Question

Titanium hydride, \(\mathrm{TiH}_{2}\), has a density of \(3.75 \mathrm{~g} / \mathrm{cm}^{3}\). (a) Calculate the density of hydrogen in \(\mathrm{TiH}_{2}\), and compare it with that in liquid \(\mathrm{H}_{2}\left(0.070 \mathrm{~g} / \mathrm{cm}^{3}\right)\). (b) How many cubic liters of \(\mathrm{H}_{2}\) at STP are absorbed in making \(1.00 \mathrm{~cm}^{3}\) of \(\mathrm{TiH}_{2} ?\)

Short answer

(a) Hydrogen density in TiH2: 0.1514 g/cm³, higher than in liquid H2. (b) 1.682 L of H2 at STP is absorbed in 1 cm³ of TiH2.

Step-by-step solution

Molar Mass of TiH2

First, calculate the molar mass of \( \text{TiH}_2 \). The atomic mass of titanium (Ti) is approximately 47.87 g/mol and hydrogen (H) is about 1.008 g/mol. Therefore, the molar mass of \( \text{TiH}_2 \) is: \[ 47.87 + 2 \times 1.008 = 49.886 \text{ g/mol}. \]

Mass Ratio of Hydrogen in TiH2

Next, find the mass of hydrogen in \( \text{TiH}_2 \) by calculating the ratio of the mass of hydrogen to the molar mass of \( \text{TiH}_2 \): \[ \frac{2 \times 1.008}{49.886} \approx 0.04038. \] So, 4.038% of the mass of \( \text{TiH}_2 \) is contributed by hydrogen.

Density of Hydrogen in TiH2

Now, calculate the density of hydrogen in \( \text{TiH}_2 \). Multiply the percentage by the density of \( \text{TiH}_2 \): \[ 0.04038 \times 3.75 \text{ g/cm}^3 \approx 0.1514 \text{ g/cm}^3. \]

Comparison with Liquid Hydrogen Density

Compare the hydrogen density in \( \text{TiH}_2 \) (0.1514 g/cm³) to that in liquid hydrogen (0.070 g/cm³). The density in \( \text{TiH}_2 \) is significantly higher.

Moles of H2 in 1 cm³ of TiH2

Determine the moles of hydrogen absorbed in 1 cm³ of \( \text{TiH}_2 \). Using the hydrogen density we find: \[ \frac{0.1514 \text{ g/cm}^3}{2.016 \text{ g/mol}} \approx 0.075062 \text{ mol/cm}^3. \]

Volume of H2 at STP

Using the ideal gas law at STP (1 mol of gas occupies 22.414 L), the volume of \( \text{H}_2 \) is: \[ 0.075062 \times 22.414 \approx 1.682 \text{ L}. \] Therefore, 1.682 L of \( \text{H}_2 \) at STP is absorbed in making 1 cm³ of \( \text{TiH}_2 \).

Key concepts

Understanding the Ideal Gas Law

The ideal gas law is a crucial concept in chemistry that relates the pressure, volume, temperature, and number of moles of a gas. It is represented by the equation \( PV = nRT \), where:

• \( P \) is the pressure of the gas in atmospheres (atm)
• \( V \) is the volume in liters (L)
• \( n \) is the number of moles of the gas
• \( R \) is the ideal gas constant, approximately \( 0.0821 \text{ L atm K}^{-1} \text{mol}^{-1} \)
• \( T \) is the temperature in Kelvin (K)

For example, when calculating the volume of hydrogen gas \(\text{H}_2\) absorbed by \(\text{TiH}_2\), at standard temperature and pressure (STP, 0°C and 1 atm), knowing that 1 mole of any ideal gas occupies 22.414 liters is essential.
This relationship allows us to determine the volume of gas needed to calculate how much \(\text{H}_2\) a certain volume of \(\text{TiH}_2\) can absorb at STP.

Explaining Density Comparison

When comparing densities, it is important to understand what density represents: the mass of a substance per unit volume, usually expressed in grams per cubic centimeter (g/cm³). Density tells us how compact a substance is, with higher densities indicating more mass in a given volume.
In the given exercise, the density of hydrogen within \(\text{TiH}_2\) is calculated to be approximately 0.1514 g/cm³. This is significantly denser than liquid hydrogen, which has a density of just 0.070 g/cm³.
This means that the hydrogen absorbed within \(\text{TiH}_2\) is much more concentrated compared to its liquid form, making \(\text{TiH}_2\) an efficient medium for storing hydrogen in a much more compact form than liquid hydrogen.

Molar Mass Calculation Insights

Molar mass is a fundamental concept in chemistry that measures the total mass of a molecule. It is calculated by summing the masses of all atoms in a compound, using their atomic masses from the periodic table, expressed in grams per mole (g/mol).
For molecules like \(\text{TiH}_2\), the molar mass is derived by adding the molar masses of its individual components: one titanium (Ti) atom and two hydrogen (H) atoms. Titanium has a molar mass of approximately 47.87 g/mol, and hydrogen is about 1.008 g/mol each.

• \(\text{Molar Mass of } \text{TiH}_2 = 47.87 + 2 \times 1.008 = 49.886\, \text{g/mol}\)

Knowing the molar mass is essential for further calculations such as determining how much hydrogen by mass is present in the compound or deducing the number of moles of components in a mixture. It lays the groundwork for understanding how different elements combine to form compounds and can be used to find strategic details like percentage composition and stoichiometry within chemical equations.