Question

The combustion of 0.1584 g benzoic acid increases the temperature of a bomb calorimeter by ${2.54}^{\circ }\mathrm{C}$ . Calculate the heat capacity of this calorimeter. (The energy released by combustion of benzoic acid is 26.42 $\mathrm{kJ}/\mathrm{g}.$ ) A $0.2130-\mathrm{g}$ sample of vanillin $\left({\mathrm{C}}_8{\mathrm{H}}_8{\mathrm{O}}_3\right)$ is then burned in the same calorimeter, and the temperature increases by ${3.25}^{\circ }\mathrm{C}$ . What is the energy of combustion per gram of vanillin? Per mole of vanillin?

Short answer

The heat capacity of the calorimeter is $1.647\frac{\mathrm{kJ}}{\text{degree}\mathrm{C}}$. The energy of combustion per gram of vanillin is $25.13\frac{\mathrm{kJ}}{\mathrm{g}}$ and per mole is $3826.75\frac{\mathrm{kJ}}{\mathrm{mol}}$.

Step-by-step solution

Calculate the heat released by benzoic acid combustion

Calculate the heat released by the combustion of 0.1584 g of benzoic acid. We are given that the heat of combustion for benzoic acid is 26.42 kJ/g. We can find the heat released by using the formula: Heat released ($q$) = mass ($m$) × heat of combustion ($H_c$) So, $q_{benzoicacid}=0.1584\mathrm{g}\times 26.42\frac{\mathrm{kJ}}{\mathrm{g}}=4.184\mathrm{kJ}$.

Calculate the heat capacity of the calorimeter

To find the heat capacity of the calorimeter, we need to divide the heat released by benzoic acid by the increase in temperature caused by the combustion. The formula for this is: Heat capacity ($C_{cal}$) = $q_{benzoicacid}÷\mathrm{\Delta }T$ Where $\mathrm{\Delta }T$ is the change in temperature. In this case, $\mathrm{\Delta }T=2.54\text{degree}\mathrm{C}$. So, $C_{cal}=\frac{4.184\mathrm{kJ}}{2.54\text{degree}\mathrm{C}}=1.647\frac{\mathrm{kJ}}{\text{degree}\mathrm{C}}$.

Determine the heat released by vanillin combustion

Since the heat capacity of the calorimeter is now known, we can use it to calculate the heat released by the combustion of vanillin. The formula for this is: Heat released ($q$) = heat capacity ($C_{cal}$) × increase in temperature ($\mathrm{\Delta }T$) In this case, $\mathrm{\Delta }T=3.25\text{degree}\mathrm{C}$. So, $q_{vanillin}=1.647\frac{\mathrm{kJ}}{\text{degree}\mathrm{C}}\times 3.25\text{degree}\mathrm{C}=5.353\mathrm{kJ}$.

Calculate the energy of combustion per gram of vanillin

To determine the energy of combustion per gram of vanillin, we need to divide the heat released by vanillin by the mass of vanillin: Energy of combustion ($H_c$) = $\frac{q_{vanillin}}{m}$ Where $m=0.2130\mathrm{g}$ So, $H_c=\frac{5.353\mathrm{kJ}}{0.2130\mathrm{g}}=25.13\frac{\mathrm{kJ}}{\mathrm{g}}$.

Calculate the energy of combustion per mole of vanillin

Lastly, we need to convert the energy of combustion per gram of vanillin to energy of combustion per mole of vanillin. First, determine the molar mass of vanillin: Molar mass of vanillin = C${}_8$H${}_8$O${}_3$ = $8\times 12.01\mathrm{g}+8\times 1.01\mathrm{g}+3\times 16.00\mathrm{g}=152.15\mathrm{g}/\mathrm{mol}$ Now, convert the energy of combustion per gram of vanillin to per mole: Energy of combustion per mole ($H_c^{\prime }\mathrm{kJ}/\mathrm{mol}$) = energy of combustion per gram ($H_c\mathrm{kJ}/\mathrm{g}$) × molar mass of vanillin ($\frac{\mathrm{g}}{\mathrm{mol}}$) $H_c^{\prime }=25.13\frac{\mathrm{kJ}}{\mathrm{g}}\times 152.15\frac{\mathrm{g}}{\mathrm{mol}}=3826.75\frac{\mathrm{kJ}}{\mathrm{mol}}$ Therefore, the energy of combustion per gram of vanillin is $25.13\frac{\mathrm{kJ}}{\mathrm{g}}$ and per mole is $3826.75\frac{\mathrm{kJ}}{\mathrm{mol}}$.

Key concepts

Heat Capacity

Heat capacity is an important concept when studying calorimetry. It measures how much energy is needed to change the temperature of an object by a certain amount. In our exercise, when benzoic acid is burned, it releases energy. This energy causes the temperature of the calorimeter to increase. The heat capacity of the calorimeter is then calculated by dividing the heat released by the temperature change.
For the benzoic acid example:

• Heat released ($q$) was found using the formula: $q=m\times H_c$
• The heat capacity of the calorimeter ($C_{cal}$) is calculated as: $C_{cal}=\frac{q_{benzoicacid}}{\mathrm{\Delta }T}$

The heat capacity tells us how much energy the calorimeter can absorb per degree of temperature change. It's crucial for determining the energy changes during chemical reactions.

Energy of Combustion

Energy of combustion is the amount of energy released when a substance is burned completely. This is a key concept when determining the energy output of fuels or food.
In the exercise:

• We find the energy released by burning benzoic acid and vanillin.
• The energy per gram helps to compare substances on a weight basis.

Using the exercise formula, we find the heat released by vanillin as $5.353\text{kJ}$. Then:

• The energy of combustion per gram of vanillin: $H_c=\frac{q_{vanillin}}{m}$

This value shows how energy-efficient the substance is per gram. It’s essential for applications like fuel selection and dietary analysis.

Enthalpy Change

Enthalpy change represents the total energy change in a system during a reaction, usually at constant pressure. For our specific example, it relates to the energy of combustion per mole.
To find the enthalpy change per mole:

• First, we determine the molar mass of vanillin.
• Convert the energy from a per-gram to a per-mole basis.

The formula used:

• $H_c^{\prime }=H_c\times \text{molar mass}$

This conversion helps in understanding the energy involved when a whole mole of substance burns. Such information is vital in chemical engineering and thermodynamic calculations, where energy efficiency and reaction planning are key.