Question

The enthalpy of combustion of \(\mathrm{CH}_{4}(g)\) when \(\mathrm{H}_{2} \mathrm{O}(l)\) is formed is \(-891 \mathrm{kJ} / \mathrm{mol}\) and the enthalpy of combustion of \(\mathrm{CH}_{4}(g)\) when \(\mathrm{H}_{2} \mathrm{O}(g)\) is formed is \(-803 \mathrm{kJ} / \mathrm{mol} .\) Use these data and Hess's law to determine the enthalpy of vaporization for water.

Short answer

The enthalpy of vaporization for water (ΔHvap) can be determined using Hess's Law and the given enthalpy of combustion values for CH4(g). By subtracting the first reaction (when H2O(l) is formed) from the second reaction (when H2O(g) is formed), we obtain the reaction 2H2O(l) -> 2H2O(g) with an enthalpy change of ΔHvap = ΔH2 - ΔH1. Substituting the given values, we find that ΔHvap = (-803 kJ/mol) - (-891 kJ/mol), which gives us an enthalpy of vaporization for water of 88 kJ/mol.

Step-by-step solution

Write down the given reactions

First, let's write down the given reactions and their enthalpies: 1. CH4(g) + 2O2(g) -> CO2(g) + 2H2O(l) ΔH1 = -891 kJ/mol 2. CH4(g) + 2O2(g) -> CO2(g) + 2H2O(g) ΔH2 = -803 kJ/mol And we need to determine the enthalpy of vaporization for water (ΔHvap).

Understand Hess's Law

Hess's Law states that the total enthalpy change for a chemical reaction is the same, no matter whether it happens in one step or multiple steps. So, if we can create a reaction that represents the enthalpy of vaporization for water using the given reactions, then we can find the ΔHvap for water.

Create a new reaction representing the enthalpy of vaporization

We can achieve this by subtracting the first reaction from the second reaction. This will give us a new reaction representing the vaporization of H2O: Reaction 3: 2H2O(l) -> 2H2O(g) ΔH3 = ΔHvap = ΔH2 - ΔH1 Now we just need to calculate ΔHvap by substituting the values of ΔH1 and ΔH2.

Calculate ΔHvap for water

ΔHvap = ΔH2 - ΔH1 ΔHvap = (-803 kJ/mol) - (-891 kJ/mol)

Solve for ΔHvap

ΔHvap = 88 kJ/mol So, the enthalpy of vaporization for water is 88 kJ/mol.

Key concepts

Enthalpy of Combustion

The enthalpy of combustion is one of the key concepts in chemical thermodynamics. It refers to the heat released when one mole of a substance reacts completely with oxygen under standard conditions. Combustion reactions are typically exothermic, meaning they release energy in the form of heat.

In the context of the exercise, we look at the enthalpy of combustion for methane (\(\mathrm{CH}_4\)), which is a commonly used fuel. We have two enthalpy values:

• \(-891 \mathrm{kJ/mol}\) when \(\mathrm{H}_2\mathrm{O}(l)\) is formed.
• \(-803 \mathrm{kJ/mol}\) when \(\mathrm{H}_2\mathrm{O}(g)\) is formed.

These values indicate the energy difference between liquid and gaseous water as they form. It's crucial to understand these differences because they relate to the physical state of water during the reactions, directly influencing the total heat exchanged.

Enthalpy of Vaporization

The enthalpy of vaporization (\(\Delta H_{vap}\)) is a term used to define the amount of energy needed to convert a liquid into a gas. It is a fundamental concept when discussing phase changes at a constant temperature.

In our problem, we are calculating the vaporization enthalpy for water (\(\mathrm{H}_2\mathrm{O}\)), demonstrating how to use Hess's Law in solving this. According to Hess's Law, the enthalpy change of this phase transition is based on the subtraction of two given reaction enthalpies:

• Combustion leading to liquid water: \(-891 \mathrm{kJ/mol}\).
• Combustion leading to gaseous water: \(-803 \mathrm{kJ/mol}\).

The difference \(88 \mathrm{kJ/mol}\) represents the enthalpy required to convert water from its liquid form into vapor under the same conditions. This makes it an important value in understanding energy changes during boiling or evaporation.

Chemical Thermodynamics

Chemical thermodynamics is the study of energy transformations in chemical processes. This discipline helps us understand how energy is absorbed or released during reactions, and it's key for predicting reaction behavior and designing chemical processes.

In terms of our exercise, the principle of Hess's Law illustrates how chemical thermodynamics can be used to calculate unknown energy changes. Hess's Law states that the total enthalpy change for a reaction is the same, irrespective of the pathway by which the reaction takes place. This is incredibly useful for calculating values like the enthalpy of vaporization, as demonstrated.

Understanding chemical thermodynamics allows scientists and engineers to harness and optimize energy usage in industrial applications. It provides the framework for processes such as combustion, phase changes like vaporization, and countless other applications that depend on controlled energy transformations.