Question

Ethanol \(\left(\mathrm{C}_{2} \mathrm{H}_{5} \mathrm{OH}\right)\) melts at \(-114^{\circ} \mathrm{C}\) and boils at \(78^{\circ} \mathrm{C}\) . 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 and \(2.3 \mathrm{J} / \mathrm{g}-\mathrm{K},\) respectively. (a) How much heat is required to convert 42.0 \(\mathrm{g}\) of ethanol at \(35^{\circ} \mathrm{C}\) to the vapor phase at \(78^{\circ} \mathrm{C} ?(\mathbf{b})\) How much heat is required to convert the same amount of ethanol at \(-155^{\circ} \mathrm{C}\) to the vapor phase at \(78^{\circ} \mathrm{C} ?\)

Short answer

(a) Total heat required to convert 42.0 g of ethanol at 35°C to the vapor phase at 78°C is given by the sum of the heat to raise the temperature from 35°C to 78°C (\(q_{1}\)) and the heat required for vaporization (\(q_{2}\)). \(q_{total} = q_1 + q_2\) (b) Total heat required to convert 42.0 g of ethanol at -155°C to the vapor phase at 78°C is given by the sum of the heat to raise the temperature from -155°C to -114°C (\(q_{3}\)), the heat required for fusion (\(q_{4}\)), the heat to raise the temperature from -114°C to 78°C (\(q_{1}\)), and the heat required for vaporization (\(q_{2}\)). \(q_{total} = q_3 + q_4 + q_1 + q_2\)

Step-by-step solution

Calculate the heat required to raise the temperature from 35°C to 78°C.

To do this, we need to use the specific heat formula: \(q = mcΔT\), where q is the heat, m is the mass, c is the specific heat, and ΔT is the change in temperature. Here, m = 42.0 g, ΔT = 78°C - 35°C, and c = 2.3 J/g-K (liquid ethanol). \(q_{1} = (42.0\,\text{g}) (2.3\,\text{J/g-K}) (78\,^\circ\text{C} - 35\,^\circ\text{C})\)

Calculate the heat required for vaporization.

Given the enthalpy of vaporization as 38.56 kJ/mol, we first need to convert the mass of ethanol (42.0 g) to moles using its molar mass (46.07 g/mol). Number of moles of ethanol = \(\frac{42.0\,\text{g}}{46.07\,\text{g/mol}}\) Next, we multiply the number of moles by the enthalpy of vaporization: \(q_{2} = n \times \Delta H_{vaporization}\)

Calculate the total heat required.

Sum the values of q1 and q2 calculated in the previous steps: Total heat required = \(q_1+ q_2\) (b)

Calculate the heat required to raise the temperature from -155°C to -114°C.

Here, m = 42.0 g, ΔT = (-114°C) - (-155°C), and c = 0.97 J/g-K (solid ethanol). \(q_3 = (42.0\,\text{g}) (0.97\,\text{J/g-K}) (-114\,^\circ\text{C} - (-155\,^\circ\text{C}))\)

Calculate the heat required for fusion.

Given the enthalpy of fusion as 5.02 kJ/mol, use the number of moles of ethanol that we calculated in the first part (a). \(q_4 = n \times \Delta H_{fusion}\)

Calculate the heat required to raise the temperature from -114°C to 78°C.

We've already done this in part (a). Total heat required in this step is \(q_1\).

Calculate the heat required for vaporization.

We've already done this in part (a). Total heat required in this step is \(q_2\).

Calculate the total heat required.

Sum the values of q3, q4, q1, and q2 calculated in the previous steps: Total heat required = \(q_3 + q_4 + q_1 + q_2\)

Key concepts

Enthalpy of Vaporization

When a liquid turns into a gas, it undergoes vaporization, a phase transition requiring energy. The enthalpy of vaporization, denoted as \( \triangle H_{vap} \), is the amount of heat needed to convert one mole of a liquid into a vapor without changing its temperature. This is a crucial concept for understanding how substances behave when they are heated.

For instance, during the boiling of ethanol, its temperature remains constant at its boiling point. The energy provided goes into breaking the intermolecular forces that hold the ethanol molecules together in the liquid phase. Once these forces are overcome, the molecules can move freely, resulting in a gaseous state. The enthalpy of vaporization is unique to each substance and is a key factor in determining how much energy is required to vaporize a given amount of material.

Enthalpy of Fusion

Similarly, the enthalpy of fusion, \( \triangle H_{fus} \), refers to the heat needed to change a substance from solid to liquid at its melting point. Also known as the heat of melting, this energy input is responsible for breaking down the more rigid structure of the solid into a less ordered liquid state.

This phase transition doesn't result in a temperature change; instead, it disrupts the fixed positions of the molecules in the solid structure, allowing them to slide past each other in the liquid state. The enthalpy of fusion varies from substance to substance and is indicative of the strength of the forces between the particles in the solid phase. For substances like ethanol, knowledge of the heat of fusion is essential to calculate the total heat involved in the process of converting it from a solid all the way to a gas.

Specific Heat Capacity

Between phase changes, substances will change temperature when heat is added or removed. This is described by the specific heat capacity, symbolized as 'c', which is defined as the amount of heat required to raise the temperature of one gram of a substance by one degree Celsius (or one Kelvin).

Different states of matter (solid, liquid, gas) possess different specific heat capacities, meaning that the amount of heat needed to change the temperature of a given mass of substance by a specific amount varies. For example, to heat liquid ethanol from one temperature to another, we would apply the formula \( q = mc\triangle T \), where 'q' is the heat energy, 'm' is the mass, 'c' is specific heat capacity, and \( \triangle T \) is the change in temperature. Understanding specific heat capacity is fundamental when calculating the energy required for heating or cooling a substance without changing its phase.

Phase Transition Heat Calculation

To find the total heat required for a substance to undergo multiple phase transitions, such as from solid to liquid (fusion) and then liquid to gas (vaporization), we need to calculate the heat for each step individually and then sum them up.

First, determine the energy needed for any temperature changes within a single phase using the specific heat capacity. Then, calculate the heat required for each phase change using the enthalpy of fusion or vaporization. By adding these quantities, we obtain the total heat energy needed for the complete transformation, considering both the phase changes and the temperature variations within each phase.

For a comprehensive understanding, it is vital to differentiate between the energy used to change the temperature of a substance and the energy used to change its phase. This distinction is critical for accurate heat calculations in physical and chemical processes.