Physics, 24.07.2019 17:30 kendricklamar29

How many photons will be required to raise the temperature of 1.8 g of water by 2.5 k ? '?

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Answer from: deaishaajennings123

Given the following data:

Mass of water = 1.8 grams

Change in temperature = 2.5 Kelvin

Wavelength of infrared = 2.9 x $10^{-4}$ cm.

Specific heat capacity of water = 4.18 J/g°K.

Conversion:

100 cm = 1 m

2.9 x $10^{-4}$ cm = 2.9 x $10^{-6}$ 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;

$Q = mc\theta$

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;

Q = 1.8(4.18)(2.5)

Q = 18.81 Joules.

Mathematically, the Planck-Einstein relation is given by the formula:

E = hf

Where:

h is Planck constant.f is photon frequency.

To find the photon frequency, we would use this formula:

$Photon\;frequency = \frac{speed}{wavelength} \\\\Photon\;frequency = \frac{3(10^8)}{2.9(10^{-6})}\\\\Photon\;frequency = \frac{300000000}{0.0000029}$

Photon frequency = 1.04 × $10^{14}$ Hertz

Applying Planck-Einstein's relation, we would determine the energy required by each of photon:

E = 6.626 × $10^{-34}$ × 1.04 × $10^{14}$

E = 6.89 × $10^{-20}$ Joules

Now, we can calculate the amount of photons required to raise the temperature of 1.8 g of water by using this expression:

$Number\;of\;photons = \frac{Q}{E} \\\\Number\;of\;photons = \frac{18.81}{6.89(10^{-20})}$

Number of photons = 2.73 × $10^{20}$ photons.

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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.

Answer from: chonawilson4

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
$Q=m C_s \Delta T$
where
m=1.8 g is the mass of the water
C_s = 4.18 J/(g K)

is the specific heat capacity of the water
$\Delta T=2.5 K$ is the increase in temperature.

Substituting the data, we find
Q=(1.8 g)(4.18 J/(gK))(2.5 K)=18.8 J=E

We know that each photon carries an energy of
E_1 = hf

where h is the Planck constant and f the frequency of the photon. Using the wavelength, we can find the photon frequency:
$\lambda = \frac{c}{f}= \frac{3 \cdot 10^8 m/s}{3 \cdot 10^{-6} m}=1 \cdot 10^{14}Hz$

So, the energy of a single photon of this frequency is
$E_1 = hf =(6.6 \cdot 10^{-34} J)(1 \cdot 10^{14} Hz)=6.6 \cdot 10^{-20} J$

and the number of photons needed is the total energy needed divided by the energy of a single photon:
$N= \frac{E}{E_1}= \frac{18.8 J}{6.6 \cdot 10^{-20} J} =2.84 \cdot 10^{20} photons$

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