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 $10.0$ 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?

Short answer

a) The heat required to raise the temperature of $80.0\mathrm{g}$ of octane from $10.0$ to ${25.0}^{\circ }\mathrm{C}$ is $2664\mathrm{J}$. b) Increasing the temperature of $1\mathrm{mol}$ of ${\mathrm{C}}_8{\mathrm{H}}_{18}(l)$ (octane) will require more heat than increasing the temperature of $1\mathrm{mol}$ of ${\mathrm{H}}_2\mathrm{O}(l)$ (water) by the same amount.

Step-by-step solution

Calculate the change in temperature

We have given that the initial temperature is ${10.0}^{\circ }\mathrm{C}$ and the final temperature is ${25.0}^{\circ }\mathrm{C}$. The change in temperature is calculated as: $\mathrm{\Delta }T=T_f-T_i$ $\mathrm{\Delta }T=25.0-10.0={15.0}^{\circ }\mathrm{C}$

Calculate the heat required for octane

Now we can use the formula $q=mC\mathrm{\Delta }T$ to find the heat required to raise the temperature of $80.0\mathrm{g}$ of octane. The specific heat for octane is $2.22\mathrm{J}/\mathrm{g}\cdot \mathrm{K}$, the mass is $80.0\mathrm{g}$, and the change in temperature is ${15.0}^{\circ }\mathrm{C}$. Plug these values into the formula: $q=(80.0\mathrm{g})(2.22\mathrm{J}/\mathrm{g}\cdot \mathrm{K})(15.0\mathrm{K})$ $q=2664\mathrm{J}$ a) The heat required is $2664\mathrm{J}$.

Compare the heat requirements for 1 mol of octane and 1 mol of water

To compare which substance requires more heat, we need to calculate the heat capacity (specific heat multiplied by the molar mass) for both substances. The specific heat of water is $4.18\mathrm{J}/\mathrm{g}\cdot \mathrm{K}$. The molar mass of octane is $12.01\cdot 8+1.01\cdot 18=114.23\mathrm{g}/\mathrm{mol}$ and the molar mass of water is $1.01\cdot 2+16.00=18.02\mathrm{g}/\mathrm{mol}$. Heat capacity of octane: $(2.22\mathrm{J}/\mathrm{g}\cdot \mathrm{K})(114.23\mathrm{g}/\mathrm{mol})=253.6\mathrm{J}/\mathrm{mol}\cdot \mathrm{K}$ Heat capacity of water: $(4.18\mathrm{J}/\mathrm{g}\cdot \mathrm{K})(18.02\mathrm{g}/\mathrm{mol})=75.3\mathrm{J}/\mathrm{mol}\cdot \mathrm{K}$ b) Increasing the temperature of $1\mathrm{mol}$ of ${\mathrm{C}}_8{\mathrm{H}}_{18}(l)$ (octane) will require more heat than increasing the temperature of $1\mathrm{mol}$ of ${\mathrm{H}}_2\mathrm{O}(l)$ (water) by the same amount.

Key concepts

Energy Required for Temperature Change

When we talk about heating a substance, what we're really speaking about is the energy required for a temperature change. This concept is essential in understanding how substances absorb heat. The amount of energy required to raise the temperature of a specific amount of a substance is calculated using the formula:

$$q=mC\mathrm{\Delta }T$$

Where:\

\
• $q$ represents the heat energy in joules (J),\
• $m$ represents the mass of the substance in grams (g),\
• $C$ is the specific heat capacity (J/g·K), and\
• $\mathrm{\Delta }T$ is the change in temperature in Kelvin (K) or degrees Celsius (°C).\

\
It's worth noting that the temperature change ($\mathrm{\Delta }T$) is the same whether you measure it in Kelvin or degrees Celsius as both scales have the same size 'degree'. The specific heat capacity, $C$, is a substance's characteristic property; it tells us how much heat is needed to raise the temperature of one gram of the substance by one degree Celsius (or one Kelvin).

In the case of our octane example, we applied this formula and found that it takes 2664 J to raise the temperature of 80 grams of octane by 15°C. This tells us that different substances require different amounts of energy to achieve the same temperature change, which is a direct consequence of their varying specific heats.

Comparing Heat Capacities

Now let's delve into the concept of comparing heat capacities of different substances. When we say 'heat capacity', we refer to the amount of heat needed to raise the temperature of one mole of a substance by one degree Celsius (or one Kelvin). Heat capacity is a bulk property, and it's related to specific heat but it takes into account the amount of substance (in moles).

To calculate it, we multiply the specific heat (C) by the molar mass (M) of the substance:

$$HeatCapacity=C\times M$$

Thus, for our octane, with a specific heat of 2.22 J/g·K and a molar mass of 114.23 g/mol, its heat capacity is lower compared to water, which means that octane would require more energy to warm up the same number of moles by the same temperature change. This difference in heat capacities signifies how substances absorb and retain heat differently and is pivotal for applications ranging from engineering to cooking.

Enthalpy and Chemistry Thermodynamics

Lastly, let's discuss the broader context of enthalpy and chemistry thermodynamics. In chemistry, enthalpy is the total heat content of a system and it's a key concept in thermodynamics. When we calculate the energy required to trigger a temperature change, we’re often considering changes in enthalpy, denoted by $\mathrm{\Delta }H$. Enthalpy change reflects the heat absorbed or released by a system at constant pressure.

For instance, in a reaction where chemical bonds are formed or broken, the change in enthalpy can tell us if the reaction is exothermic (releases heat) or endothermic (absorbs heat). This ties back to our earlier discussions because knowing a substance’s specific heat can help in calculating these enthalpy changes during chemical reactions.

Understanding how to manipulate and measure heat and temperature changes is not just academic; it's crucial for various applications in everyday life, industry, and environmental science. By mastering these principles, students can better grasp the often invisible, yet immensely powerful, forces that drive chemical transformations.