Problem 82 - Chemistry: The Central Science

Question

If you put 120 volts of electricity through a pickle, the pickle will smoke and start glowing orange-yellow. The light is emitted because sodium ions in the pickle become excited; their return to the ground state results in light emission. (a) The wavelength of this emitted light is \(589 \mathrm{~nm}\). Calculate its frequency. (b) What is the energy of \(0.10 \mathrm{~mol}\) 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?

Step-by-Step Solution

Answer

In summary, (a) the frequency of the emitted light is \(ν = \frac{3.0 × 10^8}{589 × 10^{-9}} = 5.10 × 10^{14} Hz\). (b) The energy of 1.00 mol of photons is \(E_{mol} = 6.626 × 10^{-34} × 5.10 × 10^{14} × 6.022 × 10^{23} = 2.03 × 10^2 kJ\). (c) The energy gap between sodium ion's ground state and the excited state is ΔE = 3.39 × 10^-19 J. (d) If the pickle is soaked in a different salt solution such as strontium chloride, the emitted light will likely be different due to the specific electronic configuration of strontium ions, and it would not necessarily emit light at the same wavelength of 589 nm.

Step 1: a) Frequency Calculation

To calculate the frequency of the light, we can use the formula: \(c = λν\) Where \(c\) is the speed of light (approximately \(3.0 × 10^8 \frac{m}{s}\)), \(λ\) is the wavelength, and \(ν\) is the frequency. We are given the wavelength as 589 nm, but we need to convert this value to meters: \(589 nm = 589 × 10^{-9} m\) Now we can find the frequency by rearranging the formula: \(ν = \frac{c}{λ}\)

Step 2: b) Energy of 1.00 mol of photons

To find the energy of 1.00 mol of photons, we can first calculate the energy of a single photon using the following formula: \(E = hν\) Where \(E\) is the energy, \(h\) is Planck's constant (\(6.626 × 10^{-34} Js\)), and \(ν\) is the frequency calculated in part (a). After finding the energy of a single photon, we can multiply it by Avogadro's number \(N_A (6.022 × 10^{23} mol^{-1})\) to get the energy of 1.00 mol of photons: \(E_{mol} = E × N_A\)

Step 3: c) Energy gap calculation

The energy gap between the sodium ion's ground state and excited state is equal to the energy of the emitted photon. So, the energy gap can be found using the energy calculated in part (b): \(ΔE = E_{photon}\)

Step 4: d) Observable light emission with strontium chloride

This part requires knowledge of how the electronic configuration of different elements affects the wavelengths of emitted light. Sodium's characteristic orange-yellow color is due to sodium ions in its specific electronic configuration. If the pickle were to be soaked in a different salt solution, such as strontium chloride, the emitted light would likely be different as it would correspond to the strontium ions' specific electronic configuration. It would not necessarily emit light at the same wavelength of 589 nm.

Key Concepts

Frequency Calculation

Calculating the frequency of light involves understanding the relationship between the speed of light, wavelength, and frequency. The formula we use is \[ c = \lambda u \]where \( c \) is the speed of light, \( \lambda \) is the wavelength, and \( u \) is the frequency. When the wavelength of light is given, such as in this exercise where it's 589 nm, we need to ensure units are consistent, typically converting the wavelength from nanometers to meters. This conversion is done as follows:\[ 589 \text{ nm} = 589 \times 10^{-9} \text{ m} \] Once converted, you can use the rearranged formula to find frequency:\[ u = \frac{c}{\lambda} \]Substituting the values gives the frequency part students are often required to work out in exams or problems involving photon emission.

Energy of Photons

The energy of a photon is calculated using Planck's formula:\[ E = h u \]Here, \( E \) is the energy of the photon, \( h \) is Planck's constant, and \( u \) is the frequency obtained earlier. Planck's constant is a fundamental value in physics, given as \( 6.626 \times 10^{-34} \) Js. By substituting the frequency calculated, students can determine the energy of a single photon.For practical problems, such as this, calculating the energy for a given amount of substance, such as a mole, involves using Avogadro's number to multiply the energy of a single photon. This leads to\[ E_{\text{mol}} = E \times N_A \] where \( N_A \) is Avogadro's number \( 6.022 \times 10^{23} \text{ mol}^{-1} \), giving the total energy for that amount.

Energy Gap

The energy gap in the context of photon emission refers to the difference in energy levels between the ground state and the excited state of an atom or molecule. It is essentially the energy change involved in the electron's transition. This is critical in understanding how certain colors, like the sodium ion's orange-yellow emission, occur.Since this energy is the same as the energy of the emitted photon (as the electron moves back to a lower energy state), calculating it involves using the energy value from earlier. The simple equation for this is:\[ \Delta E = E_{\text{photon}} \]This step directly links the physical observation of light color with the microscopic changes within the ion, making it clearer for students how energy changes manifest as visible light.

Light Emission in Elements

Elements emit light at specific wavelengths due to their unique electronic configurations and energy level differences. This process, called emission spectroscopy, is highly specific and serves as a fingerprint for elements, showing a distinct spectrum based on their structure. For sodium, the familiar orange-yellow glow stems from the transition of electrons between particular energy levels. When introducing different ions, such as strontium, into the pickle, the observed emitted light will change because strontium has different energy levels and electronic arrangements. Thus, in practical terms, whether you can expect the same color when different salts are used depends on the specific electronic transitions available in the ions of these salts.