Problem 38 - Chemistry

Question

A \(0.1375-\mathrm{g}\) sample of solid magnesium is burned in a constant-volume bomb calorimeter that has a heat capacity of \(3024 \mathrm{~J} /{ }^{\circ} \mathrm{C}\). The temperature increases by \(1.126^{\circ} \mathrm{C}\). Calculate the heat given off by the burning \(\mathrm{Mg},\) in \(\mathrm{kJ} / \mathrm{g}\) and in \(\mathrm{kJ} / \mathrm{mol} .\)

Step-by-Step Solution

Answer

The heat given off by burning Mg is 24.764 kJ/g and 601.65 kJ/mol.

Step 1: Determine Total Heat Released

To find the total heat released by the reaction, use the formula for heat transfer: \[ q = C \times \Delta T \]where \( C \) is the heat capacity of the calorimeter (3024 J/°C) and \( \Delta T \) is the change in temperature (1.126°C). Substitute these values into the equation:\[ q = 3024 \, \mathrm{J/°C} \times 1.126 \, °C = 3405.024 \, \mathrm{J} \]

Step 2: Convert Heat to kJ

Convert the total heat calculated in joules to kilojoules by dividing by 1000. This is because 1 kJ is equal to 1000 J.\[ q = \frac{3405.024 \, \mathrm{J}}{1000} = 3.405 \, \mathrm{kJ} \]

Step 3: Calculate Heat per Gram of Mg

Divide the heat released in kJ by the mass of the magnesium sample to find the heat given off per gram:\[ \frac{3.405 \, \mathrm{kJ}}{0.1375 \, \mathrm{g}} = 24.764 \, \mathrm{kJ/g} \]

Step 4: Calculate Molar Mass of Mg

The molar mass of magnesium (Mg) is 24.305 g/mol.

Step 5: Calculate Heat per Mole of Mg

Use the heat per gram and the molar mass to calculate the heat per mole:\[ 24.764 \, \mathrm{kJ/g} \times 24.305 \, \mathrm{g/mol} = 601.65 \, \mathrm{kJ/mol} \]

Key Concepts

Heat Capacity

Heat capacity is an important property that explains how much thermal energy a substance can hold before it changes temperature. In calorimetry, heat capacity plays a pivotal role in determining how much heat is absorbed or released during a reaction.
Essentially, heat capacity refers to the amount of heat necessary to raise the temperature of an object by 1°C. The specific formula used to calculate this in a calorimeter setting is:

• \( q = C \times \Delta T \)

where

• \( q \) is the thermal energy conducted,
• \( C \) is the heat capacity of the calorimeter, and
• \( \Delta T \) is the change in temperature.

This formula helps us understand the thermal dynamics during chemical reactions. In the case of the magnesium combustion, a heat capacity of 3024 J/°C allowed us to determine the heat produced by the reaction.

Temperature Change

Temperature change is a crucial factor in calorimetry measurements. It indicates how much the temperature of a substance has increased or decreased after absorbing or releasing energy. In the exercise, the magnesium burned and increased the temperature of the calorimeter by 1.126°C.
This value of temperature change (\( \Delta T \)) was used in the formula for heat transfer. It is essential because it helps to calculate the total energy released or absorbed by a reaction.In such exercises:

• The larger the change in temperature,
• The more heat has been transferred.

Understanding \( \Delta T \) allows students to link observed temperature changes to their theoretical calculations accurately.

Thermal Energy

Thermal energy, often associated with heat flow and temperature changes, pertains to the energy transferred between systems due to differences in temperature. It is generally measured in joules (J) or kilojoules (kJ), particularly in calorimetry.
In the given exercise, the thermal energy released from burning magnesium was first calculated as:

• \( q = 3405.024\, \text{J} \)

and then converted into kilojoules as:

• \( 3.405\, \text{kJ} \).

This conversion highlights how the amount of energy can be measured on multiple scales.
Understanding thermal energy and its conversion is key to analyzing reactions and measuring the energy differences they produce.

Molar Mass

Molar mass is a fundamental concept in chemistry that represents the mass of one mole of a given substance. It is usually expressed in grams per mole (g/mol) and acts as a bridge between the mass of the substance and the amount of substance at the molecular level.
For magnesium, the molar mass is known to be 24.305 g/mol. In the exercise's context, knowing the molar mass allows for the conversion of heat given off per gram of magnesium to heat given off per mole of magnesium.
The specific step involved multiplying the heat per gram by the molar mass, which provided:

• \( 601.65\, \text{kJ/mol} \),

thereby offering an understanding of the energy released when an entire mole of magnesium combusts. Grasping the role of molar mass helps in performing conversions crucial for understanding molecular interactions and energy transformations in chemical reactions.