Water is exposed to infrared radiation of wavelength 2.5×10−4 cm. Assume that all the radiation is absorbed and converted to heat.

How many photons will be required to raise the temperature of 2.5 g of water by 2.5 K?

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I tried solving this problem on my own, and I got 3.3 x 10^-30 photons, which was incorrect.

2.5g x 4.184J/g*K x 2.5K = 26.15 J

First step is ok.

E= 6.626E-34 J.s x 3E8 m/2.4E-6 m = 8.285 E-32
The numbers are ok that you substituted but the answer is 8.2825E-20 J. That't too many significant figure but I'll round at the end.

26.15/8.285E-32 = 3.15E-32 - incorrect answer
3.157E20

That catches two errors in two steps. I suspect, also, that you are reporting too many significant figures. I would round to 2 places since the 2.4, 2.5 and 2.5 limits it to two s.f. So I round round to 3.2E20
You may also want to check the problem. You have 2.5E-4 cm in the problem but used 2.4 in the work. I think I remember the number from your first post, however, as being 2.4E-4 cm wavelength.

I can't buy into 3.29E-20. 3.29E+20 might be ok but certainly not -20. A photon of 2.4E-6 meters is 8.28E-20 J and

8.28E-20 J/photon x 3.29E-20 photons provides 2.72E-39 which doesn't come close to providing enough energy to heat 2.5 g H2O by 2.5K.

2.5g x 4.184J/g*K x 2.5K = 26.15 J

E= 6.626E-34 J.s x 3E8 m/2.4E-6 m = 8.285 E-32
26.15/8.285E-32 = 3.15E-32 - incorrect answer

I did get the correct answer. It was actually 3.29E-20. Thank you for your help.

Well, well, well, looks like we've got a photon party going on here! Let's crunch some numbers and see if we can shed some light on this question.

To solve this problem, we need to determine how much energy is required to raise the temperature of 2.5 g of water by 2.5 K. We know that each photon carries energy given by E = hc/λ, where h is Planck's constant, c is the speed of light, and λ is the wavelength.

Given that the wavelength is 2.5×10^(-4) cm, we can convert it to meters by multiplying by 10^(-2), giving us λ = 2.5×10^(-6) m. Now, let's plug in the numbers into our energy equation:

E = (6.626 × 10^(-34) Js)(3.0 × 10^8 m/s)/(2.5×10^(-6) m)

After doing some mathematical magic, we find that each photon carries approximately 8.0 × 10^(-19) Joules of energy.

To find the total energy required to raise the temperature of water by 2.5 K, we can use the formula Q = mcΔT, where Q is the heat energy, m is the mass, c is the specific heat capacity of water, and ΔT is the change in temperature. Here, m is given as 2.5 g and ΔT is 2.5 K.

Now, plug in the numbers and calculate the heat energy:

Q = (2.5 g)(4.18 J/g·K)(2.5 K)

We find that the heat energy required is approximately 26 J.

To find the number of photons required, we can divide the total heat energy required by the energy carried by a single photon:

Number of photons = 26 J / (8.0 × 10^(-19) J/photon)

And voila! When we do the math, we discover that approximately 3.25 × 10^(19) photons are needed to raise the temperature of 2.5 g of water by 2.5 K.

So, my dear friend, it seems like you were just a few photons away from the correct answer! Better luck next time, and don't forget to keep shining bright!

To find the number of photons required to raise the temperature of water, we can use the equations related to the energy carried by a single photon and the specific heat capacity of water.

The energy carried by a single photon is given by the equation:

E = hc/λ

Where:
E is the energy of a single photon,
h is Planck's constant (6.63 x 10^(-34) J·s),
c is the speed of light (3.00 x 10^8 m/s),
λ is the wavelength of the radiation.

Now, we can calculate the energy per photon using the given wavelength:

λ = 2.5 x 10^(-4) cm = 2.5 x 10^(-6) m

E = (6.63 x 10^(-34) J·s * 3.00 x 10^8 m/s) / (2.5 x 10^(-6) m)
E ≈ 7.956 x 10^(-19) J

Next, we need to calculate the total energy required to raise the temperature of the water. The equation for heat energy is:

Q = mcΔT

Where:
Q is the heat energy required,
m is the mass of the water (2.5 g),
c is the specific heat capacity of water (4.18 J/g·K),
ΔT is the change in temperature (2.5 K).

Now, let's substitute the values:

Q = (2.5 g * 4.18 J/g·K * 2.5 K)
Q ≈ 26.125 J

To find the number of photons, we divide the total energy by the energy per photon:

Number of photons = Q / E
Number of photons ≈ 26.125 J / 7.956 x 10^(-19) J

Using a calculator, we get:

Number of photons ≈ 3.28 x 10^19 photons

Therefore, the correct answer is approximately 3.28 x 10^19 photons.

I worked this earlier. If you will show exactly what you did from my workup perhaps I can figure out what's wrong.