A certain microwave oven delivers 750 . watts (joule/s) of...

A certain microwave oven delivers 750 . watts (joule/s) of power to a coffee cup containing $50.0 \mathrm{~g}$ of water at $25.0^{\circ} \mathrm{C}$. If the wavelength of microwaves in the oven is $9.75 \mathrm{~cm}$, how long does it take, and how many photons must be absorbed, to make the water boil? The specific heat capacity of water is $4.18 \mathrm{~J} /{ }^{\circ} \mathrm{C} \cdot \mathrm{g}$ and assume only the water absorbs the energy of the microwaves.

Key Concepts

Specific Heat Capacity

Specific heat capacity is the amount of energy required to raise the temperature of a given mass of a substance by one degree Celsius (or Kelvin). It is crucial in calculating the total heat energy needed to produce a desired temperature change, as seen in the heating of water in the problem.

Energy Calculation for Temperature Change

This concept involves using the formula Q = mc?T, where Q is the heat energy required, m is the mass of the substance, c is the specific heat capacity, and ?T is the change in temperature. It allows the estimation of the total energy input needed to heat a substance from one temperature to another.

Power and Energy Relationship

Power is defined as the rate at which energy is transferred or converted, usually measured in watts (joules per second). In a scenario where a device delivers a constant power output, the total energy delivered over time can be calculated by multiplying power by the time interval, enabling the determination of the heating duration.

Photon Energy (Planck's Equation)

Photon energy is determined using Planck's relation, E = hc/?, where E is the energy of a photon, h is Planck's constant, c is the speed of light, and ? is the wavelength. This relationship is critical for calculating how many photons must be absorbed to accumulate the total energy required for a physical process such as heating water.

Transcript

00:01 For this question, we're trying to figure out how long it's going to take to heat up this water to boiling and how many photons are going to be absorbed.

00:07 So first, let's figure out the amount of energy or the amount of joules it takes to heat 50 grams of water from 25 degrees celsius to its boiling point, our final temperature, which is 100 degrees celsius.

00:23 The equation that we're going to use is q, which is our energy, is equal to m, the mass, times s, the specific heat, times the change in temperature.

00:36 So let's plug in our variables for the water.

00:41 So our q is going to be equal to the mass, 50 grams, times the specific heat, 4 .18 joules per grams times celsius, times the change in temperature.

01:02 So the final temperature, 100 degrees celsius, minus the initial temperature of 25 degrees celsius.

01:11 And then type that into your calculator and you should get 1 .57 times 10 to the fourth joules.

01:23 So that's the amount of energy that it's going to take to heat up that water to boiling.

01:30 So we're going to use this to figure out the other two questions.

01:33 So first, how long would it take? so we're told that the microwave can deliver 750 joules per second.

01:42 So how many seconds is it going to take to deliver 1 .57 times 10 to the fourth joules? so we can say for every one second, the microwave gives off 750 joules.

02:00 We'll take that times the amount of joules that we need, 1 .57 times 10 to the fourth joules.

02:08 Joules cancel out.

02:09 Type that into your calculator.

02:11 One divided by 750 times 1 .57 times 10 to the fourth.

02:16 And you should get 20 .9.

02:22 And then our units are seconds.

02:26 So it's going to take 20 .9 seconds to heat up that water...