Question

A glass plate has a mass of 0.50 kg and a specific heat capacity of 840 J/(kg·C°). The wavelength of infrared light is 18 × 10-5 m, while the wavelength of blue light is 4.6 × 10-7 m. Find the number of infrared photons and the number of blue photons needed to raise the temperature of the glass plate by 2.0 °C, assuming that all the photons are absorbed by the glass.

Answer 1

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.

$c = w_{infrared}v_{infrared}$

$v=\frac{c}{w}$

$v_{infrared}=\frac{3.00*10^8 \frac{m}{s}}{18*10^{-5} m}=1.67*10^{12} s^{-1}$

Next, let's find E.

$E_{infrared}=hv_{infrared}$

$E=(6.626*10^{-34} J*s)(1.67*10^{12} s^{-1})=1.10*10^{-21}\ J$

Now, let's find the amount of energy absorbed by the plate.

$q=mc\Delta T$

$q=0.50 kg * 840 \frac{J}{kg\ ^o C}*2.0\ ^o C = 840\ J$

Now, we can find the number of infrared photons required.

$photons\ required = \frac{energy\ absorbed}{energy\ per\ infrared\ photon}$

$photons=\frac{840\ J}{1.10*10^{-21} \frac{J}{infrared\ photon}}=7.61*10^{23} \ infrared\ photons$

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.

$c = w_{blue}v_{blue}$

$v=\frac{c}{w}$

$v_{infrared}=\frac{3.00*10^8 \frac{m}{s}}{4.6*10^{-7} m}=6.52*10^{14} s^{-1}$

Find E.

$E_{blue}=hv_{blue}$

$E=(6.626*10^{-34}\ J*s)(6.52*10^{14}\ s^{-1})=4.32*10^{-19}\ J$

The energy absorbed by the plate is the same 840 J.

Now, find the number of blue photons required.

$photons\ required = \frac{energy\ absorbed}{energy\ per\ blue\ photon}$

$photons=\frac{840\ J}{4.32*10^{-19}\ \frac{J}{blue\ photon}}=1.94*10^{21} \ blue\ photons$