Question

A microwave oven operating at 1.22 × 10⁸ nm is used to heat 165 mL of water (roughly the volume of a teacup) from 31.0 °C to 100.0 °C.

Calculate the number of photons needed if 89.5 percent of microwave energy is converted to the thermal energy of water.

Answer 1

Answer 2

To calculate the number of photons needed to heat the water, we can follow these steps:

Step 1: Calculate the energy absorbed by the water to increase its temperature. Step 2: Calculate the total energy in the microwave photons required, taking into account the efficiency of energy conversion. Step 3: Calculate the number of photons using the energy per photon.

Step 1: Calculate the energy absorbed by the water:

We can use the specific heat capacity of water (c) to calculate the energy absorbed by the water to increase its temperature:

Specific heat capacity of water (c) = 4.184 J/g°C Mass of water (m) = 165 mL = 165 g (since the density of water is approximately 1 g/mL) Initial temperature (T1) = 31.0 °C Final temperature (T2) = 100.0 °C

Energy absorbed = m * c * ΔT Energy absorbed = 165 g * 4.184 J/g°C * (100.0 °C - 31.0 °C)

Step 2: Calculate the total energy in the microwave photons:

The energy of a photon (E) is given by Planck's equation:

E = h * c / λ

Where: h = Planck's constant = 6.626 × 10^-34 J·s c = speed of light = 2.998 × 10^8 m/s λ = wavelength of the microwave photons in meters

Since the microwave oven operates at 1.22 × 10^8 nm, we need to convert the wavelength to meters:

λ (in meters) = 1.22 × 10^8 nm * (1 m / 10^9 nm) = 1.22 × 10^-1 m

Now, we can calculate the energy of a single microwave photon:

E_photon = 6.626 × 10^-34 J·s * 2.998 × 10^8 m/s / (1.22 × 10^-1 m)

Step 3: Calculate the number of photons:

Now that we have the energy absorbed by the water and the energy per photon, we can calculate the total number of photons needed:

Number of photons = Energy absorbed / E_photon

Note that we need to consider the efficiency of energy conversion, which is given as 89.5%:

Total energy in the microwave photons = Energy absorbed / 0.895

Finally, we can calculate the number of photons:

Number of photons = (Energy absorbed / 0.895) / E_photon

Substitute the values and calculate:

Energy absorbed ≈ 165 g * 4.184 J/g°C * (100.0 °C - 31.0 °C) Energy absorbed ≈ 165 g * 4.184 J/g°C * 69.0 °C Energy absorbed ≈ 47732.04 J

λ (in meters) = 1.22 × 10^-1 m

E_photon ≈ 6.626 × 10^-34 J·s * 2.998 × 10^8 m/s / (1.22 × 10^-1 m) E_photon ≈ 1.633 × 10^-24 J

Total energy in the microwave photons ≈ 47732.04 J / 0.895 ≈ 53318.09 J

Number of photons ≈ 53318.09 J / 1.633 × 10^-24 J Number of photons ≈ 3.26 × 10^28 photons

So, approximately 3.26 × 10^28 photons are needed to heat 165 mL of water from 31.0 °C to 100.0 °C, considering 89.5 percent of microwave energy converted to thermal energy.