A certain microwave oven delivers 800. watts (J/s) of power to a coffee cup containing 45.0 g of water at 25.0°C. If the wavelength of microwaves in the oven is 9.75 cm, how long does it take, and how many photons must be absorbed, to make the water boil?
heat=45*specificheatwater*(212-25) figure that out.
time=heatabove/800
energyineach photon=f*plancksconst
but f=c/lambda
so figure the energy in each photons
number of photons=heat to heat water/energyineach photon
12
To determine how long it takes, and how many photons must be absorbed to make the water boil, we need to follow several steps:
Step 1: Calculate the energy required to heat the water to its boiling point:
The specific heat capacity of water is approximately 4.18 J/(g°C).
The molar mass of water is approximately 18.02 g/mol.
The change in temperature is determined by the heating process:
ΔT = T_final - T_initial = boiling point of water - initial temperature.
Let's calculate ΔT:
Boiling point of water (at standard pressure) = 100°C
Initial temperature = 25.0°C
ΔT = 100°C - 25.0°C = 75.0°C
Now, calculate the energy required using the formula:
Energy = mass of water × specific heat capacity × ΔT
mass of water = 45.0 g
specific heat capacity of water = 4.18 J/(g°C)
Energy = 45.0 g × 4.18 J/(g°C) × 75.0°C
Step 2: Calculate the number of moles of water:
number of moles = mass of water / molar mass of water
mass of water = 45.0 g
molar mass of water = 18.02 g/mol
number of moles = 45.0 g / 18.02 g/mol
Step 3: Calculate the total energy required in units of photons:
To do this, we need to convert the energy to the number of photons absorbed by assuming each photon has a wavelength of 9.75 cm (which corresponds to the microwave wavelength):
Energy per photon = Planck's constant × speed of light / wavelength
Planck's constant (h) ≈ 6.62607004 × 10^(-34) J·s
Speed of light (c) ≈ 2.998 × 10^8 m/s
Planck's constant h = 6.62607004 × 10^(-34) J·s
Speed of light c = 2.998 × 10^8 m/s
Wavelength of microwave (λ) = 9.75 cm = 0.0975 m
Energy per photon = (6.62607004 × 10^(-34) J·s) × (2.998 × 10^8 m/s) / (0.0975 m)
Step 4: Calculate the total number of photons absorbed:
Total energy required = Energy × number of moles
Total number of photons absorbed = Total energy required / Energy per photon
The time it takes for the water to boil depends on the power delivered by the microwave oven. Let's assume it delivers 800 J/s (watts).
Step 5: Calculate the time taken to boil:
Time taken to boil = Total energy required / Power delivered by the oven
Now we can substitute the values and calculate the required quantities.
To determine how long it takes and how many photons must be absorbed to make the water boil, we need to calculate the total energy required to increase the temperature of the water from 25.0°C to its boiling point, as well as how many photons are needed to deliver that energy.
Let's break down the steps to get the answer:
Step 1: Calculate the energy required to heat the water to its boiling point.
The specific heat capacity (c) of water is 4.18 J/g°C. We can use this value to calculate the energy required to increase the temperature of the water.
Q = mcΔT
Where:
Q = energy (in Joules)
m = mass of water (in grams)
c = specific heat capacity of water (in J/g°C)
ΔT = change in temperature (in °C)
For the given problem,
m = 45.0 g
c = 4.18 J/g°C
ΔT = boiling point of water (100°C) - initial temperature (25.0°C)
Let's calculate Q:
Q = (45.0 g) * (4.18 J/g°C) * (100°C - 25.0°C)
Step 2: Convert the energy to joules to the number of photons.
To calculate the number of photons, we need to divide the energy (in joules) by the energy per photon (in joules).
The energy per photon can be calculated using the equation:
E = hc/λ
Where:
E = energy per photon (in Joules)
h = Planck's constant (6.626 x 10^-34 J.s)
c = speed of light (3.0 x 10^8 m/s)
λ = wavelength (in meters)
First, we need to convert the wavelength given in centimeters to meters:
λ = 9.75 cm * (1 m / 100 cm)
Then we can calculate E:
E = (6.626 x 10^-34 J.s) * (3.0 x 10^8 m/s) / λ
Step 3: Calculate the number of photons.
To get the number of photons, we divide the total energy (Q) by the energy per photon (E):
Number of photons = Q / E
Now you can perform the calculations to find out how long it takes and how many photons must be absorbed to make the water boil.