Question

You score this massive \(1 \mathrm{~kg}\) burrito but decide to put it in the refrigerator to eat later. It comes out at \(5^{\circ} \mathrm{C}\), and you want to heat it in the microwave up to \(75^{\circ} \mathrm{C}\) before eating it. If the microwave puts energy into the burrito at a rate of \(700 \mathrm{~W} .^{61}\) How long should you run the microwave for a high-water- content burrito having an effective specific heat capacity of \(3,000 \mathrm{~J} / \mathrm{kg} /{ }^{\circ} \mathrm{C} ?\)

Short answer

Answer: The microwave should be run for 300 seconds (5 minutes) to heat the high-water-content burrito from 5°C to 75°C.

Step-by-step solution

Calculate the change in temperature

To find the change in temperature, subtract the initial temperature from the final temperature. Change in temperature = Final temperature - Initial temperature Change in temperature = \(75^{\circ}\mathrm{C} - 5^{\circ}\mathrm{C} = 70^{\circ}\mathrm{C}\)

Calculate the energy required to heat the burrito

Use the formula: Energy = Specific heat capacity × mass × change in temperature Energy = \(3,000\frac{\mathrm{J}}{\mathrm{kg\cdot}^\circ\mathrm{C}} \times 1\mathrm{~kg} \times 70^{\circ}\mathrm{C}\) Energy = \(210,000\mathrm{~J}\)

Calculate the time to heat the burrito

Divide the energy required by the power of the microwave to find the time needed. Time = \(\frac{Energy}{Power}\) Time = \(\frac{210,000\mathrm{~J}}{700\mathrm{~W}}\) Time = \(300\mathrm{~seconds}\) So, the microwave should be run for 300 seconds (5 minutes) to heat the high-water-content burrito from \(5^{\circ}\mathrm{C}\) to \(75^{\circ}\mathrm{C}\).

Key concepts

Specific Heat Capacity

Imagine you're warming up your favorite food in the microwave. In science, we measure how much energy it takes to raise the temperature of different materials—a concept known as the specific heat capacity. This is basically a number that tells us how 'reluctant' a substance is to change its temperature. Think of it like a food's stubbornness to warm up: some items heat quickly, while others take their time. The specific heat capacity is the amount of energy, in joules, required to raise 1 kilogram of a substance by 1 degree Celsius (or Kelvin, as the size of the degree is the same in both scales).
For our burrito with high water content, its specific heat capacity is quite high because water takes a lot of energy to heat up. This scenario applies not only to microwaving burritos but also to a variety of other situations in physics and engineering, making specific heat capacity a fundamental concept.

Energy Transfer

Energy transfer is how energy moves from one object to another or changes form. For instance, when you heat up a burrito in the microwave, energy is being transferred from the microwave's electromagnetic waves to the burrito, increasing its internal energy and, as a result, its temperature. Understanding energy transfer is crucial because it allows us to calculate things like how long it will take to cook food or how much fuel you need to heat your home. In the context of thermodynamics, three main types of energy transfers exist: conduction (through direct contact), convection (via fluids), and radiation (like our microwave's method). Each type of energy transfer plays a role in how we harness and utilize energy in everyday life.

Temperature Change Calculation

Calculating temperature change is like figuring out how much warmer or colder something gets. This can be as simple as monitoring the temperature difference before and after heating your burrito—as in our exercise. But, how does this relate to energy? We can use the formula involving specific heat capacity, mass, and temperature change to calculate the energy required to achieve this temperature difference.
The process is straightforward: subtract the initial temperature from the final temperature to find the change. Then, multiply this by the specific heat capacity and the mass of the object. What you get is the energy needed. In practical applications, this helps us determine the efficiency of heating systems, predict weather patterns, or in our case, ensure your burrito is perfectly toasty warm. By understanding temperature change calculation, students can better grasp concepts related to energy usage and efficiency.