So,
We are given:
mass of plate = 0.50 kg
specific heat capacity = 840 J/(kg °C)
wavelength of infrared = 18 * 10^-5 m
wavelength of blue light = 4.6 * 10^-7 m
ΔT = +2.0 °C
Additionally, we should already know:
q = mcΔT, where q is the energy absorbed (+) or released (-) by the system, m is the mass of the system, c is the specific heat capacity, and deltaT is the change in temperature.
E = hv, where E is the energy in J, h is Planck's constant, and v is the frequency of the light
c = wv, where c is the speed of light, w is the wavelength, and v is the frequency
We need to find the number of infrared photons and the number of blue photons required to result in the given temperature change.
Key idea: if we can find the energy if each photon, we can find the number of photons required to raise the temperature of the plate.
I will start with the infrared photons. We can do this with the assumed equations. We want to find E. We have w, and we should have h and c.
First, let's find v.
![c = w_{infrared}v_{infrared}](https://tex.z-dn.net/?f=c%20%3D%20w_%7Binfrared%7Dv_%7Binfrared%7D)
![v=\frac{c}{w}](https://tex.z-dn.net/?f=v%3D%5Cfrac%7Bc%7D%7Bw%7D)
![v_{infrared}=\frac{3.00*10^8 \frac{m}{s}}{18*10^{-5} m}=1.67*10^{12} s^{-1}](https://tex.z-dn.net/?f=v_%7Binfrared%7D%3D%5Cfrac%7B3.00%2A10%5E8%20%5Cfrac%7Bm%7D%7Bs%7D%7D%7B18%2A10%5E%7B-5%7D%20m%7D%3D1.67%2A10%5E%7B12%7D%20s%5E%7B-1%7D)
Next, let's find E.
![E_{infrared}=hv_{infrared}](https://tex.z-dn.net/?f=E_%7Binfrared%7D%3Dhv_%7Binfrared%7D)
![E=(6.626*10^{-34} J*s)(1.67*10^{12} s^{-1})=1.10*10^{-21}\ J](https://tex.z-dn.net/?f=E%3D%286.626%2A10%5E%7B-34%7D%20J%2As%29%281.67%2A10%5E%7B12%7D%20s%5E%7B-1%7D%29%3D1.10%2A10%5E%7B-21%7D%5C%20J)
Now, let's find the amount of energy absorbed by the plate.
![q=mc\Delta T](https://tex.z-dn.net/?f=q%3Dmc%5CDelta%20T)
![q=0.50 kg * 840 \frac{J}{kg\ ^o C}*2.0\ ^o C = 840\ J](https://tex.z-dn.net/?f=q%3D0.50%20kg%20%2A%20840%20%5Cfrac%7BJ%7D%7Bkg%5C%20%5Eo%20C%7D%2A2.0%5C%20%5Eo%20C%20%3D%20840%5C%20J)
Now, we can find the number of infrared photons required.
![photons\ required = \frac{energy\ absorbed}{energy\ per\ infrared\ photon}](https://tex.z-dn.net/?f=photons%5C%20required%20%3D%20%5Cfrac%7Benergy%5C%20absorbed%7D%7Benergy%5C%20per%5C%20infrared%5C%20photon%7D)
![photons=\frac{840\ J}{1.10*10^{-21} \frac{J}{infrared\ photon}}=7.61*10^{23} \ infrared\ photons](https://tex.z-dn.net/?f=photons%3D%5Cfrac%7B840%5C%20J%7D%7B1.10%2A10%5E%7B-21%7D%20%5Cfrac%7BJ%7D%7Binfrared%5C%20photon%7D%7D%3D7.61%2A10%5E%7B23%7D%20%5C%20infrared%5C%20photons)
So the number of infrared photons required is 7.57 * 10^23 photons.
We can do a similar procedure for the blue light.
Find v.
![c = w_{blue}v_{blue}](https://tex.z-dn.net/?f=c%20%3D%20w_%7Bblue%7Dv_%7Bblue%7D)
![v=\frac{c}{w}](https://tex.z-dn.net/?f=v%3D%5Cfrac%7Bc%7D%7Bw%7D)
![v_{infrared}=\frac{3.00*10^8 \frac{m}{s}}{4.6*10^{-7} m}=6.52*10^{14} s^{-1}](https://tex.z-dn.net/?f=v_%7Binfrared%7D%3D%5Cfrac%7B3.00%2A10%5E8%20%5Cfrac%7Bm%7D%7Bs%7D%7D%7B4.6%2A10%5E%7B-7%7D%20m%7D%3D6.52%2A10%5E%7B14%7D%20s%5E%7B-1%7D)
Find E.
![E_{blue}=hv_{blue}](https://tex.z-dn.net/?f=E_%7Bblue%7D%3Dhv_%7Bblue%7D)
![E=(6.626*10^{-34}\ J*s)(6.52*10^{14}\ s^{-1})=4.32*10^{-19}\ J](https://tex.z-dn.net/?f=E%3D%286.626%2A10%5E%7B-34%7D%5C%20J%2As%29%286.52%2A10%5E%7B14%7D%5C%20s%5E%7B-1%7D%29%3D4.32%2A10%5E%7B-19%7D%5C%20J)
The energy absorbed by the plate is the same 840 J.
Now, find the number of blue photons required.
![photons\ required = \frac{energy\ absorbed}{energy\ per\ blue\ photon}](https://tex.z-dn.net/?f=photons%5C%20required%20%3D%20%5Cfrac%7Benergy%5C%20absorbed%7D%7Benergy%5C%20per%5C%20blue%5C%20photon%7D)
![photons=\frac{840\ J}{4.32*10^{-19}\ \frac{J}{blue\ photon}}=1.94*10^{21} \ blue\ photons](https://tex.z-dn.net/?f=photons%3D%5Cfrac%7B840%5C%20J%7D%7B4.32%2A10%5E%7B-19%7D%5C%20%5Cfrac%7BJ%7D%7Bblue%5C%20photon%7D%7D%3D1.94%2A10%5E%7B21%7D%20%5C%20blue%5C%20photons)