Problem 121 - Chemistry: The Central Science

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 \(\mathrm{H}_{2}(g)\) 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?

Step-by-Step Solution

Answer

The magnitude of the electric current in the ammeter is approximately 5.13 amperes.

Step 1: Convert volume to moles using the ideal gas law

We are given the volume (V) of hydrogen gas, the temperature (T), and the pressure (P). We can use the ideal gas law \(PV = nRT\) to find the number of moles of hydrogen gas, where n is the number of moles, R is the gas constant, and T is the temperature in Kelvin. First, let's convert the given temperature to Kelvin: T(K) = T(°C) + 273.15 T(K) = 25.5 + 273.15 T(K) = 298.65 Next, let's convert the pressure from torr to atm. 1 atm = 760 torr. P(atm) = 768 torr * (1 atm / 760 torr) ≈ 1.0105 atm Now, we can use the ideal gas law to find the number of moles of hydrogen gas (H2). \(PV = nRT\) \(n = \frac{PV}{RT}\) We are given that the volume of H2 is 12.3 mL. We need to convert this to L first. V(L) = 12.3 mL * (1 L / 1000 mL) = 0.0123 L Finally, we can plug in all the values to find n: \(n = \frac{(1.0105 \mathrm{~atm})(0.0123 \mathrm{~L})}{(0.0821\frac{\mathrm{L\cdot atm}}{\mathrm{mol\cdot K}})(298.65 \mathrm{~K})} \approx 5.07 \times 10^{-4} \mathrm{~mol}\)

Step 2: Determine the number of electrons transferred using Faraday's law

For the electrolysis of water, 2 moles of electrons are needed to produce 1 mole of hydrogen gas (H2). So, the number of moles of electrons can be found by: moles of electrons = 2 * moles of H2 moles of electrons ≈ 2 * 5.07 * 10^{-4} = 1.014 * 10^{-3} mol Now, we'll convert the moles of electrons to the number of electrons using Avogadro's number (6.022 * 10^{23} particles/mol). number of electrons = moles of electrons * Avogadro's number number of electrons ≈ 1.014 * 10^{-3} * 6.022 * 10^{23} ≈ 6.106 * 10^{20}

Step 3: Calculate the magnitude of the current

We can now calculate the magnitude of the electric current (I) using the formula I = nQ/t, where n is the number of electrons, Q is the charge of an electron (1.602 * 10^{-19} C), and t is the time in seconds. t = 2.00 min * (60 s / 1 min) = 120 s Now, calculate the electric current: I = nQ/t I ≈ (6.106 * 10^{20} electrons)(1.602 * 10^{-19} C/electron) / 120s I ≈ 5.13 A The magnitude of the electric current in the ammeter is approximately 5.13 amperes.

Key Concepts

Ideal Gas Law

The ideal gas law is a fundamental equation in chemistry that relates the four variables of a gas sample: pressure (P), volume (V), number of moles (n), and temperature (T). It is expressed as:\[PV = nRT\]where R is the ideal gas constant, with a value of 0.0821 L·atm/mol·K. This law is crucial for calculating unknown properties of gases given certain conditions. In our exercise, we used it to determine the number of moles of hydrogen (H_2) produced from the electrolysis process.
- We first converted the given temperature from Celsius to Kelvin, since Kelvin is the absolute temperature scale required in this equation.- We also converted pressure from torr to atmospheres, aligning with the units of R.- After these conversions, we rearranged the equation to solve for n, the moles of gas, using the known quantities of P, V, R, and T.
Through this application, we could quantify the hydrogen gas produced, which further connects to the next steps of determining current and electron transfer.

Faraday's Law

Faraday's law of electrolysis is a principle that relates the amount of substance transformed during electrolysis to the electric charge passed through the system. It states that the amount of substance deposited or liberated at an electrode during electrolysis is directly proportional to the amount of electric charge passed through the circuit.
In this problem, Faraday’s law helps us understand the stoichiometry of electrolysis by noting that for every 1 mole of hydrogen gas ( H_2 ) produced during water electrolysis, 2 moles of electrons are involved. This relationship is key to understanding how electric charge relates to chemical reactions in electrolysis.
- We calculated the moles of electrons needed by multiplying the moles of H_2 by two, due to the proportional relationship, given the balanced chemical equation. - This information was then used to determine the specific charge that passed through the system, assisting in calculating the electric current.

Electric Current Measurement

Measuring electric current, denoted as I in equations, involves determining the flow of electric charge during a given time period. In this exercise, we utilized the formula:\[I = \frac{nQ}{t}\]where I is the current in amperes, n is the number of electrons, Q is the charge of a single electron, and t is the time in seconds.
- We first calculated the total number of electrons, using Avogadro's number and the moles of electrons derived earlier.- Knowing the charge of one electron (1.602 \times 10^{-19} C), we calculated the total charge associated with these electrons.- By dividing this total charge by the reaction's duration in seconds, we obtained the magnitude of the current.
The ability to measure electric current accurately is fundamental in designing devices that rely on precise electrical measurements, like the student’s ammeter.

Avogadro's Number

Avogadro's number is a key constant in chemistry, providing a bridge between the macroscopic world we observe and the microscopic world of atoms and molecules. It is defined as:\[6.022 \times 10^{23} \text{ particles/mol}\]This number represents how many atoms, ions, or molecules are present in one mole of a substance. In electrolysis calculations, Avogadro's number allows us to convert between the number of moles and the number of individual particles.
- In the exercise, we used Avogadro's number to calculate the total number of electrons transferred during the electrolysis process. - Multiplying the moles of electrons by Avogadro's number gave us the actual count of electrons.
Understanding Avogadro's number is vital for all stoichiometric calculations in chemistry, confirming how large-scale quantities in the lab connect to the minute atomic scale interactions.