Example 10.1-2

The combustion of hydrogen with oxygen to produce gaseous water is extremely vigorous, producing one of the hottest flames known. Because so much energy is released for a given mass of hydrogen or oxygen, this reaction was used to fuel the NASA (National Aeronautics and Space Administration) space shuttles, which have recently been retired from service. NASA engineers calculated the exact amount of each reactant needed for the flight to make sure that the shuttles did not carry excess fuel into orbit. Calculate how many tons of hydrogen a space shuttle needed to carry for each 1.00 tn of oxygen (1 tn = 2000 lb).

The US space shuttle Discovery during liftoff. The large cylinder in the middle contains the oxygen and hydrogen that fueled the shuttle’s main engine. STS-39 Discovery, Orbiter Vehicle (OV) 103, lifts off from a Kennedy Space Center (KSC) Launch Complex (LC) Pad 39A at 7:33:14 am (Eastern Daylight Time (EDT) May 7, 1991). OV-103 atop the external tank (ET) and flanked by its two solid rocket boosters (SRBs) is partially obscured by the retracted rotating service structure (RSS) in the foreground as it leaves the mobile launcher platform and before it clears the launch tower fixed service structure (FSS). An exhaust cloud created by the SRB and space shuttle main engine (SSME) firings fills the launch pad area. This image file is in the public domain in the United States because it was solely created by NASA.

Given: reactants, products, and mass of one reactant

Asked for: mass of other reactant

Strategy:

A Write the balanced chemical equation for the reaction.

B Convert mass of oxygen to moles. From the mole ratio in the balanced chemical equation, determine the number of moles of hydrogen required. Then convert the moles of hydrogen to the equivalent mass in tons.

Solution:

We use the same general strategy for solving stoichiometric calculations as in the preceding example. Because the amount of oxygen is given in tons rather than grams, however, we also need to convert tons to units of mass in grams. Another conversion is needed at the end to report the final answer in tons.

A We first use the information given to write a balanced chemical equation. Because we know the identity of both the reactants and the product, we can write the reaction as follows:

This equation is not balanced because there are two oxygen atoms on the left side and only one on the right. Assigning a coefficient of 2 to both H₂O and H₂ gives the balanced chemical equation:

Thus 2 mol of H₂ react with 1 mol of O₂ to produce 2 mol of H₂O.

1. B To convert tons of oxygen to units of mass in grams, we multiply by the appropriate conversion factors:

   $${\text{mass of O}}_{\text{2}}=\text{1}\text{.00}\cancel{\text{tn}}\times \frac{\text{2000}\cancel{\text{lb}}}{\cancel{\text{tn}}}\times \frac{453.6\text{g}}{\cancel{\text{lb}}}=9.07\times {10}^5{\text{g O}}_{\text{2}}$$

   Using the molar mass of O₂ (32.00 g/mol, to four significant figures), we can calculate the number of moles of O₂ contained in this mass of O₂:

   $${\text{mol O}}_{\text{2}}=9.07\times {10}^5\cancel{{\text{g O}}_{\text{2}}}\times \frac{{\text{1 mol O}}_{\text{2}}}{\text{32}\text{.00}\cancel{{\text{g O}}_{\text{2}}}}=\text{2}\text{.83}\times {\text{10}}^{\text{4}}{\text{mol O}}_{\text{2}}$$

2. Now use the coefficients in the balanced chemical equation to obtain the number of moles of H₂ needed to react with this number of moles of O₂:

   $$\begin{array}{l}{\text{mol H}}_{\text{2}}={\text{mol O}}_{\text{2}}\times \frac{{\text{2 mol H}}_{\text{2}}}{{\text{1 mol O}}_{\text{2}}}\\ =\text{2}\text{.83}\times {\text{10}}^{\text{4}}\cancel{{\text{mol O}}_{\text{2}}}\times \frac{{\text{2 mol H}}_{\text{2}}}{\text{1}\cancel{{\text{mol O}}_{\text{2}}}}=5.66\times {10}^4{\text{mol H}}_{\text{2}}\end{array}$$

3. The molar mass of H₂ (2.016 g/mol) allows us to calculate the corresponding mass of H₂:

   $${\text{mass of H}}_{\text{2}}=\text{5}\text{.66}\times {\text{10}}^{\text{4}}\cancel{{\text{mol H}}_{\text{2}}}\times \frac{\text{2}{\text{.016 g H}}_{\text{2}}}{\cancel{{\text{mol H}}_{\text{2}}}}=1.14\times {10}^5{\text{g H}}_{\text{2}}$$

   Finally, convert the mass of H₂ to the desired units (tons) by using the appropriate conversion factors:

   $${\text{tons H}}_{\text{2}}=\text{1}\text{.14}\times {\text{10}}^{\text{5}}\cancel{\text{g}}{\text{H}}_{\text{2}}\times \frac{\text{1}\cancel{\text{lb}}}{\text{453}\text{.6}\cancel{\text{g}}}\times \frac{\text{1 tn}}{\text{2000}\cancel{\text{lb}}}=0.126{\text{tn H}}_{\text{2}}$$

   The space shuttle had to be designed to carry 0.126 tn of H₂ for each 1.00 tn of O₂. Even though 2 mol of H₂ are needed to react with each mole of O₂, the molar mass of H₂ is so much smaller than that of O₂ that only a relatively small mass of H₂ is needed compared to the mass of O₂.

Exercise

Alchemists produced elemental mercury by roasting the mercury-containing ore cinnabar (HgS) in air:

The volatility and toxicity of mercury make this a hazardous procedure, which likely shortened the life span of many alchemists. Given 100 g of cinnabar, how much elemental mercury can be produced from this reaction?

Answer: 86.2 g