Problem 153 - Chemistry

Question

A student placed a large unwrapped chocolate bar in a microwave oven without a rotating glass plate. After turning the oven on for less than a minute, she noticed there were evenly spaced dents (due to melting) about \(6 \mathrm{~cm}\) apart. Based on her observations, calculate the speed of light given that the microwave frequency is \(2.45 \mathrm{GHz}\). (Hint: The energy of a wave is proportional to the square of its amplitude.)

Step-by-Step Solution

Answer

The calculated speed of light, based on the observations made with the microwave oven and the chocolate bar, is approximately \(294,000,000 m/s\). This is close to the widely accepted scientific value of \(299,792,458 m/s\), thus validating the experiment conducted with the microwave oven.

Step 1: Convert frequency from GHz to Hz

The given frequency of the microwaves is 2.45 GHz. However, in scientific calculations, it is more common to use Hz (Hertz) as the unit of frequency. This requires us to convert 2.45 GHz to Hz by using the conversion factor \(1 GHz = 10^{9} Hz\). So, \(f = 2.45 GHz = 2.45 * 10^{9} Hz\).

Step 2: Determine the wavelength

The dents in the chocolate bar, due to the melting, occurred at uniformly spaced intervals of 6 cm. The distance between peaks (or troughs) of a wave is the wavelength. Therefore, the wavelength \(\lambda\) of the microwaves is twice the distance between the melting spots, as one full wave includes a peak and a trough. So, \(\lambda = 2 * 6 cm = 12 cm = 0.12 m\). The conversion from cm to m is necessary to keep the units consistent.

Step 3: Calculate the speed of light

The speed of light (\(c\)) can now be calculated using the formula \(c = \lambda f\). Plug in the values for the calculated wavelength (\(\lambda\)) and the converted frequency (\(f\)), \(c = 0.12 m * 2.45 * 10^{9} Hz = 294,000,000 m/s\).

Key Concepts

Understanding Microwave Frequency

Microwave frequency refers to how often the waves in a microwave oscillate in one second. It's measured in Hertz (Hz), where 1 Hz indicates one cycle per second. In many scientific contexts, Gigahertz (GHz) is used, especially for microwaves.

• The given microwave frequency in our problem is 2.45 GHz.
• To work with it in equations, we first convert GHz to Hz.
• This conversion uses the factor: \(1 \text{ GHz} = 10^{9} \text{ Hz}\).

So, a 2.45 GHz frequency becomes \(2.45 \times 10^{9}\) Hz. Converting units is crucial as scientific calculations often require base SI units, like meters for distance and Hertz for frequency.
This ensures formulas are applied correctly.

Wavelength Calculation Made Simple

The wavelength is the distance between consecutive peaks or troughs in a wave. Here, the interaction of microwaves with the chocolate bar creates melted spots, which are directly related to the microwave's wavelength.

• The dents appear every 6 cm; however, a full wave includes both a peak and a trough.
• This means the actual wavelength is twice the distance between dents.

Therefore, the wavelength \(\lambda\) is \(2 \times 6 \text{ cm} = 12 \text{ cm} = 0.12 \text{ m}\).
Converting to meters aligns with the standard unit for scientific measurements, ensuring consistency throughout the calculation for the speed of light.

Mastering Scientific Unit Conversion

Scientific unit conversion is a fundamental skill for precise calculations in physics. It allows us to work seamlessly within standard units like meters, grams, and seconds.

• In our example, converting the frequency from GHz to Hz ensures proper application of the speed of light formula.
• Similarly, converting the wavelength from centimeters to meters allows for consistent unit usage.

The formula \(c = \lambda f\) requires wavelength in meters and frequency in Hertz.
With these conversions, we calculate the speed of light as \(c = 0.12 \text{ m} \times 2.45 \times 10^{9} \text{ Hz} = 294,000,000 \text{ m/s}\).
These conversions are not just technicalities; they are critical for achieving accurate results.