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Chapter 7: Problem 90

The complete combustion of acetylene, \(\mathrm{C}_{2} \mathrm{H}_{2}(g),\) produces 1300. kJ of energy per mole of acetylene consumed. How many grams of acetylene must be burned to produce enough heat to raise the temperature of 1.00 gal water by \(10.0^{\circ} \mathrm{C}\) if the process is \(80.0 \%\) efficient? Assume the density of water is \(1.00 \mathrm{g} / \mathrm{cm}^{3}\).

Short Answer

Expert verified
To find the mass of acetylene needed to raise the temperature of 1.00-gal water by \(10.0^{\circ}\mathrm{C}\) with an efficiency of 80%, we first calculate the energy required (Q) to heat the water using the formula \(Q = mc\Delta T\), where m is the mass of water, c is the specific heat capacity of water (4.184 J/g°C), and ΔT is the change in temperature. After accounting for the 80% efficiency and given energy per mole of acetylene, we can calculate the moles of acetylene needed and then multiply by the molar mass (26.04 g/mol) to find the required mass of acetylene.

Step by step solution

01

Calculate the energy required to increase the temperature of water

First, we need to calculate the amount of energy (Q) required to increase the temperature of 1.00-gal water by 10 degrees Celsius by using the formula: \[Q = mcΔT\] where m is the mass of water, c is the specific heat capacity of water (4.184 J/g°C), and ΔT is the change in temperature. Since we are given the volume of water (1.00 gal), we convert it to grams using the density of water (1.00 g/cm³). 1.00 gal = 3,785.41 cm³ (using the conversion factor 1 gal = 3,785.41 cm³) Mass of water (m) = 3,785.41 cm³ × 1.00 g/cm³ = 3,785.41 g Now we can calculate Q: \[Q = (3,785.41\, \text{g})\times(4.184\, \text{J/g}^{\circ}\text{C})\times(10.0^{\circ}\text{C})\]
02

Calculate the required energy considering the efficiency

Since the process is 80% efficient, we need to account for this in our calculation. We adjust the energy required (Q) according to the efficiency: Required energy (E) = \( \frac{Q}{efficiency} \) Efficiency = 80% = 0.8 E = \( \frac {Q} {0.8} \)
03

Calculate the moles of acetylene needed

We are given that the combustion of acetylene produces 1300 kJ of energy per mole. We need to find the moles of acetylene required for the energy calculated in step 2. Moles of acetylene (n) = \( \frac{E}{energy \, per\, mole\, of\, acetylene} \) Energy per mole of acetylene = 1300 kJ/mol First, we convert the energy found in step 2 to kJ. After this, we can calculate the moles of acetylene needed.
04

Calculate the mass of acetylene needed

To find the mass of acetylene needed, we will use the formula: Mass = moles × molar mass The molar mass of acetylene (C₂H₂) is = 2 × (12.01 g/mol) + 2 × (1.01 g/mol) = 26.04 g/mol Now, we can calculate the mass of acetylene required for the process by multiplying the moles found in step 3 with the molar mass.

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Most popular questions from this chapter

For the reaction \(\mathrm{HgO}(s) \rightarrow \mathrm{Hg}(l)+\frac{1}{2} \mathrm{O}_{2}(g), \Delta H=+90.7 \mathrm{kJ}\). a. What quantity of heat is required to produce 1 mole of mercury by this reaction? b. What quantity of heat is required to produce 1 mole of oxygen gas by this reaction? c. What quantity of heat would be released in the following reaction as written? $$2 \mathrm{Hg}(l)+\mathrm{O}_{2}(g) \longrightarrow 2 \mathrm{HgO}(s)$$

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