Problem 78 - Chemistry and Chemical Reactivity

Question

To store \(2.88 \mathrm{kg}\) of gasoline with an energy equivalence of \(1.43 \times 10^{8} \mathrm{J}\) requires a volume of \(4.1 \mathrm{L} .\) In comparison, \(1.0 \mathrm{kg}\) of \(\mathrm{H}_{2}\) has the same energy equivalence. What volume is required if this quantity of \(H_{2}\) is to be stored at \(25^{\circ} \mathrm{C}\) and 1.0 atm of pressure?

Step-by-Step Solution

Answer

The required volume is approximately 12249.3 liters.

Step 1: Understand the Problem

We need to find the volume required to store 1.0 kg of hydrogen gas ( H_{2} ) at 25°C and 1.0 atm pressure. The energy equivalence of the hydrogen is provided, and we will use the ideal gas law to calculate the volume.

Step 2: Determine the number of moles of hydrogen

First, find the molar mass of hydrogen gas (H_2). Since hydrogen has an atomic mass of about 1 g/mol for each hydrogen atom, the molecular mass of H_2 is 2 g/mol. To find the number of moles: \[\text{{Number of moles}} = \frac{{\text{{mass of }} H_2}}{{\text{{molar mass}}}}\] \[\text{{Number of moles}} = \frac{{1000 \, \text{g}}}{{2 \, \text{g/mol}}} = 500 \, \text{mol}\]

Step 3: Use the Ideal Gas Law

The ideal gas law is given by: \[ PV = nRT \] where: - \(P\) is the pressure (1.0 atm), - \(V\) is the volume, - \(n\) is the number of moles (500 mol), - \(R\) is the ideal gas constant (0.0821 L·atm/mol·K), - \(T\) is the temperature in Kelvin (\(25^{\circ}\mathrm{C} = 298 \mathrm{K}\)).We need to solve for V: \[V = \frac{{nRT}}{{P}}\] \[ V = \frac{{(500 \, \text{mol})(0.0821 \, \text{L·atm/mol·K})(298 \, \text{K})}}{{1.0 \, \text{atm}}}\] \[ V \approx 12249.3 \, \text{L}\]

Step 4: Finalize the Calculation

Having computed the volume, we can conclude that to store 1.0 kg of hydrogen gas at 25°C and 1 atm, a volume of approximately 12249.3 liters is needed.

Key Concepts

Molar Mass

The concept of molar mass is important when dealing with chemical quantities because it helps convert between mass and the number of moles. Molar mass is defined as the mass of one mole of a substance. For hydrogen gas, which consists of two hydrogen atoms, the molar mass is simply the sum of the individual atomic masses:

• Each hydrogen atom has an atomic mass of approximately 1 g/mol.
• Since hydrogen gas (H\(_2\)) is diatomic, the molar mass is 2 g/mol.

This means that for every 2 grams of hydrogen gas, there is exactly one mole of molecules. This is critical in our calculations because it allows the conversion of the mass of hydrogen gas to moles using the formula:\[\text{Number of moles} = \frac{\text{mass}}{\text{molar mass}}\]Which is an essential step when applying the ideal gas law later.

Hydrogen Gas

Hydrogen gas is the lightest and most abundant chemical element, consisting of diatomic molecules (H\(_2\)). It has unique properties that make it an interesting subject for studying gas behaviors and energy storage. Here are some key characteristics:

• Being the lightest gas means it has the highest energy content by weight, which makes it efficient for energy applications.
• It is highly flammable and must be stored carefully, typically in pressurized containers.

In our exercise, we need to store hydrogen gas at standard conditions: 25°C (298 K) and 1.0 atm of pressure. The goal is to determine the volume required for this storage, using the ideal gas law. This involves understanding hydrogen's behavior under these conditions without considering intermolecular forces, as hydrogen exhibits properties close to an 'ideal' gas.

Energy Equivalence

Energy equivalence is a concept used to compare different energy forms. In this exercise, we're comparing the energy that can be stored or released by a certain mass of gasoline versus hydrogen gas. When we say 1 kg of hydrogen has the same energy equivalence as 2.88 kg of gasoline, it means that the total energy potential is equal. Here, the gasoline has an energy equivalence of \(1.43 \times 10^{8} \) Joules. This demonstrates hydrogen's high efficiency for energy storage:

• Hydrogen’s high specific energy (energy per unit mass) suggests why it is considered a potential future fuel alternative.
• This comparison allows engineers to calculate and design storage systems based on energy needs, rather than weight alone.

Calculating equivalent volumes helps in understanding the physical space these energy sources would occupy, critical for practical application in storage and transportation.