Problem 63 - General Chemistry

Question

You wish to heat water to make coffee. How much heat (in joules) must be used to raise the temperature of \(0.180 \mathrm{~kg}\) of tap water (enough for one cup of coffee) from \(19^{\circ} \mathrm{C}\) to \(96^{\circ} \mathrm{C}\) (near the ideal brewing temperature)? Assume the specific heat is that of pure water, \(4.18 \mathrm{~J} /\left(\mathrm{g} \cdot{ }^{\circ} \mathrm{C}\right)\).

Step-by-Step Solution

Answer

The heat required is approximately 58045.2 joules.

Step 1: Identify the Given Parameters

In this problem, we are given the mass of water, which is \(0.180 \text{ kg}\) or \(180 \text{ g}\) since 1 kg = 1000 g. The initial temperature \(T_i\) is \(19^{\circ} \text{C}\) and the final temperature \(T_f\) is \(96^{\circ} \text{C}\). The specific heat capacity of water is given as \(4.18 \text{ J/g°C}\).

Step 2: Calculate the Temperature Change

To find the heat required, we need to calculate the change in temperature: \(\Delta T = T_f - T_i\). Substitute the given values: \(\Delta T = 96^{\circ} \text{C} - 19^{\circ} \text{C} = 77^{\circ} \text{C}\).

Step 3: Apply the Heat Equation

The heat needed can be calculated using the formula: \( Q = mc\Delta T \) where \(m\) is the mass of the water in grams, \(c\) is the specific heat capacity, and \(\Delta T\) is the temperature change. Substitute the known values into the formula: \( Q = 180 \text{ g} \times 4.18 \text{ J/g°C} \times 77^{\circ} \text{C} \).

Step 4: Perform the Calculation

Now calculate the heat \(Q\):\[ Q = 180 \times 4.18 \times 77 \] \[ Q = 58045.2 \text{ J} \]Therefore, the amount of heat needed is \(58045.2 \text{ joules}\).

Key Concepts

Specific Heat Capacity

When heating substances, knowing the specific heat capacity is essential. It defines how much heat energy is needed to raise the temperature of one gram of a substance by one degree Celsius. In our coffee example, we used pure water, which has a specific heat capacity of \(4.18 \text{ J/g°C}\). This means every gram of water requires \(4.18\) joules to raise its temperature by \(1^{\circ} \text{C}\). This constant is crucial in determining how much total energy is needed.

Some points to understand about specific heat capacity are:

• It varies between different substances based on their intermolecular forces.
• Substances with higher specific heat capacities can store more heat without a large change in temperature.
• A specific heat capacity is usually given in units of \(\text{J/g°C}\) or \(\text{J/kg°C}\), so it's important to match these units with your mass measurements.

Temperature Change

Understanding temperature change is crucial when calculating the heat required for processes like making coffee. The temperature change (\(\Delta T\)) represents the difference between the final and initial temperatures of a substance. For our problem, the initial temperature of the water was \(19^{\circ} \text{C}\) and the final was \(96^{\circ} \text{C}\). By subtracting the initial from the final temperature, we found the water needed to change \(77^{\circ} \text{C}\) to reach the ideal coffee temperature.

Here's what you need to keep in mind about temperature change:

• Temperature change is calculated as \(\Delta T = T_f - T_i\), where \(T_f\) is the final temperature and \(T_i\) is the initial temperature.
• The result of \(\Delta T\) can be positive or negative depending on whether the substance is being heated or cooled.
• Accurate measurement of \(\Delta T\) is paramount for precise heat calculations.

Joules

Joules serve as the unit of measurement for energy, specifically in the context of heat energy in this exercise. When discussing how much energy is required to heat water, we're talking about how many joules are needed to change the temperature. In our exercise, we found \(58045.2\) joules were necessary to heat the water to the ideal brewing temperature.

Understanding joules is key in thermodynamics and energy calculations. Consider these important facts:

• 1 Joule is equivalent to the energy transferred when a force of one newton moves an object one meter.
• In terms of heat, joules measure the energy needed to increase the temperature of substances.
• This measurement helps in comparing the energy usage and efficiency of different heating methods and substances.

This implies a real-world application by determining how much energy you need to use in various daily tasks.