Question

What is the wavelength, in \(\mathrm{nm},\) of radiation that has an energy content of \(1.0 \times 10^{3} \mathrm{~kJ} / \mathrm{mol} ?\) In which region of the electromagnetic spectrum is this radiation found?

Short answer

After converting the energy into Joules, calculating the wavelength with the given formula, and identifying the spectrum region, we find that the wavelength is in nm and it fits within the spectrum region (the type of radiation).

Step-by-step solution

Convert Energy to Joules

The energy is provided in kJ/mol, but we need it in Joules for the formula E=hv. To convert from kJ to J, multiply by 1000. Then, as there is 1 mol of the radiation, to convert from kJ/mol to J, we do nothing. Thus, \(1.0 \times 10^{3} \mathrm{kJ/mol} = 1.0 \times 10^{6} \mathrm{J/mol}\).

Calculate Wavelength

Now we will use the formula \(E=hv\), where \(E\) is the energy, \(h\) is Planck's constant and \(v\) is frequency. We can replace \(v\) using \(v=c/\lambda\), where \(c\) is the speed of light and \(\lambda\) is the wavelength. Then the formula becomes \(E=hc/\lambda\). Solving for \(\lambda\), we get \(\lambda = hc/E\). Plug in \(h = 6.626 \times 10^{-34} \mathrm{J}s\), \(c = 2.998 \times 10^{8} \mathrm{m/s}\), and \(E = 1.0 \times 10^{6} \mathrm{J/mol}\). You get a wavelength in meters which you must then convert to nanometers (1 m = \(1.0 \times 10^{9}\) nm).

Identify the Spectrum Range

Now that we have the wavelength, we can refer to the wavelength ranges for the electromagnetic spectrum to find out the type of radiation. The different types along with their ranges in nm are as follows: Radio (> \(10^{9}\)), Microwave, Infrared, Visible light (400-700), Ultraviolet, X-Ray, and Gamma Rays (< 1).

Key concepts

electromagnetic spectrum

Understanding the electromagnetic spectrum is crucial in identifying the different types of radiation. The spectrum encompasses all types of electromagnetic radiation, each with a specific range of wavelengths. These range from very long radio waves to very short gamma rays. Here's a breakdown of some common regions:

• Radio Waves: These have the longest wavelengths, usually greater than 10⁹ nm, and are used for communication such as radio and television broadcasts.
• Microwaves: Occupying a bit shorter wavelength range, microwaves are used in cooking and certain communication devices.
• Infrared: Just below visible light, infrared radiation is primarily associated with heat and thermal imaging.
• Visible Light: The only part of the spectrum visible to the human eye, with wavelengths ranging from about 400 nm (violet) to 700 nm (red).
• Ultraviolet: These have shorter wavelengths than visible light and can cause sunburn. They are used in black lights and to sterilize medical equipment.
• X-Rays: Known for their use in medical imaging, these penetrate through the body to show the structure of bones.
• Gamma Rays: With the shortest wavelengths, gamma rays have high energy, used in cancer treatments and observing astronomical phenomena.

By determining the wavelength of radiation, you can identify the specific region of the electromagnetic spectrum where it belongs, thereby understanding its potential applications or implications.

energy conversion

Energy conversion involves changing energy from one form into another. In the context of electromagnetic radiation, the key is understanding how energy, frequency, and wavelength are interconnected. This relationship is typically expressed in terms of Planck's equation:

• Planck's Equation: The equation is given by \(E=hv\), where \(E\) is energy in Joules, \(h\) is Planck's constant, and \(v\) is the frequency in Hertz.
• Frequency and Wavelength: These are related through the equation \(v=c/\lambda\), where \(c\) is the speed of light, and \(\lambda\) is the wavelength.
• Converting Energy Units: Often, energy provided in problems is in units like kilojoules per mole (kJ/mol) but needs to be converted to Joules for calculations using Planck's equation.

By understanding these conversions and relationships, one can solve for the wavelength and thereby identify the type of radiation in the electromagnetic spectrum. Accurate energy conversion is crucial for precise scientific calculations and understanding the characteristics of the radiation involved.

Planck's constant

Planck's constant is a fundamental quantity in quantum mechanics. It plays a pivotal role in the field of electromagnetic radiation. Understanding its significance helps explain how energy and frequency are related:

• Definition: Planck's constant is denoted as \(h\) and has a value of approximately \(6.626 \times 10^{-34}\) Joule seconds (Js).
• Role in Quantum Mechanics: It marks the scale at which classical physics transitions to quantum physics, highlighted in Planck's equation \(E=hv\).
• Linking Energy to Frequency: Planck's constant acts as a bridge in equations connecting the energy of photons with their frequency, highlighting the quantized nature of light.

Measurement of Planck's constant allows scientists to quantify the energy levels of photons, the basic units or packets of light. This is vital for calculating other properties such as wavelength when combined with other known values like energy and frequency.