Photons of infrared radiation are responsible for much of...

Photons of infrared radiation are responsible for much of the warmth we feel when holding our hands before a fire. These photons will also warm objects. How many infrared photons with a wavelength of $1.5 \times 10^{-6} {m}$ must be absorbed by the water to warm a cup of water $(175 {g})$ from $25.0^{\circ} {C}$ ?

Key Concepts

Photon Energy and Planck's Equation

This concept highlights that electromagnetic radiation is quantized into photons, each carrying a specific amount of energy. The energy of a photon is determined by its wavelength using the formula E = hc/?, where h is Planck's constant, c is the speed of light, and ? is the wavelength.

Energy-Photon Relationship

This concept links thermodynamics and quantum mechanics by determining how many photons a certain amount of energy equates to. Once the total energy required (from thermal energy calculations) and the energy per photon (from Planck's equation) are known, the number of photons can be found by dividing the total energy by the energy per photon.

Thermal Energy and Heat Capacity

This concept involves calculating the energy required to change the temperature of a substance using its mass and specific heat capacity. The equation Q = mc?T is used here, where Q is the required thermal energy, m is the mass, c is the specific heat, and ?T is the change in temperature.

Transcript

00:01 All right.

00:01 So for this problem, we're told that the photons coming off of fire can be used to warm both our hands and to heat other objects.

00:08 And they give us the wavelength of those photons, and then want us to calculate how many of those photons it'll take to heat a cup of water that weighs 175 grams.

00:19 And to heat that cup of water from 25 degrees celsius to 40 degrees celsius.

00:25 So the basic steps that we're going to have to go through with this problem are to figure out, one, how much energy is, provided by each of these photons, how much total energy it's going to take to make that temperature change in the water, and then to just figure out how many photons it takes to meet that energy change.

00:42 So i have the essential equations that are going to need written out here.

00:45 The first two are pretty standard light equations.

00:49 This last one is one that you may or may not have seen before.

00:51 This says that the heat change on an object is equal to its specific heat times its mass times the change in temperature that you're facilitating.

01:01 So for starters, let's just go through and calculate the amount of heat needed to change the temperature, to raise the temperature of water from 25 degrees celsius to 40 degrees celsius.

01:11 So to do that, we have the heat of that change is going to be able to the specific heat of water, which in joules is going to be 4 .186 joules per gram degree celsius...

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