Question

The heat of vaporization of water at \(373 \mathrm{~K}\) is \(40.7 \mathrm{~kJ} / \mathrm{mol}\). Find \(q, w, \Delta E,\) and \(\Delta H\) for the evaporation of \(454 \mathrm{~g}\) of water at this temperature at 1 atm.

Short answer

First, 454 g of water is equivalent to 454 g / 18.015 g/mol = 25.19 mol. The heat absorbed (q) is 25.19 mol * 40.7 kJ/mol = 1025.23 kJ. Work (w) is around 0 due to negligible volume change in liquid evaporation. Thus, ∆E is approximately 1025.23 kJ, and ∆H is also 1025.23 kJ for the evaporation of 454 g of water at 373 K and 1 atm.

Step-by-step solution

Convert mass of water to moles

To find the number of moles of water, use the molar mass of water, which is approximately 18.015 g/mol. The calculation is moles of water (mol) = mass of water (g) / molar mass of water (g/mol).

Calculate the heat absorbed (q) during vaporization

Multiply the number of moles of water by the heat of vaporization to find the heat absorbed (q). The calculation is q = moles of water (mol) * heat of vaporization (kJ/mol).

Determine work done (w) during the process

Since the evaporation occurs at 1 atm pressure and there is an increase in volume, we apply the work formula for constant external pressure, w = -PΔV. However, volume change data is not provided and for ideal gases at constant pressure, w would generally be negative since gases expand upon vaporization. In the case of water vaporizing, the work done on the system is negligible or can be considered zero without information on volume change.

Calculate the change in internal energy (∆E)

The change in internal energy (∆E) can be calculated using the formula ∆E = q + w. If the work done (w) is negligible, then ∆E can be approximated to be equal to q.

Calculate the enthalpy change (∆H) for the process

The enthalpy change (∆H) for the evaporation is approximately equal to the heat absorbed during vaporization (q) at constant pressure because ∆H = q_p, where q_p is the heat transfer at constant pressure.

Key concepts

Thermochemistry

Thermochemistry is the branch of chemistry that deals with the energy changes that occur during chemical reactions and physical transformations, such as phase transitions. It's a subfield of thermodynamics, focusing specifically on heat energy. For instance, when water evaporates, it undergoes a physical transformation from a liquid to a gas, and this process involves the absorption of heat from the surroundings, known as the heat of vaporization.

In thermochemistry, we commonly measure this heat in units of joules or kilojoules per mole. When a substance changes state, like water turning into water vapor, it does so by breaking intermolecular bonds, which requires energy. This energy is often supplied in the form of heat. Understanding how much heat is needed for a substance to vaporize is crucial in many fields, including meteorology, cooking, and various industrial processes like distillation.

Enthalpy Change

Enthalpy change, commonly denoted as \(\Delta H\), is the amount of heat released or absorbed by a system at constant pressure. It is a central concept in thermochemistry because it helps us understand how much energy is involved in a reaction or a physical process like vaporization. In the case of water evaporation, the enthalpy change corresponds to the heat of vaporization since the process occurs at constant pressure.

Enthalpy change can be positive or negative, indicating whether a process is endothermic (heat absorbed) or exothermic (heat released). To find the enthalpy change for the vaporization of water, we can assume it is approximately equal to the heat of vaporization, provided we perform the process at constant pressure. It's important to note that the enthalpy change provides insights into a material’s energy requirements and is utilized in designing cooling and heating systems.

Molar Mass Calculation

Molar mass calculation is fundamental in many chemical processes and thermodynamic calculations. The molar mass is the mass of one mole of a substance and is usually expressed in grams per mole. Finding the molar mass allows us to convert between grams of a substance and the number of moles, thus bridging the gap between the macroscopic and microscopic worlds.

To calculate the number of moles from a known mass of a substance, we use the formula:\(\text{moles} = \frac{\text{mass in grams}}{\text{molar mass in grams per mole}}\).For water, with a molar mass of approximately 18.015 g/mol, this calculation enables us to determine how many moles of water are present in a given sample. Molar mass is crucial for stoichiometry in reactions, and it’s utilized in concepts like the heat of vaporization, where the energy required to vaporize a specific number of moles of a substance is measured.