Problem 86 - Chemistry Matter and Change

Question

Heating with Coal How much heat is liberated when 5.00 kg of coal is burned if the coal is 96.2\(\%\) carbon by mass and the other materials in the coal do not react? \(\Delta H_{\mathrm{comb}}\) of carbon is \(-394 \mathrm{kJ} / \mathrm{mol}\)

Step-by-Step Solution

Answer

The heat liberated is 157797 kJ when 5.00 kg of coal is burned.

Step 1: Calculate Mass of Carbon

First, calculate the mass of carbon in the coal. The coal is 96.2% carbon by mass, so the mass of carbon is 96.2% of the total mass of the coal. Calculate this by multiplying the total mass of coal by the percentage of carbon:\[ \text{Mass of carbon} = 5.00 \, \text{kg} \times 0.962 = 4.81 \, \text{kg} \]

Step 2: Convert Mass to Moles

Convert the mass of carbon to moles. Use the molar mass of carbon (12.01 g/mol) for the conversion. Since the mass is in kilograms, convert the units to grams first.\[ \text{Mass of carbon} = 4.81 \, \text{kg} \times 1000 \, \text{g/kg} = 4810 \, \text{g} \]Now, calculate the moles of carbon:\[ \text{Moles of carbon} = \frac{4810 \, \text{g}}{12.01 \, \text{g/mol}} \approx 400.50 \, \text{mol} \]

Step 3: Calculate Heat Released

Use the enthalpy of combustion (\( \Delta H_{\mathrm{comb}} \)), which is given as -394 kJ/mol. Multiply the moles of carbon by \( \Delta H_{\mathrm{comb}} \) to find the total heat released.\[ \text{Heat liberated} = 400.50 \, \text{mol} \times (-394 \, \text{kJ/mol}) \]\[ \text{Heat liberated} = -157797 \, \text{kJ} \]

Step 4: Express the Final Answer

The total heat liberated from burning 5.00 kg of coal is 157797 kJ. The negative sign indicates that it is an exothermic reaction, so heat is released.

Key Concepts

Mass to Moles Conversion

Whenever you need to convert mass to moles, remember that this is a fundamental step in many chemical calculations. The number of moles gives you the quantity of a substance based on the count of atoms, rather than just weight. This allows you to work in terms of actual particles, like atoms or molecules. Conversion involves using the molar mass of a substance, which is the mass of one mole of that substance, usually given in grams per mole.

• To convert from mass (in grams) to moles, use the formula: \[{\rm Moles} = \frac{{\rm Mass ext{ } ext{in} ext {grams}}}{{\rm Molar ext{ } Mass}}\]
• Always ensure your units match; convert kilograms to grams when necessary.

In our example of burning coal, the mass of carbon needs converting from kilograms to grams before using the molar mass of carbon (12.01 g/mol) to find the number of moles. This transformation helps align your calculations with the enthalpy units, which are usually expressed in terms of moles.

Carbon Combustion

The combustion of carbon is a typical chemical reaction involving burning carbon in the presence of oxygen, leading to carbon dioxide production and the release of energy. It's an important process not just in academic chemistry, but also in everyday applications such as energy generation.Carbon combustion can be represented by the chemical equation:\[\text{C} + \text{O}_2 \rightarrow \text{CO}_2\]Here's how it works:

• Carbon acts as the reactant combining with oxygen.
• They form carbon dioxide as the main product.
• The reaction liberates a considerable amount of heat due to the breaking and forming of chemical bonds.

In exercises involving coal, you're often seeing the effect of this reaction on a large scale, such as when calculating energy released for heating purposes. Understanding this reaction is crucial, as many fuels are carbon-based, and similar calculations are applicable.

Exothermic Reaction

An exothermic reaction is characterized by the release of heat energy into the surrounding environment. This can be observed as an increase in temperature in the surroundings, or sometimes even in the form of light or sound. Combustion reactions, like the burning of carbon, are classic examples of exothermic processes.Key characteristics of exothermic reactions include:

• The enthalpy change, \(\Delta H\), is negative, indicating energy release.
• The products are at a lower energy state compared to the reactants.
• Such processes are often spontaneous, requiring little additional energy to proceed once initiated.

In the context of carbon combustion in coal, the negative sign of \(-394 \text{kJ/mol}\) for the enthalpy of combustion reflects that the reaction imparts energy to its surroundings, a crucial consideration for applications such as heating, where energy efficiency is valued. In a way, exothermic reactions create practical benefits from chemical transformations.