# Question #2e81a

##### 1 Answer

#### Explanation:

The first thing to do here is determine how much energy in the form of heat is required to increase the temperature of that sample of coffee from

Your tool of choice here will be the following equation

#color(blue)(|bar(ul(color(white)(a/a)q = m * c * DeltaTcolor(white)(a/a)|)))" "# , where

*change in temperature*, defined as the difference between the **final temperature** and the **initial temperature**

Notice that the problem provides you with the *volume* of the sample. Use the coffee's **density** to find the **mass** of this sample

#225 color(red)(cancel(color(black)("mL"))) * overbrace("0.997 g"/(1color(red)(cancel(color(black)("mL")))))^(color(purple)("given density of coffee")) = "224.325 g"#

In you case, the *change in temperature* will be equal to

#DeltaT = 62.0^@"C" - 25.0^@"C" = 37.0^@"C"#

The specific heat of coffee is given to you in **equivalent** to *degrees Celsius* to *Kelvin*.

Plug in your values in the above equation to find the heat needed to heat the coffee

#q = 224.325 color(red)(cancel(color(black)("g"))) * 4.184"J"/(color(red)(cancel(color(black)("g")))color(red)(cancel(color(black)(""^@"C")))) * 37color(red)(cancel(color(black)(""^@"C")))#

#q = "34,727.3 J"#

Your goal now will be to find the energy of a **single photon** of wavelength **Planck - Einstein relation**, which states that the energy of a photon is proportional to its *frequency*

#color(blue)(|bar(ul(color(white)(a/a)E = h * nucolor(white)(a/a)|)))#

Here

**Planck's constant**, equal to

Now, frequency and wavelength have an **inverse relationship** described by the equation

#color(blue)(|bar(ul(color(white)(a/a)nu * lamda = c color(white)(a/a)|)))#

Here

*speed of light* in a vacuum, usually given as

Use this equation to find the frequency of a photon that has a wavelength of *meters per second*, so make sure that you convert the wavelength from *centimeters* to *meters*

#lamda * nu = c implies nu = c/(lamda)#

Plug in your value to get

#nu = (3 * 10^8 color(red)(cancel(color(black)("m")))"s"^(-1))/(12.4 * 10^(-2)color(red)(cancel(color(black)("m")))) = 2.419 * 10^9"s"^(-1)#

This means that the energy of a **single photon** of microwave radiation will be

#E = 6.626 * 10^(-34)"J" color(red)(cancel(color(black)("s"))) * 2.419 * 10^9color(red)(cancel(color(black)("s"^(-1))))#

#E = 1.603 * 10^(-24)"J"#

Now all you have to do is figure out **how many photons** of microwave radiation are needed to get a total energy of

#"34,727.3" color(red)(cancel(color(black)("J"))) * "1 photon"/(1.603 * 10^(-24)color(red)(cancel(color(black)("J")))) = color(green)(|bar(ul(color(white)(a/a)2.17 * 10^(28)"photons"color(white)(a/a)|)))#

The answer is rounded to three **sig figs**.