Question

Photons of infrared radiation are responsible for much of the warmth we feel when holding our hands before a fire. These photons will also warm other objects. How many infrared photons with a wavelength of 1.5x10^-6 m must be absorbed by the water to warm a cup of water (175g) from 25.0°C to 40°C

Answer 1

Answer:

$8.3\cdot 10^{22}$ photons

Explanation:

The energy of a photon is given by

$E_1 = \frac{hc}{\lambda}$

where

$h=6.63\cdot 10^{-34} Js$ is the Planck constant

$c=3.0\cdot 10^8 m/s$ is the speed of light

$\lambda$ is the photon wavelength

Here we have

$\lambda=1.5\cdot 10^{-6} m$

So, the energy of 1 of these infrared photons is

$E_1=\frac{(6.63\cdot 10^{-34})(3\cdot 10^8)}{1.5\cdot 10^{-6}}=1.32\cdot 10^{-19} J$

The amount of energy needed to increase the temperature of the cup of water is:

$E=mC\Delta T$

where

m = 175 g is the mass

$C=4.186 J/gC$ is the specific heat capacity

$\Delta T=40-25=15^{\circ}C$ is the increase in temperature

So,

$E=(175)(4.186)(15)=10,988 J$

Therefore, the number of photons needed is:

$n=\frac{E}{E_1}=\frac{10,988}{1.32\cdot 10^{-19}}=8.3\cdot 10^{22}$