Problem 53 - Chemistry The Central Science

Question

The specific heat of octane, \(\mathrm{C}_{8} \mathrm{H}_{18}(l),\) is \(2.22 \mathrm{~J} / \mathrm{g}-\mathrm{K}\). (a) How many J of heat are needed to raise the temperature of \(80.0 \mathrm{~g}\) of octane from \(10.0{ }^{\circ} \mathrm{C}\) to \(25.0{ }^{\circ} \mathrm{C} ?\) (b) Which will require more heat, increasing the temperature of \(1 \mathrm{~mol}\) of \(\mathrm{C}_{8} \mathrm{H}_{18}(l)\) by a certain amount or increasing the temperature of \(1 \mathrm{~mol}\) of \(\mathrm{H}_{2} \mathrm{O}(l)\) by the same amount?

Step-by-Step Solution

Answer

The amount of heat needed to raise the temperature of 80.0 g of octane from 10.0°C to 25.0°C is 2664 J. Increasing the temperature of 1 mol of octane requires more heat (253.3 J/mol) than increasing the temperature of 1 mol of water (75.3 J/mol) by the same amount.

Step 1: Find Amount of heat to raise temperature of octane

To find the amount of heat needed, we should use the formula q = mcΔT, where q is the heat, m is the mass, c is the specific heat, and ΔT is the change in temperature. Given: m = 80.0 g c = 2.22 J/g-K Initial temperature, Ti = 10.0°C Final temperature, Tf = 25.0 °C Now we should find the change in temperature (ΔT). ΔT = Tf - Ti = 25.0°C - 10.0°C = 15.0°C Now, we can compute the heat(q) using the formula: q = mcΔT = (80.0 g) × (2.22 J/g-K) × (15.0 K) q = 2664 J The amount of heat needed to raise the temperature of 80.0 g of octane from 10.0°C to 25.0°C is 2664 J.

Step 2: Compare heat for 1 mol of octane and water

First, we need to find the molar mass of octane and water. Molar mass of octane (C8H18): \(8 \times 12.01\, \text{g/mol (carbon)}+ 18 \times 1.01\, \text{g/mol (hydrogen)} = 114.23\, \text{g/mol}\) Molar mass of water (H2O): \(2 \times 1.01\, \text{g/mol (hydrogen)}+ 1 \times 16.00\, \text{g/mol (oxygen)} = 18.02\, \text{g/mol}\) Now, let's find the heat required per mole of octane and water for the same change in temperature. Specific heat of water = 4.18 J/g-K Heat for 1 mole of octane: q(octane) = Molar mass x Specific heat x ΔT q(octane) = (114.23 g/mol) × (2.22 J/g-K) × (1 K) q(octane) = 253.3 J/mol Heat for 1 mole of water: q(water) = Molar mass x Specific heat x ΔT q(water) = (18.02 g/mol) × (4.18 J/g-K) × (1 K) q(water) = 75.3 J/mol Since 253.3 J/mol (octane) > 75.3 J/mol (water), increasing the temperature of 1 mol of octane requires more heat than increasing the temperature of 1 mol of water by the same amount.

Key Concepts

Thermochemistry

Thermochemistry is the branch of chemistry concerned with the study of energy and heat associated with chemical reactions and physical transformations. A fundamental concept in thermochemistry is the law of conservation of energy, which states that energy cannot be created or destroyed, only transferred or transformed from one form to another.

For example, when a substance like octane heats up, the energy provided to increase its temperature is stored within the substance as thermal energy. This storage and transfer of energy are quantified through specific heat calculations, which tell us how much energy is required to raise the temperature of a given mass of a substance by a certain amount. Understanding thermochemistry is crucial for a myriad of applications, including designing engines, predicting product yields in reactions, and even cooking.

Enthalpy

Enthalpy, symbolized as H, is a measure of the total heat content of a thermodynamic system at constant pressure. It is a state function, meaning its value depends only on the current state of the system, not the path taken to reach that state. The change in enthalpy, \( \Delta H \), is essential in thermochemistry as it represents the heat absorbed or released during a chemical reaction or physical process.

In the exercise above, the change in enthalpy can be related to the heat required to raise the temperature of octane. The specific heat capacity is a component in the calculation of enthalpy change. Calculating \( \Delta H \) allows scientists and engineers to predict how much energy is needed to initiate a reaction or cause a substance to undergo a physical change without worrying about the intricacies of the processes.

Heat Transfer

Heat transfer is the movement of thermal energy from one object or substance to another, driven by a temperature difference. In the context of our exercise, when we provide heat to octane, we are facilitating the transfer of energy that increases the kinetic energy of its particles and thereby raises its temperature. This process can be described quantitatively by the equation \( q = mc \Delta T \), linking the heat transferred to the mass of the substance, its specific heat capacity, and the change in temperature.

Understanding heat transfer principles is vital in various practical situations, such as engineering, environmental science, and even in everyday activities like cooking or insulating homes. Knowledge about how heat is transferred allows for the efficient design of systems that either need to dissipate heat quickly or retain it effectively.

Molar Heat Capacity

Molar heat capacity is the amount of heat required to raise the temperature of one mole of a substance by one degree Celsius. It is an intrinsic property that varies from substance to substance and takes into account the molecular structure and bonding. In specific heat calculations, knowing the molar heat capacity allows us to predict how different substances will react to heat energy.

For example, in the exercise, we compared the molar heat capacity of octane and water. Despite octane having a lower specific heat capacity than water, the amount of heat needed to raise the temperature of a mole of octane was greater due to its higher molar mass. This kind of information is crucial when designing systems that involve temperature control, selecting materials for heating applications, or understanding the energy changes in chemical reactions.