Enormous numbers of microwave photons are needed to warm macroscopic samples of matter. A portion of soup containing 177 g of water is heated in a microwave oven from 20degrees Celsius to 98 degrees Celsius, with readiation of wavelength 1.55x10^-2m. How many photons are absorbed by the water in the soup?

Calculate q for the heat required to heat the soup.

q = mass x specific heat H2O x delta T.
q = ?? in Joules.
Then use E = hc/wavelength to determine the energy per photon. Go from there. Post your work if you get stuck. Check my thinking.

ok. whew. just put in my answer into webassign and i got it correct.

Ill show the example with the numbers I was given. That way you can actually do the problem with your numbers.

I found the way to do this at the link below.

so my water mass was 360 g
my temp went from 28C to 93C
the radiation of wavelength is 1.55 x 10-2 m (same)

k so the first thing to do is find the work needed.
Q=mcΔT
this is mass times (4.184J/g°C) ΔT. so (360)(4.184)(93-28) = 97905.6 J. The 4.184 changes things to Angstroms. The overall equation will have the units Joules.

k so we also have the equations
E'=hv and c=λv
h = 6.626 x 10^-34
c = 3 x 10^8
solve and replace so that somehow you get this
Q (97905.6 J) / (hc/1.55 x 10^-2)

so 97905.6/((6.626e-34)(3e8)/(1.55e-2))
I got 7.6e27
webassign verified it.

Well, let me just say, that's one enthusiastic soup you've got there! So, let's calculate the number of photons it takes to warm it up.

To start, we need to calculate the energy of each photon using the formula E = hc/λ, where:
- E is the energy of each photon,
- h is Planck's constant (approximately 6.626 x 10^-34 J·s),
- c is the speed of light (approximately 3.0 x 10^8 m/s), and
- λ is the wavelength of the radiation (given as 1.55 x 10^-2 m).

Now we can calculate the energy of each photon:
E = (6.626 x 10^-34 J·s)(3.0 x 10^8 m/s) / (1.55 x 10^-2 m)
E ≈ 1.28 x 10^-19 J

Next, we can calculate the total energy needed to raise the temperature of the water:
ΔE = mcΔT, where:
- ΔE is the change in energy,
- m is the mass of the water (177 g, which is also 177 x 10^-3 kg),
- c is the specific heat of water (approximately 4.18 J/g°C),
- ΔT is the change in temperature (98°C - 20°C = 78°C).

ΔE = (177 x 10^-3 kg)(4.18 J/g°C)(78°C)
ΔE ≈ 58.78 J

Now, to find the number of photons, we divide the total energy by the energy of each photon:
Number of photons = ΔE / E
Number of photons ≈ (58.78 J) / (1.28 x 10^-19 J)

Hold on to your clown hat because here comes the answer... it's approximately 4.59 x 10^20 photons!

That's a whole lot of photons just to heat up some soup! It seems like they're having quite the party in there.

To calculate the number of photons absorbed by the water in the soup, we need to use the equation:

E = nhν

Where:
E is the energy of the photons,
n is the number of photons,
h is Planck's constant (6.626 x 10^-34 J.s), and
ν is the frequency of the radiation.

First, let's find the frequency (ν) of the radiation. We can use the speed of light (c) and the wavelength (λ) of the radiation:

c = λν

Rearranging the equation, we find:

ν = c / λ

Given:
λ = 1.55 x 10^-2 m
c = 3 x 10^8 m/s

Substituting the values into the equation, we can find the frequency (ν):

ν = (3 x 10^8 m/s) / (1.55 x 10^-2 m)
ν = 1.935 x 10^10 Hz

Now, let's calculate the energy (E) of each photon:

E = hν

Substituting the values:

E = (6.626 x 10^-34 J.s) x (1.935 x 10^10 Hz)
E = 1.281 x 10^-23 J

Next, we need to calculate the energy change (ΔE) in the water. We can use the equation:

ΔE = mcΔT

Where:
m is the mass of the water (177 g),
c is the specific heat capacity of water (4.18 J/g°C), and
ΔT is the change in temperature.

Given:
m = 177 g
c = 4.18 J/g°C
ΔT = (98°C - 20°C) = 78°C

Substituting the values:

ΔE = (177 g) x (4.18 J/g°C) x (78 °C)
ΔE = 5.18 x 10^4 J

Now, to find the number of photons (n), we can rearrange the equation:

n = ΔE / E

Substituting the calculated values:

n = (5.18 x 10^4 J) / (1.281 x 10^-23 J)
n = 4.05 x 10^27 photons

Therefore, approximately 4.05 x 10^27 photons are absorbed by the water in the soup.

its so easy