Question

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

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

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: \[Q = (3,785.41\, \text{g})\times(4.184\, \text{J/g}^{\circ}\text{C})\times(10.0^{\circ}\text{C})\]

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 = \( \frac {Q} {0.8} \)

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.

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.

Key concepts

Energy Conversion Efficiency

Energy conversion efficiency is a fundamental concept in physics and chemistry. It defines how effective a process is at converting input energy into useful output energy. In any energy conversion process, some energy is lost, typically due to heat, friction, or other forms of resistance. For example, if a combustion process is 80% efficient, it means that only 80% of the energy from the fuel is turned into useful work, while the remaining 20% is lost.

In this problem, the combustion reaction of acetylene operates at 80% efficiency. This means that to find out the actual energy needed for the process, we must divide the calculated energy required to heat the water by 0.8 (the efficiency factor).

• The formula used is: \[ \text{Required Energy (E)} = \frac{Q}{\text{Efficiency}} \]
• The efficiency in decimals is 0.8 for 80% efficiency.

This adjustment is crucial as it helps one understand that more fuel might be necessary due to energy losses, thus impacting how much acetylene needs to be burned.

Molar Mass Calculation

Molar mass calculation is essential in stoichiometry. It allows us to convert between moles of a substance and grams, which is practical for lab work and chemical reactions. For acetylene (\( \text{C}_2\text{H}_2 \)), finding the molar mass involves summing the atomic masses of its constituent atoms.

Acetylene consists of two carbon atoms, each with an atomic mass of approximately 12.01 g/mol, and two hydrogen atoms, each with an atomic mass of approximately 1.01 g/mol. Therefore:

• Carbon: \( 2 \times 12.01 = 24.02 \text{ g/mol} \)
• Hydrogen: \( 2 \times 1.01 = 2.02 \text{ g/mol} \)

Adding these together gives a total molar mass for acetylene of \( 26.04 \text{ g/mol} \).

This value is then used to convert moles of acetylene into grams, which is necessary for determining how much physical material is needed to provide the required energy in the reaction.

Specific Heat Capacity of Water

The specific heat capacity of water is a crucial concept in thermodynamics. It quantifies the amount of heat required to change the temperature of a unit mass of a substance by 1 degree Celsius. For water, this value is relatively high at 4.184 J/g°C.

This means that 4.184 Joules of energy are needed to raise the temperature of 1 gram of water by 1°C. Given its high specific heat capacity, water is effective in absorbing energy without undergoing a rapid increase in temperature. This property is why water is often used as a coolant and in thermal systems to store energy.

• In the given exercise, the formula \[ Q = mcΔT \] is used to calculate the energy required for temperature change.
• Here, "m" represents the mass of water, "c" is the specific heat capacity (4.184 J/g°C), and \(ΔT\) is the temperature change in degrees Celsius.

The calculation involves multiplying the mass of water by the specific heat capacity and the change in temperature (10°C in this case) to find the energy required.