Question

A wad of steel wool (specific heat \(\left.=0.45 \mathrm{~J} / \mathrm{g} \cdot{ }^{\circ} \mathrm{C}\right)\) at \(25^{\circ} \mathrm{C}\) is misplaced in a microwave oven, which later is accidentally turned on high. The oven generates microwaves of \(13.5 \mathrm{~cm}\) wavelength at the rate of 925 moles of those photons per second. All of the photons are converted to high- heat energy, raising the temperature of the steel wool. Steel wool reacts explosively with the oxygen in the oven when the steel wool reaches \(400.0^{\circ} \mathrm{C}\). If the steel wool explodes after 1.55 seconds, how many grams of steel wool were accidentally put into the oven?

Short answer

Answer: To find the mass of the steel wool, follow the steps outlined in the solution above. First, calculate the energy of one photon and then determine the total energy generated by the microwaves during the given time. Finally, use the specific heat formula to find the mass of the steel wool. The mass can be found using the equation: \(m = \frac{E_\text{total}}{c\Delta T}\), where \(E_\text{total}\) is the total energy generated by the microwaves, \(c\) is the specific heat, and \(\Delta T\) is the change in temperature.

Step-by-step solution

Find the energy of one photon

The energy of a photon can be calculated using the formula: \(E = hf\) where \(E\) is the energy, \(h\) is the Planck's constant (\(6.63 \times10^{-34} \text{ J s}\)), and \(f\) is the frequency. Given the wavelength of the photon (\(\lambda\)) as 13.5 cm or 0.135 m, we can find the frequency using the formula: \(f = \frac{c}{\lambda}\) where \(c\) is the speed of light (\(3 \times 10^8 \text{ m/s}\)). Now, let's find the frequency: \(f = \frac{3 \times 10^8}{0.135} = 2.222 \times 10^9 \text{ Hz}\) Using the frequency, we can find the energy of a single photon: \(E = (6.63 \times10^{-34})(2.222 \times 10^9) = 1.473 \times 10^{-24} \text{ J}\)

Calculate the total energy generated by the microwaves

Given that the microwave oven generates 925 moles of photons per second, we can find the total number of photons generated during 1.55 seconds: \(N_\text{total photons} = 925 \times 6.022 \times 10^{23} \times 1.55\). Now, we can calculate the total energy generated by these photons: \(E_\text{total} = N_\text{total photons} \times E_\text{single photon}\).

Calculate the mass of the steel wool

We are given that all of the energy from the microwaves is converted to heat. The temperature of the steel wool increases from \(25^\circ \text{C}\) to \(400^\circ \text{C}\). Using the specific heat formula, we can find the mass of the steel wool: \(q = mc\Delta T\) where \(q\) is the heat gained, \(m\) is the mass of the steel wool, \(c\) is the specific heat (\(0.45 \text{ J/g} \cdot { }^{\circ} \text{ C}\)), and \(\Delta T\) is the change in temperature (\(375^\circ \text{C}\)). Given that \(q = E_\text{total}\), we can re-arrange the equation to get the mass: \(m = \frac{E_\text{total}}{c\Delta T}\). Using the total energy calculated in step 2, we can then find the mass of the steel wool. By following this step-by-step process, we can solve the problem and find the mass of the steel wool that was accidentally placed in the microwave oven.

Key concepts

Microwave Photon Energy

Understanding the energy of a microwave photon is essential when studying the interaction between electromagnetic radiation and matter. Microwaves are a form of electromagnetic radiation with wavelengths longer than infrared light but shorter than radio waves.

In the given exercise, the steel wool's temperature change is a result of the energy transfer from microwave photons. The energy (\( E \)) of a single photon is defined by the equation \( E = hf \) where \( h \) represents Planck's constant, and \( f \) is the frequency of the photon. Planck's constant is a fundamental constant in physics, representing the quantization of energy at the microscopic level. Its precise value is \( h = 6.63 \times 10^{-34} \text{ J s} \).

The frequency can be determined from the wavelength using the speed of light, \( f = \frac{c}{\lambda} \) with \( c \) representing the speed of light, and \( \lambda \) being the wavelength of the photon. For microwaves with a wavelength of 13.5 cm, the frequency can be calculated and thus, the energy of a single microwave photon can be computed. Knowing the energy per photon is crucial for calculating the total energy that the microwave oven imparts to the steel wool.

Planck's Constant Application

Planck's constant is not just a number; it is the bridge between the quantum world and the classical physics we observe. By applying Planck's constant in our calculations, we uncover the discrete nature of energy in the realm of quantum mechanics.

In our problem, Planck's constant helps us determine the energy of each photon emitted by the microwave. While the wavelength provides information about the photonic energy scale, it is the application of Planck's constant that allows us to translate this into a precise energy quantity.

Each photon releases its discrete packet of energy when absorbed by the steel wool, which, in turn, raises the temperature of the wool. The entire process relies on the principle that energy can only be transferred in fixed amounts defined by \( h \) and the frequency of the electromagnetic wave in question. Therefore, by utilizing Planck's constant, we can understand and calculate the energy absorbed by any material exposed to electromagnetic radiation.

Heat Capacity

Heat capacity, a concept intimately related to temperature changes in substances, is a measure of how much energy is required to raise the temperature of a given mass of a substance by a certain temperature interval. The specific heat is a type of heat capacity that is normalized to the units of mass of the substance.

In the context of the exercise, the specific heat of steel wool is given, \( c = 0.45 \text{ J/g} \cdot{ }^{\circ} \text{C} \) which means that 0.45 joules of energy are needed to raise the temperature of one gram of steel wool by one degree Celsius.

To find out how much energy has been transferred to the steel wool, and consequently its mass, we use the specific heat equation \( q = mc\Delta T \). Here, \( q \) represents the heat absorbed, \( m \) the mass of steel wool, and \( \Delta T \) the temperature change. Given the total energy provided by microwave photons and the temperature change of steel wool to reach the point of explosion, we can re-arrange the equation to solve for \( m \) and determine the grams of steel wool in the oven.