Question

Microwave ovens are highly efficient, compared to other cooking appliances. A 1100 -watt microwave oven, running at full power for 90 seconds, will raise the temperature of 1 cup of water \((225 \mathrm{mL})\) from \(20^{\circ} \mathrm{C}\) to \(67^{\circ} \mathrm{C} .\) As a rough measure of the efficiency of the microwave oven, compare its energy consumption with the energy required to raise the water temperature.

Short answer

The microwave oven's efficiency is approximately 44.6%.

Step-by-step solution

Calculate the Energy Consumed by the Microwave

The power of the microwave is given as 1100 watts, and it runs for 90 seconds. The energy consumed can be calculated using the formula \(E = P \times t\), where \(E\) is energy in joules, \(P\) is power in watts, and \(t\) is time in seconds.\[E = 1100 \times 90 = 99000 \text{ joules}\]

Calculate the Energy Required to Heat the Water

To calculate the energy required to raise the temperature of water, use the formula \(Q = m \cdot c \cdot \Delta T\), where \(Q\) is the heat energy in joules, \(m\) is the mass of the water, \(c\) is the specific heat capacity of water \((4.18 \, \text{J/g}^\circ\text{C})\), and \(\Delta T\) is the change in temperature.The mass \(m\) can be calculated using the density of water, which is approximately 1 g/mL, thus for 225 mL, the mass \(m = 225\) g.The change in temperature \(\Delta T = 67 - 20 = 47^\circ C\).Now, calculate \(Q\):\[Q = 225 \times 4.18 \times 47 = 44171.5 \, \text{joules}\]

Calculate the Efficiency of the Microwave

Efficiency can be calculated as the ratio of the energy required to heat the water to the energy consumed by the microwave, multiplied by 100 to get a percentage.\[\text{Efficiency } = \left(\frac{Q}{E} \right) \times 100 = \left(\frac{44171.5}{99000} \right) \times 100\approx 44.6\%\]

Interpret the Results

The microwave oven's efficiency is approximately 44.6%. This means that about 44.6% of the total energy consumed by the microwave is effectively used to raise the water's temperature.

Key concepts

Energy Consumption Calculation

When you want to find out how much energy an electrical appliance, like a microwave oven, consumes, you can use a simple formula. The energy consumed (\(E\)) in joules is the product of the power (\(P\)) in watts and the duration (\(t\)) the appliance is used, in seconds. This calculation will provide you with an estimate of the total energy used during operation.- **Formula**: \[E = P \times t\]Let's say you have a 1100-watt microwave oven, and it's running for 90 seconds. Using the formula:- **Calculation**: \[E = 1100 \times 90 = 99000 \text{ joules}\]Hence, while using such an appliance, it's clear that a considerable amount of energy is consumed for even a short period. Knowing this helps evaluate the cost-effectiveness and efficiency of using a microwave compared to other appliances.

Specific Heat Capacity of Water

Understanding the specific heat capacity of materials, like water, is crucial for calculating energy requirements in heating or cooling tasks. Specific heat capacity (\(c\)) is defined as the amount of energy needed to raise the temperature of 1 gram of a substance by 1 degree Celsius.For water, the specific heat capacity is relatively high at 4.18 Joules per gram per degree Celsius. This means that water can absorb a lot of heat without a significant rise in temperature. It's why water is often used in cooking and cooling processes.- **Example Calculation**: - **Mass of water (\(m\))**: In this exercise, we deal with 225 grams of water (assuming the density of water is 1 g/mL). - **Specific heat (\(c\))**: 4.18 J/g°C - **Temperature change (\(\Delta T\))**: As calculated, from 20°C to 67°C, which is 47°C.With these values, you can calculate the energy required to heat this amount of water:\[Q = m \cdot c \cdot \Delta T = 225 \cdot 4.18 \cdot 47 = 44171.5 \text{ joules}\]

Temperature Change Calculation

Another essential concept is calculating how much the temperature of a substance will change when a certain amount of energy is applied. This is fundamental to understanding how heating mechanisms work, especially in everyday appliances like microwaves.Here, the change in temperature (\(\Delta T\)) is simply the difference between the final temperature (\(T_f\)) and the initial temperature (\(T_i\)).In our example, the goal is to raise water temperature from 20°C to 67°C:- **Initial temperature (\(T_i\))**: 20°C- **Final temperature (\(T_f\))**: 67°C- **Temperature change (\(\Delta T\))**: \[\Delta T = T_f - T_i = 67 - 20 = 47 \text{°C}\]This calculation shows how much any substance needs to be heated or cooled to reach a desired temperature. By understanding these concepts, you gain a deeper insight into the workings of everyday technologies and the efficiency of heating systems.