Question

56White phosphorus, \(\mathrm{P}_{4}\), ignites in air to produce \(\mathrm{P}_{4} \mathrm{O}_{10}\). $$ \mathrm{P}_{4}(\mathrm{~s})+5 \mathrm{O}_{2}(\mathrm{~g}) \longrightarrow \mathrm{P}_{4} \mathrm{O}_{10}(\mathrm{~s}) $$ When \(3.56 \mathrm{~g} \mathrm{P}_{4}\) is burned, \(85.8 \mathrm{~kJ}\) of thermal energy is evolved at constant pressure. Calculate the combustion enthalpy of \(\mathrm{P}_{4}\).

Short answer

The combustion enthalpy of \(\mathrm{P}_4\) is \(-2990 \mathrm{~kJ/mol}\).

Step-by-step solution

Understand the problem

We need to calculate the combustion enthalpy (enthalpy change) of the reaction where white phosphorus (\(\mathrm{P}_4\)) burns to form \(\mathrm{P}_4\mathrm{O}_{10}\). We are given the mass of \(\mathrm{P}_4\) and the thermal energy evolved.

Calculate Moles of \(\mathrm{P}_4\)

First, calculate the molar mass of \(\mathrm{P}_4\). Since the atomic mass of phosphorus \( (\mathrm{P}) \) is approximately \(30.97\, \mathrm{g/mol}\), the molar mass of \(\mathrm{P}_4\) is \(4 \times 30.97 = 123.88 \, \mathrm{g/mol}\). Now, calculate the moles of \(\mathrm{P}_4\) using the given mass of \(3.56\, \mathrm{g}\): \[ \text{moles of } \mathrm{P}_4 = \frac{3.56 \mathrm{~g}}{123.88 \mathrm{~g/mol}} \approx 0.0287 \mathrm{~mol} \]

Calculate Enthalpy Change per Mole

We know that \(85.8\, \mathrm{kJ}\) of energy is released when \(0.0287\, \text{mol}\) of \(\mathrm{P}_4\) is combusted. To find the enthalpy change per mole, we calculate:\[ \Delta H = \frac{85.8 \mathrm{~kJ}}{0.0287 \mathrm{~mol}} \approx 2990 \mathrm{~kJ/mol} \]

Present the Combustion Enthalpy

Since the enthalpy pertains to combustion (which is exothermic), the combustion enthalpy is negative. Therefore, the combustion enthalpy of \(\mathrm{P}_4\) is \(-2990 \mathrm{~kJ/mol}\).

Key concepts

Thermodynamics

Thermodynamics is a branch of physics that deals with the relationships and conversions between heat and other forms of energy. The core of thermodynamics is the concept of energy transfer, especially when it involves heat and work. In the realm of chemistry, thermodynamics helps us understand how energy changes during chemical reactions.
In the case of the combustion of white phosphorus, we are particularly interested in enthalpy, which is a measure of the total energy of a thermodynamic system. This includes internal energy plus the product of pressure and volume.
The reaction of phosphorus with oxygen is exothermic, meaning it releases energy. The enthalpy change for such a reaction is usually negative, indicating a loss of energy in the form of heat to the surroundings. We describe this as combustion enthalpy. The easy breakdown of energy content changes is crucial for practical applications such as energy production, engine design, and environmental studies.

Chemical Reactions

Chemical reactions involve the transformation of reactants into products. During a chemical reaction, bonds are broken and new bonds are formed, which involves energy changes. These changes are the reason behind the absorption or release of energy.
In the exercise, white phosphorus reacts with oxygen to form phosphorus pentoxide. This reaction can be represented with the chemical equation:

• \( \text{P}_4(\text{s}) + 5 \text{O}_2(\text{g}) \rightarrow \text{P}_4\text{O}_{10}(\text{s}) \)

This balanced equation shows the reactants (white phosphorus and oxygen) and the product (phosphorus pentoxide). To trigger this reaction, white phosphorus needs to ignite, or burn, which implies an exothermic process. The energy released in an exothermic reaction is vital for calculating the combustion enthalpy. Understanding these reactions helps predict how much thermal energy might be involved and how substances interact.

Molar Mass Calculation

Calculating molar mass is fundamental in chemistry and plays a critical role in determining the outcomes of chemical reactions. It involves adding the atomic masses of all atoms in a molecular formula. For white phosphorus, with the molecular formula \(\text{P}_4\), the molar mass is calculated based on the atomic mass of phosphorus.
Given that the atomic mass of phosphorus is approximately \(30.97\, \text{g/mol}\), the molar mass of \(\text{P}_4\) is calculated as:

• \(4 \times 30.97 = 123.88 \, \text{g/mol}\)

Knowing the molar mass allows us to convert between grams and moles—an essential step when measuring substances for reactions. In the problem, converting the mass of \(3.56\, \text{g}\) of \(\text{P}_4\) into moles was necessary to determine the amount of substance reacting and the related heat energy release. This step is key to calculating the combustion enthalpy and understanding how substances quantifiably interact in reactions.