To determine the mass of HF that must react to produce 690 kJ of energy, let's follow a detailed, step-by-step approach.
### Step 1: Understand the Thermochemical Equation
Given thermochemical equation is:
[tex]\[
SiO_2 (s) + 4 HF (g) \rightarrow SiF_4 (g) + 2 H_2O (l) \quad \Delta H ^{\circ}_{\text{rxn}} = -184 \text{ kJ}
\][/tex]
This equation tells us that when 4 moles of HF react with SiO_2, the reaction releases 184 kJ of energy. Here, the sign of [tex]\(\Delta H\)[/tex] is negative, indicating an exothermic reaction (energy is released).
### Step 2: Calculate the Number of Moles of HF Required to Produce 690 kJ
We know from the thermochemical equation that 4 moles of HF release 184 kJ of energy. To find out how many moles of HF are needed to produce 690 kJ, we set up a proportion:
[tex]\[
\frac{\text{Energy produced by 4 moles of HF}}{\text{Energy required}} = \frac{184 \text{ kJ}}{690 \text{ kJ}}
\][/tex]
This can be rewritten to find the moles of HF:
[tex]\[
\text{Moles of HF} = \frac{690 \text{ kJ}}{\frac{184 \text{ kJ}}{4 \text{ moles}}}
\][/tex]
Simplify the fraction:
[tex]\[
\text{Moles of HF} = \frac{690 \text{ kJ}}{46 \text{ kJ/mole}} = 15 \text{ moles}
\][/tex]
### Step 3: Calculate the Molar Mass of HF
The molar mass of HF can be calculated by adding the atomic masses of hydrogen (H) and fluorine (F):
[tex]\[
\text{Molar mass of HF} = \text{Mass of H} + \text{Mass of F} = 1 \text{ g/mol} + 19 \text{ g/mol} = 20 \text{ g/mol}
\][/tex]
### Step 4: Calculate the Mass of HF Required
Now that we know the number of moles of HF required and the molar mass of HF, we can calculate the mass of HF needed:
[tex]\[
\text{Mass of HF} = \text{Number of moles of HF} \times \text{Molar mass of HF}
\][/tex]
[tex]\[
\text{Mass of HF} = 15 \text{ moles} \times 20 \text{ g/mol} = 300 \text{ g}
\][/tex]
### Conclusion
To produce 690 kJ of energy, [tex]\( 300 \)[/tex] grams of HF must react.
