Problem 22 - Chemistry: The Central Science

Question

If you put 120 volts of electricity through a pickle, the pickle will smoke and start glowing an orange-yellow color. The light is emitted because the sodium ions in the pickle become excited; their return to the ground state results in light emission (see Figure \(6.13 \mathrm{~b}\) and Sample Exercise 6.3). (a) The wavelength of this emitted light is \(589 \mathrm{~nm}\). Calculate its frequency. (b) What is the energy of \(0.10\) mole of these photons? (c) Calculate the energy gap between the excited and ground states for the sodium ion. (d) If you soaked the pickle for a long time in a different salt solution, such as strontium chloride, would you still observe \(589 \mathrm{~nm}\) light emission? Why or why not?

Step-by-Step Solution

Answer

The frequency of the emitted light is 5.09 x 10^14 Hz. The energy of 0.10 moles of these photons is 20.3 kJ. The energy gap between the excited and ground states for the sodium ion is 203 kJ/mol. Soaking the pickle in a different salt solution, such as strontium chloride, would result in a different wavelength of the emitted light due to the different energy levels of strontium ions.

Step 1: Calculate the frequency of the emitted light

Using the relationship between the speed of light (c), wavelength (λ), and frequency (ν), we can find the frequency of the emitted light: c = λν ν = c/λ First, convert the wavelength from nm to meters: \(589 nm = 589 \times 10^{-9} m\) Now, calculate the frequency: ν = \(\frac{3.00\times10^8 m/s}{589\times10^{-9} m}\) = 5.09 x 10^14 Hz.

Step 2: Calculate the energy of 0.10 moles of these photons

Using the relationship between the photon energy (E), Planck's constant (h), and frequency (ν): E = hν First, calculate the energy of one photon of emitted light: E = \((6.626\times10^{-34} Js)(5.09\times10^{14} Hz)\) = 3.37 x 10^{-19} J Now, multiply the energy per photon by the number of photons in 0.10 moles to find the total energy: Energy = \((3.37 \times 10^{-19} J/photon)(0.10 mol)(6.022 \times 10^{23} photons/mol)\) = 20.3 kJ.

Step 3: Calculate the energy gap between the excited and ground states for the sodium ion

Divide the energy calculated in Step 2 by the number of moles of photons to find the energy gap between the excited and ground states for one sodium ion: Energy gap = \(\frac{20.3 kJ}{0.10 mol}\) = 203 kJ/mol

Step 4: Determine whether soaking in a different salt solution would affect the emission wavelength

Soaking the pickle in a different salt solution, such as strontium chloride, would introduce different ions into the pickle. The energy levels of these ions would not be the same as those of sodium ions. When the excited state of strontium ions fall back to the ground state, the emitted light would have a different wavelength due to the different energy levels of strontium ions. Therefore, the emitted light would not have a wavelength of 589 nm when soaked in strontium chloride.

Key Concepts

Photon Energy

When it comes to understanding light emission in chemistry, photon energy is a key player. Photons are particles of light, and their energy can be calculated using Planck's equation:

• \( E = hu \)

This formula tells us that the energy \( E \) of a photon is directly related to both its frequency \( u \) and Planck's constant \( h \), which is approximately \( 6.626 \times 10^{-34} \) Js.
This relationship is vital in exercises like the glowing pickle experiment, where the emitted light color indicates the energy released when sodium ions return to their ground states. By calculating photon energy, one can determine the exact amount of energy emitted as light.

Wavelength and Frequency

Wavelength and frequency are two fundamental concepts that help describe light waves. They're deeply interconnected, as light's speed \( c \) is always constant in vacuum, approximated as \( 3.00 \times 10^8 \) meters per second. The equation governing this relationship is:

• \( c = \lambda u \)

Here, \( \lambda \) is the wavelength, and \( u \) is the frequency of the light.
In practical exercises, such as the one involving sodium ions in a pickle, we first convert the wavelength into meters for consistency with the speed of light. This allows us to accurately calculate the frequency and further explore the energies associated with specific wavelengths of light.
For instance, with a wavelength of 589 nm, converting to meters yields \( 589 \times 10^{-9} \) m, providing the necessary units to compute frequency.

Excited and Ground States

Understanding excited and ground states is crucial when discussing light emission in chemistry. Atoms or ions can move between these states based on energy absorption or emission.
When an atom absorbs energy, its electrons jump to a higher energy level, or excited state. This is generally an unstable state. When it releases that energy, typically as light, the electrons return to their lower energy, or ground state.

• The energy of the light emitted during this transition reflects the energy difference between these two states.

For example, in sodium ions, the orange-yellow glow observed is due to electrons returning to the ground state and emitting photons of that specific energy level.
By calculating the energy difference from light emission, one can determine the energy gap, highlighting the importance of excited and ground states in these transitions.

Sodium Ions

Sodium ions play a fascinating role in chemical light emission. In this context, we're looking at sodium ions infused within a pickle. When electric current flows through, these ions become excited, jump into higher energy levels, and eventually settle back to their ground states.
This exercise illustrates how specific ions, like sodium, have characteristic emissions. The 589 nm wavelength observed is particular to sodium because its unique energy levels create a distinct light color upon emission.
If we change the ion, for instance, by soaking the pickle in strontium chloride, the emission won't be the same. Different ions have different electronic structures, energy levels, and therefore, different wavelengths of emitted light. This demonstrates how the type of ion affects the color and properties of the emitted light.