Respuesta :
Missing part in the text of the problem:
"Water is exposed to infrared radiation of wavelength 3.0×10^−6 m"
First we can calculate the amount of energy needed to raise the temperature of the water, which is given by
[tex]Q=m C_s \Delta T[/tex]
where
m=1.8 g is the mass of the water
[tex]C_s = 4.18 J/(g K)[/tex] is the specific heat capacity of the water
[tex]\Delta T=2.5 K[/tex] is the increase in temperature.
Substituting the data, we find
[tex]Q=(1.8 g)(4.18 J/(gK))(2.5 K)=18.8 J=E[/tex]
We know that each photon carries an energy of
[tex]E_1 = hf[/tex]
where h is the Planck constant and f the frequency of the photon. Using the wavelength, we can find the photon frequency:
[tex]\lambda = \frac{c}{f}= \frac{3 \cdot 10^8 m/s}{3 \cdot 10^{-6} m}=1 \cdot 10^{14}Hz [/tex]
So, the energy of a single photon of this frequency is
[tex]E_1 = hf =(6.6 \cdot 10^{-34} J)(1 \cdot 10^{14} Hz)=6.6 \cdot 10^{-20} J[/tex]
and the number of photons needed is the total energy needed divided by the energy of a single photon:
[tex]N= \frac{E}{E_1}= \frac{18.8 J}{6.6 \cdot 10^{-20} J} =2.84 \cdot 10^{20} photons [/tex]
"Water is exposed to infrared radiation of wavelength 3.0×10^−6 m"
First we can calculate the amount of energy needed to raise the temperature of the water, which is given by
[tex]Q=m C_s \Delta T[/tex]
where
m=1.8 g is the mass of the water
[tex]C_s = 4.18 J/(g K)[/tex] is the specific heat capacity of the water
[tex]\Delta T=2.5 K[/tex] is the increase in temperature.
Substituting the data, we find
[tex]Q=(1.8 g)(4.18 J/(gK))(2.5 K)=18.8 J=E[/tex]
We know that each photon carries an energy of
[tex]E_1 = hf[/tex]
where h is the Planck constant and f the frequency of the photon. Using the wavelength, we can find the photon frequency:
[tex]\lambda = \frac{c}{f}= \frac{3 \cdot 10^8 m/s}{3 \cdot 10^{-6} m}=1 \cdot 10^{14}Hz [/tex]
So, the energy of a single photon of this frequency is
[tex]E_1 = hf =(6.6 \cdot 10^{-34} J)(1 \cdot 10^{14} Hz)=6.6 \cdot 10^{-20} J[/tex]
and the number of photons needed is the total energy needed divided by the energy of a single photon:
[tex]N= \frac{E}{E_1}= \frac{18.8 J}{6.6 \cdot 10^{-20} J} =2.84 \cdot 10^{20} photons [/tex]
Given the following data:
Mass of water = 1.8 grams
Change in temperature = 2.5 Kelvin
Wavelength of infrared = [tex]2.9[/tex] x [tex]10^{-4}[/tex] cm.
Specific heat capacity of water = 4.18 J/g°K.
Conversion:
100 cm = 1 m
[tex]2.9[/tex] x [tex]10^{-4}[/tex] cm = [tex]2.9[/tex] x [tex]10^{-6}[/tex] cm
To find the amount of photons required to raise the temperature of 1.8 g of water:
First of all, we would determine the quantity of energy required to raise the temperature of water:
Mathematically, quantity of energy is given by the formula;
[tex]Q = mc\theta[/tex]
Where:
- Q represents the quantity of energy.
- m represents the mass of an object.
- c represents the specific heat capacity.
- ∅ represents the change in temperature.
Substituting the given parameters into the formula, we have;
[tex]Q = 1.8(4.18)(2.5)[/tex]
Q = 18.81 Joules.
Mathematically, the Planck-Einstein relation is given by the formula:
[tex]E = hf[/tex]
Where:
- h is Planck constant.
- f is photon frequency.
To find the photon frequency, we would use this formula:
[tex]Photon\;frequency = \frac{speed}{wavelength} \\\\Photon\;frequency = \frac{3(10^8)}{2.9(10^{-6})}\\\\Photon\;frequency = \frac{300000000}{0.0000029}[/tex]
Photon frequency = [tex]1.04[/tex] × [tex]10^{14}[/tex] Hertz
Applying Planck-Einstein's relation, we would determine the energy required by each of photon:
[tex]E =[/tex] [tex]6.626[/tex] × [tex]10^{-34}[/tex] × [tex]1.04[/tex] × [tex]10^{14}[/tex]
E = [tex]6.89[/tex] × [tex]10^{-20}[/tex] Joules
Now, we can calculate the amount of photons required to raise the temperature of 1.8 g of water by using this expression:
[tex]Number\;of\;photons = \frac{Q}{E} \\\\Number\;of\;photons = \frac{18.81}{6.89(10^{-20})}[/tex]
Number of photons = [tex]2.73[/tex] × [tex]10^{20}[/tex] photons.
Read more: https://brainly.com/question/16901506
Complete Question:
Water is exposed to infrared radiation of wavelength 2.9x10-4 cm. Assume that all the radiation is absorbed and converted to heat. How many photons will be required to raise the temperature of 1.8 g of water by 2.5 K? Express your answer using two significant figures.
