Question

Ethanol \(\left(\mathrm{C}_{2} \mathrm{H}_{5} \mathrm{OH}\right)\) melts at \(-114^{\circ} \mathrm{C}\) and boils at \(78^{\circ} \mathrm{C}\). Its density is \(0.789 \mathrm{~g} / \mathrm{mL}\). The enthalpy of fusion of ethanol is \(5.02 \mathrm{~kJ} / \mathrm{mol}\), and its enthalpy of vaporization is \(38.56 \mathrm{~kJ} / \mathrm{mol}\). The specific heats of solid and liquid ethanol are \(0.97 \mathrm{~J} / \mathrm{g}-\mathrm{K}\) and \(2.3 \mathrm{~J} / \mathrm{g}-\mathrm{K}\), respectively. (a) How much heat is required to convert \(25.0 \mathrm{~g}\) of ethanol at \(25^{\circ} \mathrm{C}\) to the vapor phase at \(78^{\circ} \mathrm{C} ?\) (b) How much heat is required to convert \(5.00 \mathrm{~L}\) of ethanol at \(-140^{\circ} \mathrm{C}\) to the vapor phase at \(78^{\circ} \mathrm{C}\) ?

Short answer

(a) The heat required to convert \(25.0\, \mathrm{g}\) of liquid ethanol at \(25^{\circ} \mathrm{C}\) to vapor phase at \(78^{\circ} \mathrm{C}\) is \(23912.5\, \mathrm{J}\). (b) The heat required to convert \(5.00\, \mathrm{L}\) of solid ethanol at \(-140^{\circ} \mathrm{C}\) to vapor phase at \(78^{\circ} \mathrm{C}\) is \(4057449.2\, \mathrm{J}\).

Step-by-step solution

Compute the heat to raise the temperature of liquid ethanol to its boiling point

Since the initial temperature of ethanol is 25°C, we can calculate the heat required to raise the temperature of liquid ethanol to its boiling point (78°C) using the formula \(q = mcΔT\), where \(m\) is the mass, \(c\) is the specific heat, and \(ΔT\) is the change in temperature. \(q = (25.0 \,\mathrm{g}) \times (2.3 \, \mathrm{J/g \cdot K}) \times (78 - 25) \, \mathrm{K}\) \(q = 3022.5 \, \mathrm{J}\)

Compute the heat to vaporize liquid ethanol

Now, we calculate the heat required to vaporize the liquid ethanol using the formula \(q = nΔH_v\), where \(n\) is the number of moles and \(ΔH_v\) is the enthalpy of vaporization. First, calculate the number of moles: \(n = \dfrac{25.0\, \mathrm{g}}{46.07\, \mathrm{g/mol}}\) (Divide by the molar mass of ethanol, which is 46.07 g/mol.) \(n = 0.542\, \mathrm{mol}\) Next, compute the heat for vaporization: \(q = (0.542\, \mathrm{mol}) \times (38.56\, \mathrm{kJ/mol})\) \(q = 20.89\, \mathrm{kJ}\) Convert this value to joules: \(q = 20.89\, \mathrm{kJ} \times \dfrac{1000\, \mathrm{J}}{1\, \mathrm{kJ}} = 20890\, \mathrm{J}\)

Calculate the total heat required

Finally, we add the heat required to raise the temperature (Step 1) and the heat required to vaporize the ethanol (Step 2): Total heat = \(3022.5 \, \mathrm{J} + 20890 \, \mathrm{J} = 23912.5 \, \mathrm{J}\) (b) Calculate the heat required to convert 5.00 L of solid ethanol at -140°C to vapor phase at 78°C.

Compute the heat to raise the temperature of solid ethanol to its melting point

Since the initial temperature of ethanol is -140°C, we can calculate the heat required to raise the temperature of solid ethanol to its melting point (-114°C) using the formula \(q = mcΔT\): \(q = m \times c \times ΔT\) Convert volume to mass using the given density: \(m = (5.00 \,\mathrm{L}) \times (0.789 \, \mathrm{g/mL}) \times \dfrac{1000\, \mathrm{mL}}{1\, \mathrm{L}} = 3945\, \mathrm{g}\) \(q = (3945 \, \mathrm{g}) \times (0.97 \, \mathrm{J/g \cdot K}) \times (-114 - (-140)) \, \mathrm{K}\) \(q = 97258.5 \, \mathrm{J}\)

Compute the heat to melt solid ethanol

Now, we calculate the heat required to melt the solid ethanol using the formula \(q = nΔH_f\): First, calculate the number of moles: \(n = \dfrac{3945\, \mathrm{g}}{46.07\, \mathrm{g/mol}}\) \(n = 85.6\, \mathrm{mol}\) Next, compute the heat for melting: \(q = (85.6\, \mathrm{mol}) \times (5.02\, \mathrm{kJ/mol})\) \(q = 429.512\, \mathrm{kJ}\) Convert this value to joules: \(q = 429.512\, \mathrm{kJ} \times \dfrac{1000\, \mathrm{J}}{1\, \mathrm{kJ}} = 429512\, \mathrm{J}\)

Compute the heat to raise the temperature of liquid ethanol to its boiling point

Calculate the heat required to raise the temperature of liquid ethanol to its boiling point (78°C): (Similar to part (a), step 1) \(q = (3945 \,\mathrm{g}) \times (2.3 \, \mathrm{J/g \cdot K}) \times (78 - (-114)) \, \mathrm{K}\) \(q = 336902.7 \, \mathrm{J}\)

Compute the heat to vaporize liquid ethanol

Calculate the heat required to vaporize liquid ethanol: (Similar to part (a), step 2) \(q = (85.6\mathrm{mol}) \times (38.56\, \mathrm{kJ/mol})\) \(q = 3299.776\, \mathrm{kJ}\) Convert this value to joules: \(q = 3299.776\, \mathrm{kJ} \times \dfrac{1000\, \mathrm{J}}{1\, \mathrm{kJ}} = 3299776\, \mathrm{J}\)

Calculate the total heat required

Finally, add the heat for each step: Total heat = \(97258.5\, \mathrm{J} + 429512\, \mathrm{J} + 336902.7\, \mathrm{J} + 3299776\, \mathrm{J} = 4057449.2\, \mathrm{J}\)

Key concepts

Specific Heat

Specific heat is a property that describes how much heat is needed to raise the temperature of a unit mass of a substance by one degree. It's an important factor in thermodynamics that helps us understand how substances absorb and transfer heat. For ethanol, the specific heat values are given for both solid and liquid states:

• Solid ethanol: 0.97 J/g·K
• Liquid ethanol: 2.3 J/g·K

These values indicate that it takes more energy to raise the temperature of liquid ethanol as compared to its solid form. The formula for specific heat is \( q = mcΔT \),where:

• \( q \) is the heat added,
• \( m \) is the mass of the substance,
• \( c \) is the specific heat capacity,
• \( ΔT \) is the change in temperature.

This equation is used to calculate the heat involved during the heating or cooling of a substance without any phase change, as seen in the solution steps for calculating heat with ethanol.

Enthalpy of Vaporization

Enthalpy of vaporization refers to the amount of energy required to change a given quantity of a substance from the liquid to the gas phase at constant pressure. This concept is significant in understanding energy transformations during phase changes. In the exercise, the enthalpy of vaporization for ethanol is given as 38.56 kJ/mol. This means that to vaporize one mole of ethanol, 38.56 kJ of energy is needed.The formula used to determine the heat required for vaporization:\[ q = nΔH_v \]where:

• \( q \) is the heat absorbed,
• \( n \) is the number of moles of substance,
• \( ΔH_v \) is the enthalpy of vaporization.

This formula is essential for calculating the energy needed to completely vaporize a liquid, as demonstrated in various steps of the solution when converting ethanol to vapor.

Phase Transitions

Phase transitions occur when a substance changes from one state of matter to another, such as from solid to liquid or liquid to gas. These transitions involve heat energy, but during the transition, the temperature of the substance remains constant. Two main types of phase transitions are highlighted in the problem:

• Melting (solid to liquid)
• Vaporization (liquid to gas)

The enthalpy of fusion and vaporization are key factors during these transitions. For ethanol:

• The enthalpy of fusion is 5.02 kJ/mol, required to melt solid ethanol.
• The enthalpy of vaporization is 38.56 kJ/mol for converting liquid ethanol to vapor.

Understanding these transitions is crucial for solving problems involving heat changes across phases, as the solution steps illustrate the need to factor in these transitions while calculating total heat.

Moles and Molar Mass

Understanding moles and molar mass is fundamental to performing calculations in thermodynamics. The concept of moles allows us to quantify the amount of a substance, while molar mass provides the connection between mass and the amount in moles. In the exercise using ethanol, the molar mass is provided as 46.07 g/mol. This value lets us convert grams into moles, facilitating the use of equations that require the number of moles, such as those for enthalpy calculations.The formula to find the number of moles is:\[ n = \frac{m}{M} \]where:

• \( n \) is the number of moles,
• \( m \) is the mass of the substance,
• \( M \) is the molar mass of the substance.

This conversion step is critical in any thermodynamic calculation involving enthalpy changes, as it provides the needed amount of substance for accurate energy assessment.