To solve the problem of finding how many grams of lithium hydroxide (LiOH) are needed to react with 509 liters of carbon dioxide (CO₂) at standard temperature and pressure (STP), follow these detailed steps:
1. Identify the molar mass of lithium hydroxide:
The molar mass of lithium hydroxide (LiOH) is [tex]\( \boxed{23.95} \text{ g/mol} \)[/tex].
2. Determine the volume of CO₂ given:
The volume of CO₂ is 509 liters.
3. Use the molar volume of a gas at STP:
The molar volume of a gas at STP (Standard Temperature and Pressure) is [tex]\( 22.4 \text{ liters/mol} \)[/tex].
4. Calculate the number of moles of CO₂:
To find the number of moles of CO₂, divide the given volume of CO₂ by the molar volume:
[tex]\[
\text{moles of CO₂} = \frac{509 \text{ liters}}{22.4 \text{ liters/mol}} = \boxed{22.72321428571429} \text{ mol}
\][/tex]
5. Use the stoichiometry of the reaction to find moles of LiOH:
According to the given chemical equation, 2 moles of LiOH react with 1 mole of CO₂.
Therefore, the moles of LiOH needed will be:
[tex]\[
\text{moles of LiOH} = 2 \times \text{moles of CO₂} = 2 \times 22.72321428571429 = \boxed{45.44642857142858} \text{ mol}
\][/tex]
6. Calculate the mass of LiOH needed:
Finally, to find the mass of LiOH needed, multiply the number of moles of LiOH by its molar mass:
[tex]\[
\text{mass of LiOH} = \text{moles of LiOH} \times \text{molar mass of LiOH} = 45.44642857142858 \text{ mol} \times 23.95 \text{ g/mol} = \boxed{1088.4419642857144} \text{ grams}
\][/tex]
Therefore, the mass of lithium hydroxide needed to react with 509 liters of carbon dioxide at STP is:
[tex]\[
\boxed{1088.4419642857144} \text{ grams}
\][/tex]
