Problem 113 - General Chemistry: Principles and Modern Applications

Question

Several factors are involved in determining the cooking times required for foods in a microwave oven. One of these factors is specific heat. Determine the approximate time required to warm \(250 \mathrm{mL}\) of chicken broth from \(4^{\circ} \mathrm{C}\) (a typical refrigerator temperature) to \(50^{\circ} \mathrm{C}\) in a \(700 \mathrm{W}\) microwave oven. Assume that the density of chicken broth is about \(1 \mathrm{g} / \mathrm{mL}\) and that its specific heat is approximately \(4.2 \mathrm{Jg}^{-1}\) \(^{\circ} \mathrm{C}^{-1}\).

Step-by-Step Solution

Answer

It takes approximately 69 seconds to warm the chicken broth.

Step 1: Calculate the mass

The first step is to calculate the mass of the chicken broth. The density of the chicken broth is given as \( 1 \, \text{g/mL} \), and the volume is \( 250 \, \text{mL} \). Therefore, the mass \( m \) can be calculated using the formula \( \text{mass} = \text{density} \times \text{volume} \). This results in \( m = 1 \, \text{g/mL} \times 250 \, \text{mL} = 250 \, \text{g} \).

Step 2: Calculate the temperature change

The next step is to calculate the temperature change \( \Delta T \). This is done by subtracting the initial temperature from the final temperature. This results in \( \Delta T = 50^{\circ}C - 4^{\circ}C = 46^{\circ}C \).

Step 3: Calculate the heat required

The heat \( q \) required to warm the chicken broth can now be calculated using the formula \( q = m \cdot c \cdot \Delta T \). The specific heat \( c \) of the chicken broth is given as \( 4.2 \, \text{Jg}^{-1}\text{C}^{-1} \), the mass \( m \) is 250 g, and the temperature change \( \Delta T \) is 46 C. This results in \( q = 250 \, \text{g} \times 4.2 \, \text{Jg}^{-1}\text{C}^{-1} \times 46^{\circ}C = 48300 \, \text{J} \).

Step 4: Calculate the time

The final step is to calculate the time \( t \) it takes to warm the chicken broth. The power \( P \) of the microwave is given as \( 700 \, \text{W} = 700 \, \text{J/s} \). The time can be calculated using the formula \( t = \frac{q}{P} \). This results in \( t = \frac{48300 \, \text{J}}{700 \, \text{J/s}} = 69 \, \text{s} \). Therefore, it takes approximately 69 seconds to warm the chicken broth.

Key Concepts

Microwave Cooking

Microwave ovens use a unique method to heat food quickly and efficiently. They do this by using electromagnetic waves which interact with water molecules in food. This creates heat through a process called dielectric heating.
This type of cooking is different from traditional methods, where heat gradually moves from the outside to the inside. In a microwave, the entire food item gets heated fairly uniformly at once.
One of the key factors determining how long it takes to cook or heat something in a microwave is the power of the microwave oven. Common household microwaves range between 600 to 1200 watts.

• The higher the power, the quicker food can be heated.
• Always consider the volume and properties of the food being cooked, such as its water content or specific heat, as they significantly affect cooking time.

Understanding how microwaves work is crucial for efficient use and even cooking.

Thermal Energy

Thermal energy is the energy that comes from the heat present in a substance. When you heat something in a microwave, you are essentially increasing its thermal energy.
This process is key for changing the temperature of a substance, as higher thermal energy corresponds to a higher temperature.
In microwaving chicken broth, thermal energy is increased by transferring energy from the microwave's electromagnetic waves to the water molecules in the broth.

• The amount of thermal energy needed is calculated by knowing the specific heat, mass, and the temperature change of the substance.
• Materials with high specific heat will require more energy to change temperature.

Grasping the concept of thermal energy helps in controlling and predicting heating processes.

Temperature Change

Temperature change is a measure of how much the temperature of a substance has increased or decreased during heating or cooling.
In our exercise example, the initial temperature of the chicken broth is 4°C, and it needs to be warmed to 50°C, resulting in a temperature change of 46°C.
Calculating the temperature change involves simple subtraction: 1. Final Temperature (50°C) 2. Initial Temperature (4°C)
This change is an essential part of calculating the thermal energy required because energy is directly related to how much the temperature needs to change.

• Knowledge of initial and final temperatures is crucial for correctly computing heat energy in thermodynamics.
• Always ensure units are correctly converted and applied in calculations.

Understanding temperature change helps in effectively using formulas to find needed heat energy.

Power Calculation

Power calculation is an important step when determining how long it will take for an appliance like a microwave to perform a task, such as heating food.
Power is defined as the rate at which energy is transferred or converted, measured in watts (W). In our case, the microwave has a power of 700 W, meaning it can transfer 700 joules of energy per second.
Using the formula for time, given by \( t = \frac{q}{P} \), where \( q \) is the thermal energy required and \( P \) is the power, allows you to calculate how long it takes to heat something.

• The higher the power, the shorter the time needed to achieve the desired temperature.
• Accurate power calculation ensures food is neither undercooked nor overheated.
• Practical knowledge of power helps in setting correct timing for cooking processes.

By understanding power calculations, you make informed decisions on cooking times and temperatures.