Problem 98 - Chemistry: Principles and Reactions

Question

A hydrogen-oxygen fuel cell operates on the reaction: $$ \mathrm{H}_{2}(g)+\frac{1}{2} \mathrm{O}_{2}(g) \longrightarrow \mathrm{H}_{2} \mathrm{O}(l) $$ If the cell is designed to produce \(1.5\) amp of current and if the hydrogen is contained in a 1.0-L tank at 200 atm pressure and \(25^{\circ} \mathrm{C}\), how long can the fuel cell operate before the hydrogen runs out? Assume that oxygen gas is in excess.

Step-by-Step Solution

Answer

Answer: The fuel cell can operate for approximately 2916.67 hours before the hydrogen runs out.

Step 1: Calculate the number of moles of hydrogen available in the tank

To do this, we will use the Ideal Gas Law formula, which is given by: $$ PV = nRT $$ Where: P = pressure (200 atm) V = volume (1.0 L) n = number of moles R = gas constant (0.0821 L atm K⁻¹ mol⁻¹) T = temperature (298 K, equivalent to 25°C) We will solve for n: $$ n = \frac{PV}{RT} $$

Step 2: Plug in the values and calculate the number of moles

Now we will substitute the given values into the formula: $$ n = \frac{(200 \,\text{atm})(1.0\, \text{L})}{(0.0821\, \text{L atm K}^{-1} \text{mol}^{-1})(298\, \text{K})} $$ Calculate the value of n: $$ n = 81.55\, \text{moles} $$

Step 3: Calculate the total charge produced in the fuel cell

We know that 1 mole of hydrogen produces 2 moles of electrons, and 1 mole of electrons contains \(6.022\times10^{23}\) elementary charges. Therefore, the total charge produced by the available hydrogen can be calculated using Faraday's constant (F), which is \(9.65\times10^4 \,\text{C} \,\text{mol}^{-1}\): $$ Q = n \times 2 \times F $$

Step 4: Plug in the values and calculate the total charge produced in the fuel cell

Now we will substitute the values into the formula: $$ Q = (81.55\, \text{moles}) \times 2 \times (9.65\times10^4 \,\text{C} \,\text{mol}^{-1}) $$ Calculate the value of Q: $$ Q = 1.57 \times 10^7\, \text{C} $$

Step 5: Calculate the time it takes for the hydrogen to run out

Now, we are given that the cell produces \(1.5\) amps of current. Recall that current (in Amperes) is defined as the amount of charge (in Coulombs) passing through a point per unit of time (in seconds). So, we have: $$ I = \frac{Q}{t} $$ Solving for t, we get: $$ t = \frac{Q}{I} $$

Step 6: Plug in the values and calculate the time

Now we will substitute the values into the formula: $$ t = \frac{1.57 \times 10^7\, \text{C}}{1.5\, \text{A}} $$ Calculate the value of t: $$ t = 1.05 \times 10^7\, \text{s} $$

Step 7: Convert time into hours

We can convert the time into hours by dividing it by the number of seconds in an hour (3600 seconds): $$ t = \frac{1.05 \times 10^7\, \text{s}}{3600\, \text{s/h}} $$ Calculate the value of t: $$ t = 2916.67\, \text{hours} $$ The fuel cell can operate for approximately \(2916.67\) hours before the hydrogen runs out.

Key Concepts

Ideal Gas Law

The Ideal Gas Law is a fundamental equation in chemistry and physics that helps us understand how gases behave under different conditions of pressure (P), volume (V), temperature (T), and number of moles (n). It is expressed as:
\[PV = nRT\]
Where R represents the ideal gas constant. For this exercise, the Ideal Gas Law is used to calculate the number of moles of hydrogen gas (H2) given the pressure, volume, and temperature of the gas. Understanding the amount of hydrogen is essential to determining how long the fuel cell can run. Since real gases behave like ideal gases at high temperature or low pressure, this assumption is fully applicable to this scenario where the tank pressure is 200 atm and the temperature is 25°C. This step is crucial as it sets the foundation for further calculations concerning the fuel cell's operation time.

Faraday's Constant

Faraday's constant is an important value in electrochemistry representing the total charge of one mole of electrons. It is denoted by the symbol F and is equal to approximately \(9.65 \times 10^4 \,\text{C mol}^{-1}\), where C is coulombs per mole of electrons. This constant is utilized in the exercise to calculate the total charge that the fuel cell can generate. In the context of a fuel cell, knowing the charge helps us predict the total amount of electricity that the fuel cell will produce before the hydrogen is exhausted.

Bridge Between Chemistry and Electricity

Understanding Faraday's constant allows students to make connections between moles of hydrogen and the electrical charge, thus bridging a gap between the principles of chemistry and the practical application of electricity. Every mole of hydrogen yields two moles of electrons, which according to Faraday's constant, translates to a particular amount of charge that can be harnessed for electrical work.

Electrochemistry

Electrochemistry is the branch of chemistry that deals with the relationship between electrical energy and chemical changes. In your fuel cell problem, electrochemistry principles are at play in converting chemical energy from the reaction between hydrogen and oxygen into electric energy. The heart of a hydrogen-oxygen fuel cell is its ability to perform this conversion efficiently.

Oxidation and Reduction Reactions

At the anode, the oxidation of hydrogen gas occurs, liberating electrons that travel through an external circuit. Meanwhile, at the cathode, oxygen gas is reduced by these electrons, forming water. This redox reaction generates an electric current utilized by the fuel cell. By solving this exercise, you're not only calculating time based on chemical consumption, but also applying electrochemical concepts to understand how clean energy is generated, which is vital in today's push towards sustainable energy sources.