Problem 23 - Chemistry and Chemical Reactivity

Question

Ethanol, \(\mathrm{C}_{2} \mathrm{H}_{5} \mathrm{OH},\) boils at \(78.29^{\circ} \mathrm{C} .\) How much energy, in joules, is required to raise the temperature of \(1.00 \mathrm{kg}\) of ethanol from \(20.0^{\circ} \mathrm{C}\) to the boiling point and then to change the liquid to vapor at that temperature? (The specific heat capacity of liquid ethanol is \(2.44 \mathrm{J} / \mathrm{g} \cdot \mathrm{K},\) and its enthalpy of vaporization is \(855 \mathrm{J} / \mathrm{g} .\) )

Step-by-Step Solution

Answer

The total energy required is approximately 997,076 J.

Step 1: Calculate Temperature Change

First, find the temperature change needed to heat the ethanol to its boiling point. Subtract the initial temperature from the boiling temperature: \( 78.29^{\circ} \mathrm{C} - 20.0^{\circ} \mathrm{C} = 58.29^{\circ} \mathrm{C} \). We will use this temperature difference (\( \Delta T \)) in our calculation of thermal energy required.

Step 2: Calculate Mass in Grams

Convert the mass of ethanol from kilograms to grams since the given specific heat capacity and enthalpy of vaporization are in \( \mathrm{g} \). Given the mass is \( 1.00 \mathrm{kg} \), then \( 1.00 \mathrm{kg} = 1000 \mathrm{g} \).

Step 3: Calculate Energy to Heat Ethanol

Use the formula to calculate the energy required to increase the temperature to boiling: \( q = m \cdot c \cdot \Delta T \), where: \( m = 1000 \mathrm{g} \) (mass), \( c = 2.44 \mathrm{J/g \cdot K} \) (specific heat capacity), and \( \Delta T = 58.29^{\circ} \mathrm{C} \) (temperature change). Substitute these values to get: \( q = 1000 \cdot 2.44 \cdot 58.29 = 142075.6 \mathrm{J} \).

Step 4: Calculate Energy to Vaporize Ethanol

To find the energy required to vaporize the ethanol, use the formula: \( q = m \cdot \Delta H_{vap} \), where \( m = 1000 \mathrm{g} \) (mass) and \( \Delta H_{vap} = 855 \mathrm{J/g} \) (enthalpy of vaporization). Substitute these values to get: \( q = 1000 \cdot 855 = 855000 \mathrm{J} \).

Step 5: Calculate Total Energy Required

Add the energy required to heat the ethanol and the energy required to vaporize it: \( 142075.6 \mathrm{J} + 855000 \mathrm{J} = 997075.6 \mathrm{J} \). This is the total energy required.

Key Concepts

Understanding Specific Heat Capacity

Specific heat capacity is the amount of energy needed to raise the temperature of 1 gram of a substance by 1 Celsius degree or 1 Kelvin. It's a crucial concept when studying how substances heat up or cool down.
In this exercise, ethanol has a specific heat capacity of \(2.44 \, \text{J/g} \cdot \text{K}\). This means that to increase the temperature of 1 gram of ethanol by 1 Kelvin, you would need 2.44 Joules of energy.

• To calculate the energy needed to heat a given mass of a substance, use the formula \(q = m \cdot c \cdot \Delta T\), where \(m\) is the mass, \(c\) is the specific heat capacity, and \(\Delta T\) is the temperature change.
• In the given problem, we calculated this energy for ethanol to reach its boiling point.

This concept is vital in thermodynamics as it influences how different substances respond to heat input.

Calculating Temperature Change

Temperature change (\(\Delta T\)) is the difference between the final and initial temperatures of a substance. It tells you how much a substance's temperature needs to be increased or decreased.
For ethanol in this exercise, the temperature change needed to reach the boiling point is calculated as \(78.29^{\circ} \text{C} - 20.0^{\circ} \text{C} = 58.29^{\circ} \text{C}\).

• This temperature difference is crucial for calculating the energy required to raise the ethanol's temperature.
• The formula \(q = m \cdot c \cdot \Delta T\) uses this \(\Delta T\) to find out how much energy is needed.

Understanding how to calculate and use temperature change is fundamental in solving problems involving heat transfer.

Exploring Phase Transition

Phase transition occurs when a substance changes from one state of matter to another, such as from liquid to gas. This process requires or releases energy without changing the temperature of the substance.
In our problem, ethanol transitions from liquid at its boiling point to vapor. The energy needed for this phase change is called the enthalpy of vaporization.

• Ethanol's enthalpy of vaporization is \(855 \, \text{J/g}\). This means 855 Joules are needed to vaporize 1 gram of ethanol.
• To calculate the total energy for vaporization, use the formula \(q = m \cdot \Delta H_{vap}\).

This concept helps in understanding processes like boiling and condensation, which are essential in various scientific and industrial applications.