A 900-W microwave generator in an oven generates energy quanta of frequency 2560 MHz. (a) How many energy quanta does it emit per second? (b) How many energy quanta must be absorbed by a pasta dish placed in the radiation cavity to increase its temperature by \(45.0 \mathrm{K}\) ? Assume that the dish has a mass of \(0.5 \mathrm{kg}\) and that its specific heat is \(0.9 \mathrm{kcal} / \mathrm{kg} \cdot \mathrm{K} .\) (c) Assume that all energy quanta emitted by the generator are absorbed by the pasta dish. How long must we wait until the dish in (b) is ready?

Planck's equation is at the heart of quantum mechanics, providing a relationship between the energy of a photon and its frequency. Mathematically, the equation is expressed as
\[E = h \times f\]
where \(E\) represents the energy of the photon, \(h\) is Planck's constant (approximately \(6.63 \times 10^{-34} \mathrm{J} \text{·} \mathrm{s}\)), and \(f\) is the frequency of the photon in Hertz.
Planck's constant is a fundamental value that arises in the quantization of light and energy. The concept of quantization is key to understanding that energy is not continuous; it exists in discrete 'packets' known as quanta. Therefore, Planck's equation provides a way to calculate the discrete energy levels that a photon can possess given its frequency.
Relevance to Microwave Generators:When we consider microwave ovens, the microwaves generated are essentially photons with specific frequencies. Planck's equation allows us to determine the energy associated with each of these photons (or microwaves). In the problem provided, a frequency of 2560 MHz (megahertz) was converted into Hertz to find the energy per photon produced by a microwave generator, a direct application of Planck's equation.

The specific heat formula is crucial in thermodynamics for calculating the energy required to change the temperature of a substance. The formula is given by
\[Q = m \times c \times \Delta T\]
where \(Q\) is the total heat energy transferred, \(m\) represents the mass of the substance, \(c\) is the specific heat capacity, and \(\Delta T\) is the change in temperature.
Specific heat capacity is a property that indicates how much heat energy is needed to raise the temperature of one kilogram of a substance by one degree Kelvin (or Celsius). Different materials have different specific heat capacities. This formula is essential in our problem because it helps us determine the amount of energy required to increase the temperature of the pasta dish by 45.0 K.
Practical Scenario:In the microwave oven scenario, converting the specific heat capacity of the pasta dish from kcal/kg⋅K to J/kg⋅K allows us to work consistently with SI units. This conversion is vital to accurately determine the energy needed to reach the desired temperature change, directly influencing how many energy quanta will be absorbed by the pasta dish from the microwave generator.

Energy quanta refer to the smallest, indivisible units of energy as proposed by Max Planck during his work on black-body radiation. A single quantum of energy, particularly in the context of electromagnetic radiation like light or microwaves, is called a photon. Each photon carries a discrete amount of energy determined by its frequency, as described by Planck's equation.
The idea that energy is quantized revolutionized physics by introducing the concept that energy transfer occurs in fixed, small packets rather than in a continuous flow. This concept is pivotal when considering processes such as the absorption and emission of energy by atoms and molecules.
Role in Cooking:When a microwave generator operates, it emits these energy quanta that the food absorbs. The specific energy quanta's frequency (in the microwave range for microwave ovens) corresponds to energy levels that efficiently interact with water molecules in food, causing them to heat up. In the provided exercise, each photon emitted by the microwave has a specific energy, and knowing this allows us to compute the total number of photons (quanta of energy) needed to raise the temperature of the pasta dish. This concept ties directly to how microwave ovens cook food and provides a quantifiable method to calculate cooking times based on energy absorption.