<u>Answer:</u> The number of photons are ![3.7\times 10^8](https://tex.z-dn.net/?f=3.7%5Ctimes%2010%5E8)
<u>Explanation:</u>
We are given:
Wavelength of microwave =
(Conversion factor:
)
- To calculate the energy of one photon, we use Planck's equation, which is:
![E=\frac{hc}{\lambda}](https://tex.z-dn.net/?f=E%3D%5Cfrac%7Bhc%7D%7B%5Clambda%7D)
where,
h = Planck's constant = ![6.625\times 10^{-34}J.s](https://tex.z-dn.net/?f=6.625%5Ctimes%2010%5E%7B-34%7DJ.s)
c = speed of light = ![3\times 10^8m/s](https://tex.z-dn.net/?f=3%5Ctimes%2010%5E8m%2Fs)
= wavelength = 0.122 m
Putting values in above equation, we get:
![E=\frac{6.625\times 10^{-34}J.s\times 3\times 10^8m/s}{0.122m}\\\\E=1.63\times 10^{-24}J](https://tex.z-dn.net/?f=E%3D%5Cfrac%7B6.625%5Ctimes%2010%5E%7B-34%7DJ.s%5Ctimes%203%5Ctimes%2010%5E8m%2Fs%7D%7B0.122m%7D%5C%5C%5C%5CE%3D1.63%5Ctimes%2010%5E%7B-24%7DJ)
Now, calculating the energy of the photon with 88.3 % efficiency, we get:
![E=1.63\times 10^{-24}\times \frac{88.3}{100}=1.44\times 10^{-24}J](https://tex.z-dn.net/?f=E%3D1.63%5Ctimes%2010%5E%7B-24%7D%5Ctimes%20%5Cfrac%7B88.3%7D%7B100%7D%3D1.44%5Ctimes%2010%5E%7B-24%7DJ)
- To calculate the mass of water, we use the equation:
![Density=\frac{Mass}{Volume}](https://tex.z-dn.net/?f=Density%3D%5Cfrac%7BMass%7D%7BVolume%7D)
Density of water = 1 g/mL
Volume of water = 165 mL
Putting values in above equation, we get:
![1g/mL=\frac{\text{Mass of water}}{165mL}\\\\\text{Mass of water}=165g](https://tex.z-dn.net/?f=1g%2FmL%3D%5Cfrac%7B%5Ctext%7BMass%20of%20water%7D%7D%7B165mL%7D%5C%5C%5C%5C%5Ctext%7BMass%20of%20water%7D%3D165g)
- To calculate the amount of energy of photons to raise the temperature from 23°C to 100°C, we use the equation:
![q=mc\Delta T](https://tex.z-dn.net/?f=q%3Dmc%5CDelta%20T)
where,
m = mass of water = 165 g
c = specific heat capacity of water = 4.184 J/g.°C
= change in temperature = ![T_2-T_1=100^oC-23^oC=77^oC](https://tex.z-dn.net/?f=T_2-T_1%3D100%5EoC-23%5EoC%3D77%5EoC)
Putting values in above equation, we get:
![q=165g\times 4.184J/g.^oC\times 77^oC\\\\q=53157.72J](https://tex.z-dn.net/?f=q%3D165g%5Ctimes%204.184J%2Fg.%5EoC%5Ctimes%2077%5EoC%5C%5C%5C%5Cq%3D53157.72J)
This energy is the amount of energy for 'n' number of photons.
- To calculate the number of photons, we divide the total energy by energy of one photon, we get:
![n=\frac{q}{E}](https://tex.z-dn.net/?f=n%3D%5Cfrac%7Bq%7D%7BE%7D)
q = 53127.72 J
E = ![1.44\times 10^{-24}J](https://tex.z-dn.net/?f=1.44%5Ctimes%2010%5E%7B-24%7DJ)
Putting values in above equation, we get:
![n=\frac{53157.72J}{1.44\times 10^{-24}J}=3.7\times 10^{28}](https://tex.z-dn.net/?f=n%3D%5Cfrac%7B53157.72J%7D%7B1.44%5Ctimes%2010%5E%7B-24%7DJ%7D%3D3.7%5Ctimes%2010%5E%7B28%7D)
Hence, the number of photons are ![3.7\times 10^8](https://tex.z-dn.net/?f=3.7%5Ctimes%2010%5E8)