Microwave ovens work by irradiating food with microwave radiation, which is absorbed and converted into heat. Assum-ing that radiation with l = 15.0 cm is used, that all the energy is converted to heat, and that 4.184 J is needed to raise the temperature of 1.00 g of water by 1.00 °C, how many photons are necessary to raise the temperature of a 350 mL cup of water from 20 °C to 95 °C?

Verified step by step guidance

1. 1. First, we need to calculate the total energy required to raise the temperature of the water. The specific heat capacity of water is given as 4.184 J/g°C. The mass of the water can be calculated from its volume (350 mL) and the density of water (1 g/mL). The temperature change is the final temperature (95°C) minus the initial temperature (20°C). The total energy (Q) can be calculated using the formula Q = mcΔT, where m is the mass, c is the specific heat capacity, and ΔT is the temperature change.
2. 2. Next, we need to calculate the energy of a single photon. The energy (E) of a photon can be calculated using the formula E = hc/λ, where h is Planck's constant (6.626 x 10^-34 J.s), c is the speed of light (3.00 x 10^8 m/s), and λ is the wavelength. The wavelength is given as 15.0 cm, but we need to convert this to meters by multiplying by 0.01.
3. 3. Finally, we can calculate the number of photons required. This is done by dividing the total energy required by the energy of a single photon. This will give us the number of photons necessary to raise the temperature of the water from 20°C to 95°C.

Key Concepts

Photon Energy

The energy of a photon is determined by its wavelength, given by the equation E = hc/λ, where E is energy, h is Planck's constant (6.626 x 10^-34 J·s), c is the speed of light (3.00 x 10^8 m/s), and λ is the wavelength in meters. For microwaves with a wavelength of 15.0 cm (0.15 m), this relationship allows us to calculate the energy of each photon, which is essential for determining how many photons are needed to transfer a specific amount of energy to the water.

Heat Transfer and Specific Heat Capacity

Specific heat capacity is the amount of heat required to raise the temperature of a unit mass of a substance by one degree Celsius. For water, this value is 4.184 J/g·°C. To find the total energy needed to raise the temperature of 350 mL (or 350 g) of water from 20 °C to 95 °C, we use the formula Q = mcΔT, where Q is the heat energy, m is the mass, c is the specific heat capacity, and ΔT is the change in temperature.

Energy Conservation in Photon Absorption

In the context of microwave heating, energy conservation dictates that the total energy absorbed by the water must equal the energy provided by the photons. This means that the total energy required to heat the water, calculated from the specific heat capacity, must be divided by the energy per photon to find the total number of photons needed. This principle is crucial for understanding how energy is transferred from the microwave radiation to the water.