Question

The speed of a computer chip is measured by its frequency. Advanced Micro Devices (ADM) has developed a microprocessor, "The Piledriver," that operates at 8.429 GHz \(\left(1 \mathrm{GHz}=1 \times 10^{9} \mathrm{~Hz}\right)\) (a) What is the wavelength (in \(\mathrm{nm}\) ) of the frequency? (b) In what spectral range does it belong? (c) What is the energy (in \(\mathrm{kJ} / \mathrm{mol}\) ) associated with this frequency?

Short answer

Based on the given frequency of 8.429 GHz, the calculated wavelength is approximately 35.602 nm, which falls within the Ultraviolet spectrum. The energy associated with this frequency is approximately 3.3628 x \(10^{-4} \mathrm{kJ/mol}\).

Step-by-step solution

Convert frequency to Hz

First, we need to convert the given frequency from GHz to Hz: \(\text{Frequency (f)} = 8.429 \: \text{GHz} = 8.429 \times 10^9 \: \text{Hz}\)

Calculate the wavelength

Now, we can use the formula for the relationship between speed, frequency, and wavelength: \(v = f\lambda\) \(\lambda = \frac{v}{f}\) Replacing the values: \(\lambda = \frac{3 \times 10^8 \mathrm{\:m/s}}{8.429 \times 10^9 \: \text{Hz}} = 3.5602 \times 10^{-11} \mathrm{\:m}\)

Convert wavelength to nm

Finally, we will convert the wavelength from meters to nanometers (1 m = \(10^9\) nm): \(\lambda = 3.5602 \times 10^{-11}\mathrm{\:m} \times \frac{10^9 \: \text{nm}}{1 \: \text{m}} = 35.602 \: \text{nm}\) So, the wavelength is approximately 35.602 nm. b) Determining the spectral range:

Review the electromagnetic spectrum

We will look at the electromagnetic spectrum to determine the range corresponding to the wavelength calculated. Here are some ranges: - Radio waves: \(> 1 \: \text{mm}\) - Microwaves: \(1 \: \text{mm}\) - \(0.7 \: \text{cm}\) - Infrared: \(700 \: \text{nm}\) - \(1 \: \text{mm}\) - Visible light: \(400 \: \text{nm}\) - \(700 \: \text{nm}\) - Ultraviolet: \(10 \: \text{nm}\) - \(400 \: \text{nm}\) - X-rays: \(0.01 \: \text{nm}\) - \(10 \: \text{nm}\) - Gamma rays: \(< 0.01 \: \text{nm}\)

Determine the range

Our calculated wavelength is 35.602 nm, which falls within the Ultraviolet spectrum. c) Calculating energy:

Calculate energy in Joules

To determine the energy associated with this frequency, we will use Planck's equation: \(E = h\nu\) Replacing the values: \(E = (6.626 \times 10^{-34} \mathrm{Js})(8.429 \times 10^9 \: \text{Hz}) = 5.5844 \times 10^{-24} \mathrm{J}\)

Convert energy to kJ/mol

Finally, we will convert the energy from Joules to kilojoules per mole using Avogadro's number: \(E = 5.5844 \times 10^{-24}\mathrm{J} \times \frac{1 \: \text{kJ}}{10^3 \: \text{J}} \times \frac{6.022 \times 10^{23} \mathrm{mol^{-1}}}{1\: \text{molecule}}\) \(E = 3.3628 \times 10^{-4} \mathrm{kJ/mol}\) So, the energy associated with this frequency is approximately 3.3628 x \(10^{-4} \mathrm{kJ/mol}\).

Key concepts

Spectral Range Explained

The electromagnetic (EM) spectrum encompasses all forms of electromagnetic radiation, ranging from very long radio waves to very short gamma rays. The spectral range refers to the specific region of the electromagnetic spectrum that an electromagnetic wave falls into based on its wavelength. Understanding spectral ranges is crucial in various fields, from astrophysics to telecommunications.

For instance, the spectral range determines the type of technology that will be used to detect or generate the wave. Microwaves, used in radar systems and microwave ovens, have longer wavelengths than visible light, but shorter than radio waves. Similarly, the ultraviolet spectrum, just beyond the visibility to the human eye, has implications for everything from studying stars to designing sunglasses that block harmful UV rays.

Wavelength Calculation Steps

To understand how devices like the 'Piledriver' microprocessor function, we often need to calculate the wavelength of their operating frequencies. To do this, we follow a simple wavelength calculation method that uses the fundamental formula:
\[\begin{equation}\text{Speed of light }(c) = \text{Frequency }(f) \times \text{Wavelength }(\text{\textlambda})\rightarrow \text{\textlambda} = \frac{c}{f}\right).\right).\right).\ \right).\right).\right).\right).\right).\right).\right).\right).\right).\right).\right).\right).\right).\right).\right).\right).\right).\right).\right).\right).\right).\right).\right).\right).\right).\right).\right).\right).\right).\right).\right).\right).\right).\right).\right).\right).\right).\right).\right).\right).\right).\right).\right).\right).\right).}\right). \end{equation}\]where c is the speed of light (approximately 3 \times 10^{\text{8}} m/s). This formula allows for the conversion of the frequency of any electromagnetic wave into its corresponding wavelength. Calculating the wavelength is essential not just for understanding properties of radiation but also for practical applications such as designing antennas or optical components.

Understanding Planck's Equation

Planck's equation is a foundational pillar in quantum mechanics, describing the energetics of photons, the particles of light. It is represented by the formula: \[\begin{equation}E = hu\end{equation}\]where E stands for energy, h is Planck's constant (6.626 x 10^{-34} J·s), and u (nu) is the frequency of the electromagnetic wave. The energy determined by Planck's equation is quantized, meaning it comes in discrete amounts.

In practical terms, using this equation, scientists can calculate the energy carried by a single photon given its frequency. This is crucial for applications ranging from determining the energy requirements of a microprocessor to understanding the detailed processes within photosynthesis. It also provides the gateway to working out reactions in photochemistry and the diagnosis of medical conditions using various forms of radiation.