Final answer:
The heat energy required to convert 51.4 g of solid ethanol at -114.5 °C to gaseous ethanol at 135.7 °C is 148.75 kJ.
Step-by-step explanation:
To convert the solid ethanol at -114.5 °C to gaseous ethanol at 135.7 °C, we need to consider the different phase changes it undergoes. First, we need to calculate the heat energy required to raise the temperature of the solid ethanol from -114.5 °C to its melting point of -114.5 °C. We can use the specific heat capacity of the solid ethanol to calculate this heat energy:
Q1 = mass * specific heat capacity * temperature change
Q1 = 51.4 g * 2.45 J/g °C * (0 °C - (-114.5 °C))
Q1 = 51.4 g * 2.45 J/g °C * 114.5 °C
Q1 = 14,159.15 J
Next, we need to calculate the heat energy required to melt the solid ethanol at its melting point. We can use the molar heat of fusion to calculate this:
Q2 = moles * molar heat of fusion
Since we know the molar mass of ethanol is 46 g/mol, we can calculate the number of moles:
moles = mass / molar mass
moles = 51.4 g / 46 g/mol
moles = 1.12 mol
Q2 = 1.12 mol * 4.60 kJ/mol
Q2 = 5.15 kJ
Finally, we need to calculate the heat energy required to vaporize the liquid ethanol at its boiling point. We can use the molar heat of vaporization to calculate this:
Q3 = moles * molar heat of vaporization
Q3 = 1.12 mol * 38.56 kJ/mol
Q3 = 43.15 kJ
The total heat energy required is the sum of Q1, Q2, and Q3:
Total heat energy = Q1 + Q2 + Q3
Total heat energy = 14,159.15 J + 5.15 kJ + 43.15 kJ
Total heat energy = 48,454.15 J + 48.30 kJ
Total heat energy = 148.75 kJ