101.325\ \mathrm{kPa/atm}}\]

Step 3: Plug in values and solve for moles of hydrogen gas

Now let's calculate the moles of hydrogen gas using the modified ideal gas law formula: \[n=\frac{P_\mathrm{atm}\cdot V}{R\cdot T}\]

Step 4: Calculate the number of electrons transferred

The electrolysis of water produces hydrogen gas, so the balanced reaction is: \[\mathrm{2H_2O\to 2H_2 + O_2}\] For every 1 mole of hydrogen gas produced, 2 moles of electrons are transferred. Hence, we need to find the number of electrons transferred during the entire process. \[\mathrm{Electrons\:transferred} = 2\cdot n\]

Step 5: Calculate the total charge transferred

We can calculate the total charge transferred using the Faraday constant, which is \(96485\ \mathrm{C/mol}\) (Coulombs per mole of electrons): \[\mathrm{Total\:charge\:transferred} = \mathrm{Electrons\:transferred} \cdot 96485\ \mathrm{C/mol}\]

Step 6: Calculate the average current

Finally, we can find the average current by dividing the total charge transferred by the time duration in seconds: \[\mathrm{Average\:current} = \frac{\mathrm{Total\:charge\:transferred}}{\mathrm{Time\:duration}}\] The time duration given is \(90\ \mathrm{min} = 5400\ \mathrm{s}\). Now we can calculate the average current and express it in amperes, where 1 A = 1 C/s.

Key Concepts

Ideal Gas Law

The Ideal Gas Law is an essential tool in understanding the behavior of gases under various conditions. The formula is given by:\[ PV = nRT \]where:

• \( P \) is the pressure of the gas in atmospheres (atm),
• \( V \) is the volume of the gas in liters (L),
• \( n \) is the number of moles of the gas,
• \( R \) is the universal gas constant \( (8.314 \, \text{J} \cdot \text{K}^{-1} \cdot \text{mol}^{-1}) \),
• \( T \) is the temperature in Kelvin (K), which can be converted from Celsius by adding 273.15.

In the exercise, the Ideal Gas Law is used to find out how many moles of hydrogen (\( \mathrm{H_2} \)) are produced from the electrolysis of water. This requires careful conversion between units, such as transforming pressure from kilopascals to atmospheres, and temperature from Celsius to Kelvin. These conversions ensure that the calculation of moles is accurate.

Electrolysis

Electrolysis is a process that drives a chemical reaction using electricity, often used to split compounds into simpler substances. In this context, water is split into hydrogen (\( \mathrm{H_2} \)) and oxygen (\( \mathrm{O_2} \)) gases.The balanced chemical equation for the electrolysis of water is:\[ \mathrm{2H_2O \to 2H_2 + O_2} \]This shows that two molecules of water are decomposed to form two molecules of hydrogen and one molecule of oxygen. During electrolysis, electrons are transferred from one reactant to another, and the quantity of these transferred electrons is directly related to the amount of hydrogen gas produced. Understanding this process is key to linking the volume of gas produced with the current that passed through the solution.

Faraday Constant

The Faraday Constant is a fundamental number in electrochemistry, representing the charge of one mole of electrons, approximately\( 96485 \, \mathrm{C/mol} \) (Coulombs per mole).During electrolysis, the Faraday Constant helps calculate the total charge transferred in the process. As hydrogen gas forms, its production is associated with electron exchange. The relationship can be expressed as:

• For every mole of hydrogen produced, two moles of electrons are transferred.

Using this constant allows us to translate moles of electrons into total charge transferred, crucial for calculating current.

Average Current Calculation

The concept of average current in an electrolysis setup refers to the steady flow of electric charge over time. Current (I) is the rate of charge flow, measured in amperes (A), and it can be calculated using the formula:\[ \mathrm{Average\ current\ (A)} = \frac{\mathrm{Total\ charge\ transferred\ (C)}}{\mathrm{Time\ duration\ (s)}} \]In the exercise scenario, the electrolysis lasts 90 minutes, equivalent to 5400 seconds.Knowing the total charge, obtained from the Faraday Constant and the amount of electrons, allows one to find the average current. This involves:

• Converting the collected gas volume to moles.
• Calculating the corresponding charge using the Faraday Constant.
• Dividing by the total time the current was applied.

This straightforward calculation bridges the gap between theoretical electrochemical reactions and real-world measurement of electric current.