How many liters of oxygen gas (O_(2)) will produce Mograms of carbon dicxide (CO_(2)) at STP? C_(3)H_(8)+5O_(2)arrow 3CO_(2)+4H_(2)O

To determine how many liters of oxygen gas (O₂) are needed to produce a certain mass of carbon dioxide (CO₂) at standard temperature and pressure (STP), we need to follow these steps:<br /><br />1. **Write down the balanced chemical equation:**<br /> \[<br /> C_3H_8 + 5O_2 \rightarrow 3CO_2 + 4H_2O<br /> \]<br /><br />2. **Calculate the molar masses of the reactants and products:**<br /> - Molar mass of \(C_3H_8\) (propane): \(3 \times 12.01 + 8 \times 1.01 = 44.11 \, \text{g/mol}\)<br /> - Molar mass of \(O_2\): \(2 \times 16.00 = 32.00 \, \text{g/mol}\)<br /> - Molar mass of \(CO_2\): \(12.01 + 2 \times 16.00 = 44.01 \, \text{g/mol}\)<br /><br />3. **Determine the number of moles of CO₂ produced:**<br /> Given that we have \(M\) grams of CO₂:<br /> \[<br /> \text{Moles of } CO_2 = \frac{M \, \text{g}}{44.01 \, \text{g/mol}}<br /> \]<br /><br />4. **Use the stoichiometry of the balanced equation to find the moles of O₂ needed:**<br /> According to the balanced equation, 5 moles of \(O_2\) produce 3 moles of \(CO_2\). Therefore, the moles of \(O_2\) needed are:<br /> \[<br /> \text{Moles of } O_2 = \frac{5}{3} \times \text{Moles of } CO_2<br /> \]<br /><br />5. **Convert the moles of \(O_2\) to liters at STP:**<br /> At STP (Standard Temperature and Pressure), 1 mole of any gas occupies 22.4 liters. Therefore, the volume of \(O_2\) in liters is:<br /> \[<br /> \text{Volume of } O_2 = \text{Moles of } O_2 \times 22.4 \, \text{L/mol}<br /> \]<br /><br />Putting it all together, we get:<br />\[<br />\text{Volume of } O_2 = \frac{5}{3} \times \frac{M}{44.01} \times 22.4<br />\]<br /><br />Simplifying the expression:<br />\[<br />\text{Volume of } O_2 = \frac{5 \times 22.4 \times M}{3 \times 44.01}<br />\]<br /><br />\[<br />\text{Volume of } O_2 = \frac{112 \times M}{132.03}<br />\]<br /><br />\[<br />\text{Volume of } O_2 \approx 0.847 \times M \, \text{liters}<br />\]<br /><br />So, the volume of oxygen gas needed to produce \(M\) grams of carbon dioxide at STP is approximately \(0.847 \times M\) liters.