The heat of fusion of ice is 6.00 kJ/mol.
Find the number of photons of wavelength = 6.78×10^−6 m that must be absorbed to melt 6.80 g of ice.
6.8 g ice x heat fusion ice = ?joules needed to melt the ice.
How much do we have in a single photon.
E = hc/wavelength = x joules in one photon.
xjoules/photon x # photons = ?J to melt ice.
To solve this problem, we need to follow these steps:
Step 1: Convert the mass of ice from grams to moles.
Step 2: Use the molar mass of water to convert moles of ice to moles of water.
Step 3: Use Avogadro's number to convert moles of water to the number of molecules of water.
Step 4: Use the equation E = hc/λ to calculate the energy of one photon.
Step 5: Use the energy per photon to calculate the number of photons needed to melt the ice.
Let's go through each step in detail:
Step 1: Convert the mass of ice from grams to moles.
Given:
Mass of ice = 6.80 g
To convert from grams to moles, we need the molar mass of water, which is 18.01528 g/mol.
Moles of ice = (mass of ice) / (molar mass of water)
Moles of ice = 6.80 g / 18.01528 g/mol
Moles of ice ≈ 0.3772 mol
Step 2: Convert moles of ice to moles of water.
Since melting ice converts it from a solid to a liquid, we need to convert the moles of ice to moles of water. The ratio of moles of ice to moles of water is 1:1.
Moles of water = 0.3772 mol
Step 3: Convert moles of water to the number of molecules of water.
We use Avogadro's number (6.022 × 10^23 molecules/mol) to convert moles of water to the number of molecules of water.
Number of molecules of water = (moles of water) × (Avogadro's number)
Number of molecules of water = 0.3772 mol × 6.022 × 10^23 molecules/mol
Number of molecules of water ≈ 2.27 × 10^23 molecules
Step 4: Calculate the energy of one photon.
The equation for the energy of a photon is given by E = hc/λ, where h is the Planck constant (6.62607015 × 10^-34 J·s), c is the speed of light (2.998 × 10^8 m/s), and λ is the wavelength of the photon (6.78 × 10^-6 m).
E = (Planck constant × speed of light) / (wavelength)
E = (6.62607015 × 10^-34 J·s × 2.998 × 10^8 m/s) / (6.78 × 10^-6 m)
E ≈ 2.924 × 10^-19 J
Step 5: Calculate the number of photons needed to melt the ice.
To find the number of photons, we divide the energy needed to melt the ice by the energy of one photon.
Number of photons = (energy needed to melt the ice) / (energy of one photon)
Number of photons = (heat of fusion) × (moles of water) / (energy of one photon)
Given:
Heat of fusion of ice = 6.00 kJ/mol
Energy of one photon ≈ 2.924 × 10^-19 J
Number of photons = (6.00 kJ/mol × 10^3 J/1 kJ) × (0.3772 mol) / (2.924 × 10^-19 J)
Number of photons ≈ 7.74 × 10^22 photons
Therefore, approximately 7.74 × 10^22 photons with a wavelength of 6.78 × 10^-6 m must be absorbed to melt 6.80 g of ice.
To find the number of photons of a given wavelength that must be absorbed to melt a certain amount of ice, we need to follow these steps:
Step 1: Convert the mass of ice to moles.
Given:
Mass of ice = 6.80 g
Molar mass of water (H2O) = 18.015 g/mol
To convert the mass of ice to moles, we use the formula:
moles = mass / molar mass
moles of ice = 6.80 g / 18.015 g/mol
Step 2: Calculate the energy required to melt the ice.
Given:
Heat of fusion of ice = 6.00 kJ/mol
To calculate the energy required to melt the ice, we use the formula:
energy = moles × heat of fusion
energy = moles of ice × heat of fusion
energy = (6.80 g / 18.015 g/mol) × (6.00 kJ/mol)
Step 3: Convert the energy required to joules.
1 kJ = 1000 J
To convert the energy from kilojoules to joules, we multiply the energy by 1000:
energy = (6.80 g / 18.015 g/mol) × (6.00 kJ/mol) × (1000 J/kJ)
Step 4: Calculate the number of photons.
Given:
Wavelength of a photon = 6.78×10^−6 m
Planck's constant = 6.63 × 10^−34 J·s
Speed of light = 3.00 × 10^8 m/s
To calculate the number of photons, we use the formula:
energy of photons = energy / energy per photon
energy per photon = (Planck's constant × speed of light) / wavelength
number of photons = energy of photons / energy per photon
Substituting the values:
energy per photon = (6.63 × 10^−34 J·s × 3.00 × 10^8 m/s) / (6.78 × 10^−6 m)
Finally, divide the energy of photons by the energy per photon to get the number of photons.
Keep in mind that solving these calculations requires a calculator or appropriate software, as they involve several steps and numerical values.