Question

When a \(6.00-\mathrm{g}\) sample of coal is burned, it releases enough heat to raise the temperature of \(2010 \mathrm{~g}\) of water from \(24.0^{\circ} \mathrm{C}\) to \(41.5^{\circ} \mathrm{C}\). (a) How much heat did the coal release as it burned? (b) Calculate the heat of combustion of coal in units of \(\mathrm{kJ} / \mathrm{g}\).

Short answer

(a) The heat released by the coal is approximately \(146.065 \mathrm{kJ}\). (b) The heat of combustion for the coal is approximately \(24.344 \mathrm{kJ/g}\).

Step-by-step solution

Calculate the Heat Released (A)

To calculate the heat released by the coal, you have to use the formula \(q = mc\Delta T\), where \(q\) is the heat, \(m\) is the mass, \(c\) is the specific heat, and \(\Delta T\) is the change in temperature. For water, \(c\) is typically \(4.18 \mathrm{J/g\cdot C}\). The mass of the water is given as \(2010 \mathrm{g}\) and the temperature increase (\(\Delta T\)) is \(41.5^{\circ} \mathrm{C} - 24.0^{\circ} \mathrm{C} = 17.5^{\circ} \mathrm{C}\). So, the heat released (\(q\)) is \(2010 \mathrm{g} \cdot 4.18 \mathrm{J/g\cdot C} \cdot 17.5^{\circ} \mathrm{C}\).

Convert Joules to Kilojoules

The result from the previous calculation will be in joules (J). However, we need the result in kilojoules for the next step. To convert joules to kilojoules, simply divide the result by 1000.

Calculate the Heat of Combustion (B)

To calculate the heat of combustion per gram, you use the formula \(q/m_{coal}\), where \(q\) is the heat released by the coal (in kJ) and \(m_{coal}\) is the mass of the burned coal. Alert: the mass of coal should in grams which is given as \(6 \mathrm{g}\). Therefore, the heat of combustion would be calculated by taking the heat released (obtained from previous parts and converted to kJ) and dividing it by the mass of the coal.

Key concepts

Thermochemistry

Thermochemistry is the study of the heat energy involved in chemical reactions and physical changes. It's a fundamental concept that helps us understand how energy is absorbed or released when substances react or change their physical state. The central theme in thermochemistry is the law of conservation of energy, stating that energy can neither be created nor destroyed. In the context of our coal combustion example, the energy released when coal burns is absorbed by the surrounding water, causing the water temperature to rise.

Specific Heat Capacity

Specific heat capacity, often simply referred to as specific heat, is an intrinsic property of a substance that indicates how much heat energy is required to raise the temperature of one gram of the substance by one degree Celsius. It is denoted by the symbol 'c' and is measured in units of joules per gram per degree Celsius (J/g°C). In our coal burning scenario, water has a specific heat capacity of 4.18 J/g°C, which means that 4.18 joules are needed to raise the temperature of 1 gram of water by 1°C. This property is crucial for calculating the amount of heat absorbed by the water as it is heated by the energy released from coal combustion.

Energy Conversion

Energy conversion is the process of changing one form of energy into another. In chemical reactions, such as the combustion of coal, chemical energy stored in the bonds of the reactants is converted into heat energy. This conversion is central to many industrial and everyday processes, including the generation of electricity and the heating of homes. In the exercise, the chemical energy in coal is converted into thermal energy that is transferred to water, increasing its temperature. It's key to recognize that during these conversions, the total amount of energy is conserved, aligning with the principle of energy conservation.

Chemical Thermodynamics

Chemical thermodynamics deals with the relationship between heat, work, and chemical reactions. It is the science that quantifies the energy transformations in systems as they undergo physical changes or chemical reactions. The heat of combustion, as seen in our example, is a thermodynamic property and it represents the heat released when a substance undergoes complete combustion in the presence of oxygen at a constant pressure. The calculated heat of combustion for coal provides valuable information about its energy content, which is essential for assessing its efficiency as a fuel.