Problem 19 - Chemical Principles

Question

Sodium vapor lamps, used for public lighting, emit yellow light of wavelength \(589 \mathrm{~nm}\). How much energy is emitted by (a) an excited sodium atom when it generates a photon; (b) \(5.00 \mathrm{mg}\) of sodium atoms emitting light at this wavelength; (c) \(1.00 \mathrm{~mol}\) of sodium atoms emitting light at this wavelength?

Step-by-Step Solution

Answer

The energy emitted (a) by an excited sodium atom when it generates a photon is approximately 3.37 x 10^{-19} joules; (b) by 5.00 mg of sodium atoms is approximately 1.61 x 10^{-3} joules; (c) by 1.00 mol of sodium atoms is approximately 203.58 kJ.

Step 1: Calculate the energy of a single photon

First, find the energy of one photon using the equation from Planck's relation: \(E = hu\), where \(h\) is Planck's constant and \(u\) the frequency of the light. To find \(u\), use the relation \(c = \lambdau\), where \(c\) is the speed of light and \(\lambda\) is the wavelength. Thus, \(u = \frac{c}{\lambda}\). Substitute the values for \(c\) and \(\lambda\) to find the frequency. Then, use \(E = hu\) to calculate the energy of a single photon.

Step 2: Determine the number of sodium atoms in 5.00 mg

To determine the number of sodium atoms in a 5.00 mg sample, first, convert the mass of sodium to grams (mg to g). Then use the molar mass of sodium (\textasciitilde{23} g/mol) to convert the mass of sodium to moles. Use Avogadro's number (\(6.022 \times 10^{23}\) atoms/mol) to calculate the total number of sodium atoms.

Step 3: Calculate the total energy for 5.00 mg of Sodium

Multiply the energy of one photon (obtained in Step 1) by the total number of atoms (calculated in Step 2) to get the total energy emitted by the 5.00 mg sample.

Step 4: Calculate the total energy for 1.00 mol of Sodium atoms

Use the energy of a single photon and Avogadro's number to calculate the total energy emitted by 1.00 mole of sodium atoms. Multiply the energy of one photon by Avogadro's number to get the total energy.

Key Concepts

Planck's Relation

Planck's relation is a fundamental concept in quantum mechanics, establishing a direct link between the energy of a photon and its corresponding frequency. The formula, expressed as \( E = h u \), where \( E \) stands for energy, \( h \) is Planck's constant (approximately \( 6.626 \times 10^{-34} \) Joule seconds), and \( u \) is the frequency of the photon.

This equation is vital in understanding the quantized nature of light. Essentially, it implies that light can be absorbed or emitted in discrete packets, or 'quanta', of energy called photons. It is the foundation for photon energy calculation, which is intrinsic to understanding phenomena such as the emissions from a sodium vapor lamp.

Photon Energy Calculation

The process of calculating a photon's energy is straightforward when using Planck’s relation. To determine the energy of a photon given by a source like a sodium vapor lamp, one needs to know the frequency of the light. If the wavelength \( (\lambda) \) is given instead, the speed of light \( c \) can be utilized to find the frequency with the relationship \( u = \frac{c}{\lambda} \), where \( c \) is a constant value, approximately \( 3.00 \times 10^{8} \) meters per second.

With the frequency determined, one can simply substitute it into Planck's equation to get the energy. This quantitative approach is essential in areas ranging from the analysis of atomic spectra to the design of electronic devices.

Speed of Light

The speed of light, denoted by \( c \), is a physical constant important in many areas of physics. It’s value is roughly \( 3.00 \times 10^{8} \) meters per second (m/s). The constancy of the speed of light in a vacuum is one of the postulates of Einstein's theory of relativity and is used in photon energy calculations.

It is used to calculate the frequency of electromagnetic waves when their wavelength is known, through the equation \( c = \lambda u \). This parameter is crucial when determining the energy of photons emitted or absorbed by substances such as the sodium in sodium vapor lamps.

Avogadro's Number

Avogadro's number, approximately \( 6.022 \times 10^{23} \), represents the quantity of elementary entities (usually atoms or molecules) in one mole of a substance. This constant translates the microscopic scale of atoms and molecules to a macroscopic scale that we can work with in the laboratory.

Knowing Avogadro’s number allows us to calculate the number of atoms in any given mass of a substance, like sodium, which is exactly what is needed when calculating the total energy emitted from a specific quantity of sodium as is required in problems dealing with sodium vapor lamps.

Molar Mass

Molar mass is the mass of one mole of a substance and it is expressed in grams per mole (g/mol). This measure provides a bridge between the atomic world and the laboratory scale. For sodium, the molar mass is approximately 23 g/mol. When dealing with exercises involving the emission of light from sodium vapor lamps, molar mass allows one to convert between the mass of sodium and the number of moles, and subsequently to the quantity of sodium atoms by using Avogadro's number.

This step is essential because it determines the scale of the reaction or phenomenon - that is, how many photons will be emitted, and consequently, the total energy output which could be calculated by multiplying the energy of a single photon by the number of photons emitted.