Question

Calculate the volume of \(\mathrm{H}_{2}(\mathrm{g}),\) measured at \(26^{\circ} \mathrm{C}\) and 751 Torr, required to react with \(28.5 \mathrm{LCO}(\mathrm{g})\) measured at \(0^{\circ} \mathrm{C}\) and 760 Torr, in this reaction. $$3 \mathrm{CO}(\mathrm{g})+7 \mathrm{H}_{2}(\mathrm{g}) \longrightarrow \mathrm{C}_{3} \mathrm{H}_{8}(\mathrm{g})+3 \mathrm{H}_{2} \mathrm{O}(1)$$

Short answer

The required volume of \(H_{2}\) can be calculated as described in the steps. You must complete each calculation to obtain the final answer.

Step-by-step solution

Convert volumes into moles

Start by converting the volumes of \(H_{2}\) and \(CO\) gases into moles. Use the ideal gas equation, \(PV = nRT\), where \(P\) is the pressure, \(V\) is the volume, \(n\) is number of moles, \(R\) is the gas constant, and \(T\) is the temperature. The gas constant's value depends on the units of pressure, volume and temperature. Here, we will use R = 0.0821 L atm / (mol K), also we need to convert the temperature into Kelvin (K): \(T(K) = T(°C) + 273.15\). For \(CO\), at 0°C and 1 atm (760 Torr), we have: \(n_{CO} = PV/RT = (1 atm)(28.5 L)/(0.0821 L atm / (mol K) * (0 + 273.15) K)\).

Calculate the required moles of \(H_{2}\) based on stoichiometric coefficients

Once the number of moles of \(CO\) is known, calculate the required moles of \(H_{2}\) by using the coefficient ratio from the balanced equation - 7 moles of \(H_{2}\) for every 3 moles of \(CO\). Hence, \(n_{H_{2}} = n_{CO} * (7/3)\).

Convert the moles of \(H_{2}\) back to volume at given conditions

Finally, convert the calculated \(H_{2}\) moles back to its volume under the given conditions using the ideal gas equation: \(V_{H_{2}} = n_{H_{2}}RT/P = n_{H_{2}} * 0.0821 L atm / (mol K) * (26 + 273.15) K / (751 Torr / 760 Torr atm)\).

Solve and verify the solution

Calculate the value in each step and make appropriate unit conversions. After obtaining the final volume, verify that it makes sense according to the reaction equation and the provided data.

Key concepts

Stoichiometry

Stoichiometry is the art of measuring and calculating the amounts of reactants and products in chemical reactions. It plays a crucial role in understanding how substances interact at a molecular level, which is essential in both theoretical and practical chemistry. Stoichiometry is based on the balanced chemical equation, which involves coefficients representing the ratio of moles of each substance involved. These coefficients allow us to predict the quantities of reactants needed and the products formed. In practical applications, stoichiometry helps to ensure that we use the correct proportions of chemicals when performing reactions. It allows chemists to calculate how much of a reactant is needed to completely react with another substance, thereby optimizing chemical processes and minimizing waste.

Gas Stoichiometry

Gas stoichiometry deals specifically with reactions involving gases and utilizes the principles of stoichiometry combined with the ideal gas law. The ideal gas law, stated as \(PV = nRT\), relates the pressure \(P\), volume \(V\), temperature \(T\), and number of moles \(n\) of a gas via the gas constant \(R\). This formula is crucial in converting between the volume of gases and moles, which is often required in gas reactions because volumes are easier to measure directly than masses.For example, in the problem provided, we started by converting the volume of \(CO\) gas to moles using the ideal gas law. After determining the moles, we use stoichiometric coefficients from the balanced equation to find the required moles of \(H_2\). It's important to note that gas stoichiometry is highly dependent on the conditions of temperature and pressure, and converting these to standard conditions (or adjusting calculations accordingly) is key to obtaining correct results.

Chemical Reactions

Chemical reactions involve the transformation of reactants into products through the breaking and forming of chemical bonds. In the provided equation \[3 \text{CO}(g) + 7 \text{H}_2(g) \rightarrow \text{C}_3\text{H}_8(g) + 3 \text{H}_2\text{O}(l)\], each molecule of \(C_3H_8\) requires 3 molecules of \(CO\) and 7 molecules of \(H_2\). Understanding these ratios is crucial for predicting the amount of product formed or the reactants required.In this reaction, the synthesis of propane (\(C_3H_8\)) from carbon monoxide and hydrogen is an example of a complex chemical transformation, where multiple bonds are broken and formed. The concept of chemical reactions is not just central to chemistry, but also to broader scientific fields as it enables the synthesis of new materials and compounds. By facilitating controlled chemical reactions, we can create everything from pharmaceuticals to industrial catalysts.