Problem 91 - Chemistry The Central Science

Question

(a) How many grams of \(\mathrm{H}_{2}\) can be stored in 100.0 \(\mathrm{kg}\) of the alloy FeTi if the hydride FeTiH_ is formed? (b) What volume does this quantity of \(\mathrm{H}_{2}\) occupy at STP? (c) If this quantity of hydrogen was combusted in air to produce liquid water, how much energy could be produced?

Step-by-Step Solution

Answer

\(Approximately 963 \: grams \: of \: H_2 \: can \: be \: stored \: in \: 100.0 \: kg \: of \: the \: alloy \: FeTi. \: This \: quantity \: of \: H_2 \: occupies \: a \: volume \: of \: 21,357.7 \: L \: at \: STP. \: When \: combusted \: in \: air \: to \: produce \: liquid \: water, \: this \: hydrogen \: can \: produce \: approximately \: 136,182.3 \: kJ \: of \: energy. \)

Step 1: Calculate the mass of hydrogen stored in FeTi alloy

We know that the hydride formed is FeTiH. To calculate the hydrogen stored in 100 kg of alloy, find the proportion of hydrogen in the FeTiH compound. First, we need to find the molar mass of FeTiH, Fe, and Ti: Fe: The molar mass of Fe is 55.85 g/mol. Ti: The molar mass of Ti is 47.87 g/mol. H: The molar mass of H is 1.01 g/mol. Now we find the molar mass of FeTiH by adding the molar masses of Fe, Ti, and H: Molar mass of FeTiH = Molar mass of Fe + Molar mass of Ti + Molar mass of H = 55.85 g/mol + 47.87 g/mol + 1.01 g/mol = 104.73 g/mol Next, find the proportion of hydrogen in the FeTiH compound: Proportion of hydrogen = Molar mass of H / Molar mass of FeTiH = 1.01 g/mol / 104.73 g/mol ~ 0.00963 Now, to find the mass of hydrogen stored in 100 kg of FeTi alloy, multiply the mass of the alloy by the proportion of hydrogen: Mass of hydrogen = Mass of FeTi alloy * Proportion of hydrogen = 100,000 g * 0.00963 ≈ 963 g So, around 963 grams of hydrogen can be stored in 100 kg of the FeTi alloy.

Step 2: Calculate the volume of hydrogen at STP

Now that we know the mass of hydrogen, we can find the volume it occupies at STP (standard temperature and pressure, defined as 0°C and 1 atm). The molar volume of an ideal gas at STP = 22.4 L/mol. First, convert the mass of hydrogen to moles: Number of moles = Mass / Molar Mass = 963 g / (1.01 g/mol) ≈ 953.47 moles Now, calculate the volume of hydrogen at STP: Volume of hydrogen = Number of moles × Molar volume at STP = 953.47 moles × 22.4 L/mol ≈ 21,357.7 L So, the volume occupied by the hydrogen at STP is approximately 21,357.7 L.

Step 3: Calculate the energy produced during combustion

The combustion reaction of hydrogen with oxygen to produce liquid water can be represented as: 2 H₂(g) + O₂(g) → 2 H₂O(l) The standard enthalpy of formation for liquid water is -285.83 kJ/mol. This means that each mole of hydrogen gas will produce -285.83 kJ/mol / 2 = -142.915 kJ/mol of energy when combusted. Now, calculate the energy produced: Energy produced = Number of moles of hydrogen × Energy produced per mole = 953.47 moles × (-142.915 kJ/mol) = -136,182.3 kJ The energy produced by combusting this quantity of hydrogen is approximately -136,182.3 kJ. The negative sign indicates that the reaction is exothermic (releases heat).

Key Concepts

Molar Mass Calculation

Understanding molar mass is essential in calculations involving chemical reactions, as it allows you to convert between mass and moles. Molar mass is the mass of one mole of a substance, typically expressed in grams per mole (g/mol). To calculate the molar mass of a compound, sum up the molar masses of all the individual elements present in that compound.
For example, in the case of FeTiH:

• The molar mass of Fe (iron) is 55.85 g/mol.
• The molar mass of Ti (titanium) is 47.87 g/mol.
• The molar mass of H (hydrogen) is 1.01 g/mol.

By adding these up, the molar mass of FeTiH is calculated as: \[\text{Molar mass of FeTiH} = 55.85\, \text{g/mol} + 47.87\, \text{g/mol} + 1.01\, \text{g/mol} = 104.73\, \text{g/mol}\]Next, to find how much hydrogen a 100 kg FeTi alloy can hold, determine the proportion of hydrogen in the compound: \[\text{Proportion of hydrogen} = \frac{1.01\, \text{g/mol}}{104.73\, \text{g/mol}} \approx 0.00963\]Finally, multiply this proportion by the mass of the alloy: \[\text{Mass of hydrogen} = 100,000\, \text{g} \times 0.00963 \approx 963\, \text{g}\]This calculation shows that around 963 grams of hydrogen can be stored in 100 kg of the FeTi alloy.

Gas Volume at STP

When dealing with gases, understanding their behavior under standard conditions is often necessary. Standard Temperature and Pressure (STP) is an agreed-upon set of conditions where the temperature is 0°C and the pressure is 1 atm. Under these conditions, one mole of an ideal gas occupies a volume of 22.4 liters. This is a constant that simplifies calculations involving gases at STP.
To find the volume that a specific mass of gas occupies at STP, first convert the mass to moles using the gas's molar mass:

• For hydrogen, which has a molar mass of 1.01 g/mol, 963 grams would contain \( \frac{963\, \text{g}}{1.01\, \text{g/mol}} \approx 953.47 \) moles.
• The volume at STP can then be calculated using the relation \( \text{Volume} = \text{Number of moles} \times \text{Molar volume at STP}\).

Hence:\[ \text{Volume of hydrogen at STP} = 953.47\, \text{moles} \times 22.4\, \text{L/mole} \approx 21,357.7\, \text{L} \]Therefore, 963 grams of hydrogen would occupy approximately 21,357.7 liters at STP. This knowledge is particularly helpful in industrial and laboratory settings where gas volume needs to be precisely managed.

Combustion Reaction Energy

Combustion reactions release or absorb energy in the form of heat. In the case of hydrogen, combustion with oxygen to form water is highly exothermic, meaning it releases a significant amount of energy.
The balanced combustion reaction for hydrogen is:

• 2 \(\text{H}_2(g) + \text{O}_2(g) \rightarrow 2 \text{H}_2\text{O}(l)\)
• Here, the standard enthalpy change for forming water (\(\Delta H_f^0\)) is -285.83 kJ/mol.

Because two moles of hydrogen gas produce two moles of water, the energy released per mole of hydrogen combusted is half the standard enthalpy of formation: \(-\frac{285.83\, \text{kJ/mol}}{2} = -142.915\, \text{kJ/mol}\)For 953.47 moles of hydrogen, the total energy released upon combustion is:\[\text{Energy produced} = 953.47\, \text{moles} \times (-142.915\, \text{kJ/mol}) \approx -136,182.3\, \text{kJ}\]The negative sign indicates it's exothermic, releasing approximately 136,182.3 kJ of energy. Such calculations are essential in fields like energy production, where understanding the energy potential of fuels is crucial.