Microwave ovens use microwave radiation to heat food. The energy of the microwaves is absorbed by water molecules in food and then transferred to other components of the food. (a) Suppose that the microwave radiation has a wavelength of 10 cm. How many photons are required to heat 200 mL of water from 25 to 75 °C?

Verified step by step guidance

Step 1: Calculate the energy required to heat the water. Use the formula for heat energy: \( q = m \cdot c \cdot \Delta T \), where \( m \) is the mass of the water, \( c \) is the specific heat capacity of water (4.18 J/g°C), and \( \Delta T \) is the change in temperature.

Step 2: Convert the volume of water to mass. Since the density of water is approximately 1 g/mL, 200 mL of water has a mass of 200 g.

Step 3: Calculate the change in temperature \( \Delta T \) as the final temperature minus the initial temperature: \( 75°C - 25°C = 50°C \).

Step 4: Calculate the energy of a single photon using the formula \( E = \frac{hc}{\lambda} \), where \( h \) is Planck's constant (6.626 \times 10^{-34} \text{ J s}), \( c \) is the speed of light (3.00 \times 10^8 \text{ m/s}), and \( \lambda \) is the wavelength of the microwave radiation (10 cm converted to meters).

Step 5: Determine the number of photons required by dividing the total energy calculated in Step 1 by the energy of a single photon calculated in Step 4.

Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Photon Energy

The energy of a photon is determined by its wavelength, described by the equation E = hc/λ, where E is energy, h is Planck's constant, c is the speed of light, and λ is the wavelength. For microwaves with a wavelength of 10 cm, this relationship allows us to calculate the energy of each photon, which is essential for determining how many photons are needed to heat the water.

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 approximately 4.18 J/g°C. Understanding this concept is crucial for calculating the total energy needed to heat 200 mL of water from 25 to 75 °C, as it directly influences the total energy required for the temperature change.

Energy Transfer in Heating

When microwaves heat food, the energy absorbed by water molecules increases their kinetic energy, leading to a rise in temperature. This energy transfer is key to understanding how the absorbed energy from photons translates into thermal energy, allowing us to calculate the total number of photons needed based on the energy required to achieve the desired temperature increase in the water.

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