Sure! Let's solve this problem step-by-step:
1. Understand the balanced chemical equation:
The balanced chemical equation for the decomposition of potassium nitrate ([tex]\( KNO_3 \)[/tex]) is:
[tex]\[
4 \, \text{KNO}_3 (s) \rightarrow 2 \, \text{K}_2\text{O} (s) + 2 \, \text{N}_2 (g) + 5 \, \text{O}_2 (g)
\][/tex]
This tells us that 4 moles of [tex]\( KNO_3 \)[/tex] produce 5 moles of [tex]\( O_2 \)[/tex].
2. Given information:
- We need to produce 58 kg of oxygen ([tex]\( O_2 \)[/tex]).
- The molar mass of oxygen ([tex]\( O_2 \)[/tex]) is 32 g/mol.
3. Conversion of mass of oxygen to grams:
Since the given mass is in kilograms, we need to convert it to grams:
[tex]\[
58 \, \text{kg} = 58 \times 1000 = 58000 \, \text{grams}
\][/tex]
4. Calculate moles of [tex]\( O_2 \)[/tex]:
To find out how many moles of [tex]\( O_2 \)[/tex] are produced, we use the molar mass of [tex]\( O_2 \)[/tex]:
[tex]\[
\text{Moles of } O_2 = \frac{\text{mass of } O_2}{\text{molar mass of } O_2}
\][/tex]
[tex]\[
\text{Moles of } O_2 = \frac{58000 \, \text{grams}}{32 \, \text{g/mol}} = 1812.5 \, \text{moles}
\][/tex]
5. Determine the moles of [tex]\( KNO_3 \)[/tex] needed:
According to the balanced chemical equation, 4 moles of [tex]\( KNO_3 \)[/tex] produce 5 moles of [tex]\( O_2 \)[/tex]. We can set up a ratio to find the moles of [tex]\( KNO_3 \)[/tex] required:
[tex]\[
\frac{4 \, \text{moles of } KNO_3}{5 \, \text{moles of } O_2} = \frac{x \, \text{moles of } KNO_3}{1812.5 \, \text{moles of } O_2}
\][/tex]
Solving for [tex]\( x \)[/tex]:
[tex]\[
x = 1812.5 \times \left(\frac{4}{5}\right) = 1450.0 \, \text{moles}
\][/tex]
6. Conclusion:
The number of moles of [tex]\( KNO_3 \)[/tex] that must be heated to produce 58 kg of oxygen is:
[tex]\[
1450.0 \, \text{moles}
\][/tex]
