Ethanol (C2H5OH) has been produced since antiquity by the fermentation of fruits and vegetables. Given the following data, if 5.87 kJ of energy are transferred to 13.3 g of frozen ethanol at -130.0 °C, what is the final temperature of the sample?
Heat capacity (solid) = 0.97 J/°C • g
Heat capacity (liquid) = 2.72 J/°C • g
Heat capacity (gas) = 2.42 J/°C • g
Normal melting point = −114.0 °C
Normal boiling point = 78.4 °C
Heat of fusion = 7.61 kJ/mol
Heat of vaporization = 39.3 kJ/mol
How much heat is required to raise T of solid ethanol from -130 to -114? That will be mass ethanol x specific heat solid ethanol x (Tf-Ti) = ?
How much heat has is left?
5870 J - ? = x
How much heat is required to melt the ethanol. That is mass ethanol x heat fusion = y
x-y = z = heat remaining from the original 5870 J.
z = mass liquid ethanol x specific heat liquid ethanol x (Tf - Ti). Substitute and solve for Tf.
To find the final temperature of the frozen ethanol sample, we need to calculate the amount of energy required to heat the sample from its initial temperature to its final temperature.
First, let's calculate the energy required to raise the temperature of the frozen ethanol from -130.0 °C to its melting point, -114.0 °C. Since ethanol is in the solid state, we will use the heat capacity of the solid form.
Energy (Q1) = mass (m) × heat capacity (solid) × change in temperature (ΔT1)
Substituting the values given:
Q1 = 13.3 g × 0.97 J/°C • g × (-114.0 °C - (-130.0 °C))
Next, let's calculate the energy required for ethanol to undergo a phase change from solid to liquid at its melting point. We will use the heat of fusion.
Energy (Q2) = mass (m) × heat of fusion
Substituting the values given:
Q2 = 13.3 g × (7.61 kJ/mol / molar mass of ethanol)
Since the molar mass of ethanol is 46.07 g/mol, we can calculate the energy required for the phase change.
Then, let's calculate the energy required to raise the temperature of the liquid ethanol from its melting point to its final temperature. We will use the heat capacity of the liquid form.
Energy (Q3) = mass (m) × heat capacity (liquid) × change in temperature (ΔT2)
Substituting the values given:
Q3 = 13.3 g × 2.72 J/°C • g × (final temperature - (-114.0 °C))
Lastly, sum up all the energies to get the total energy transferred to the sample:
Total energy (Q total) = Q1 + Q2 + Q3
Now, we have the total energy transferred to the sample. We can rearrange the equation to solve for the final temperature:
final temperature = (Q total / (mass (m) × heat capacity (liquid))) + (-114.0 °C)
Substitute the values and calculate to find the final temperature.