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 \(25.0g\) of ethanol from \(25°C\) to the vapor phase at \(78°C\) is approximately \(23930 J\). b) The heat required to convert \(5.00L\) of ethanol from \(-140°C\) to the vapor phase at \(78°C\) is approximately \(5,475,919 J\).

Step-by-step solution

Heat Liquid Ethanol from \(25°C\) to \(78°C\)

We will start by heating the liquid ethanol. The formula to calculate the heat (\(q\))required is given by: \(q = mcΔT\) where \(m\) is the mass of ethanol, \(c\) is the specific heat of ethanol, and Δ\(T\) is the change in temperature. Here, \(m = 25.0g\), \(c = 2.3\frac{J}{gK}\), and Δ\(T = T_{final} - T_{initial} = 78°C - 25°C = 53°C\). Now, we can calculate the heat: \(q_{HeatLiquid} = mcΔT = (25.0g)(2.3\frac{J}{gK})(53K) = 3032.5 J\)

Vaporize Ethanol

After heating the liquid ethanol, we need to vaporize it. The formula to calculate the heat required is: \(q = nΔH\) where \(n\) is the number of moles of ethanol and Δ\(H\) is the enthalpy of vaporization. First, we need to find the number of moles (\(n\)) of ethanol. The molecular weight of ethanol is \(C_2H_5OH = 46.07\frac{g}{mol}\). Therefore, \(n = \frac{25.0g}{46.07\frac{g}{mol}} = 0.542mol\). The enthalpy of vaporization is given as Δ\(H = 38.56\frac{kJ}{mol}\) or \(38560\frac{J}{mol}\). Now, we can calculate the heat: \(q_{Vaporization} = nΔH = (0.542mol)(38560\frac{J}{mol}) = 20897.52 J\)

Calculate Total Heat

Now, we can calculate the total heat required to convert \(25.0g\) of ethanol at \(25°C\) to the vapor phase at \(78°C\): \(q_{Total} = q_{HeatLiquid} + q_{Vaporization} = 3032.5 J + 20897.52 J = 23930 J\) b) Convert \(5.00L\) of ethanol at \(-140°C\) to the vapor phase at \(78°C\): We follow the same steps as in part (a), adding the steps to heat solid ethanol and melt it.

Calculate Mass of Ethanol

We know that the density of ethanol is given as \(ρ = 0.789\frac{g}{mL}\). We can use this to calculate the mass of \(5.00L\) of ethanol: \(m = ρV = (0.789\frac{g}{mL})(5000mL) = 3945g\)

Heat Solid Ethanol from \(-140°C\) to \(-114°C\)

We will start by heating the solid ethanol. Using the formula \(q = mcΔT\), where \(m = 3945g\), \(c = 0.97\frac{J}{gK}\), and Δ\(T = T_{final} - T_{initial} = -114°C -(-140°C) = 26°C\): \(q_{HeatSolid} = mcΔT = (3945g)(0.97\frac{J}{gK})(26K) = 99396.7 J\)

Melt Solid Ethanol

After heating the solid ethanol, we need to melt it. Using the formula \(q = nΔH\), we first find the number of moles (\(n\)) of ethanol: \(n = \frac{3945g}{46.07\frac{g}{mol}} = 85.63mol\) The enthalpy of fusion is given as Δ\(H = 5.02\frac{kJ}{mol}\) or \(5020\frac{J}{mol}\). Now, we can calculate the heat: \(q_{Melting} = nΔH = (85.63mol)(5020\frac{J}{mol}) = 429961.26 J\) Next, we will heat liquid ethanol, similar to part (a), and then vaporize ethanol.

Heat Liquid Ethanol from \(-114°C\) to \(78°C\)

Using the formula \(q = mcΔT\), where \(m = 3945g\), \(c = 2.3\frac{J}{gK}\), and Δ\(T = T_{final} - T_{initial} = 78°C - (-114°C) = 192°C\): \(q_{HeatLiquid} = mcΔT = (3945g)(2.3\frac{J}{gK})(192K) = 1733916 J\)

Vaporize Ethanol

Using the formula \(q = nΔH\), where \(n = 85.63mol\) and Δ\(H = 38560\frac{J}{mol}\): \(q_{Vaporization} = nΔH = (85.63mol)(38560\frac{J}{mol}) = 3299644.8 J\)

Calculate Total Heat

Now, we can calculate the total heat required to convert \(5.00L\) of ethanol at \(-140°C\) to the vapor phase at \(78°C\): \(q_{Total} = q_{HeatSolid} + q_{Melting} + q_{HeatLiquid} + q_{Vaporization} = 99396.7 J + 429961.26 J + 1733916 J + 3299644.8 J = 5475918.76 J\) In conclusion: a) The heat required to convert \(25.0g\) of ethanol from \(25°C\) to the vapor phase at \(78°C\) is approximately \(23930 J\). b) The heat required to convert \(5.00L\) of ethanol from \(-140°C\) to the vapor phase at \(78°C\) is approximately \(5,475,919 J\).

Key concepts

Enthalpy of Fusion

The enthalpy of fusion, often simply called the heat of fusion, is a measure of the energy required to change a substance from solid to liquid at its melting point, without changing its temperature. This energy input helps overcome intermolecular forces, allowing molecules to move more freely as they transition to a liquid state. It is an intrinsic property of the substance and is usually expressed in kilojoules per mole (kJ/mol).

For ethanol, the enthalpy of fusion is given as 5.02 kJ/mol. This value indicates that 5.02 kJ of energy is needed to melt one mole of solid ethanol at its melting point, which is -114°C.

• This value is important when calculating the total energy required to melt ethanol as part of a phase change.
• The enthalpy of fusion is crucial in understanding the energy balance in phase change processes.

Understanding the concept of enthalpy of fusion is important when dealing with any process involving phase transitions like melting in thermochemistry.

Enthalpy of Vaporization

The enthalpy of vaporization is the energy needed to convert a liquid to a gas at its boiling point, without changing temperature. This energy is used to break intermolecular attractions between liquid molecules, allowing them to enter the gaseous phase. Like the enthalpy of fusion, it is also expressed in kilojoules per mole (kJ/mol).

For ethanol, the enthalpy of vaporization is 38.56 kJ/mol. This property means you need 38.56 kJ of energy to vaporize one mole of liquid ethanol at its boiling point of 78°C.

• Understanding this concept is key when calculating the heat required to vaporize ethanol in thermochemical equations.
• The enthalpy of vaporization plays a significant role in energy balance calculations involving evaporation or boiling.

Proper understanding of this concept is essential when you're performing calculations involving phase changes from liquid to gas in your thermochemistry problems.

Specific Heat Capacity

Specific heat capacity is the amount of heat required to raise the temperature of 1 gram of a substance by 1°C (or 1 K). Each substance has a unique specific heat capacity, an intrinsic property that indicates how it responds to heat input.

For ethanol, the specific heat is 0.97 J/g-K for the solid state and 2.3 J/g-K for the liquid state. This variation highlights how ethanol's phase affects its heat absorption ability.

• In the context of ethanol, knowing the specific heat is crucial for calculating how much energy is needed to heat the substance.
• Specific heat capacity is used in the energy calculations for both solid and liquid phases of ethanol during heating.

Understanding specific heat capacity aids in predicting how substances under different phases absorb and respond to energy.

Phase Changes

Phase changes involve transitions between solid, liquid, and gaseous states. These processes require either absorption or release of energy, which involves thermochemical equations to quantify the heat transfer needed.

In the context of ethanol, we look at transitions from a solid at -140°C to a gaseous state at 78°C. This path involves:

• Raising the temperature of solid ethanol to its melting point, requiring energy calculated using specific heat.
• The melting phase, where energy is added equal to the enthalpy of fusion.
• Further warming the resulting liquid ethanol to its boiling point, using the liquid's specific heat.
• Vaporizing ethanol, which requires energy equal to the enthalpy of vaporization.

Understanding these phase changes is crucial to comprehensively applying thermochemistry concepts in real-world scenarios like calculating the total required heat for ethanol phase transitions.