Question

The energy difference between the \(5 d\) and \(6 s\) sublevels in gold accounts for its color. Assuming this energy difference is about \(2.7 \mathrm{eV}\) (electron volts; \(1 \mathrm{eV}=1.602 \times 10^{-19} \mathrm{~J}\) ), explain why gold has a warm yellow color.

Short answer

The energy difference produces light with a wavelength of about 459 nm, meaning gold absorbs blue-violet light and reflects yellow, giving it a warm yellow color.

Step-by-step solution

Understand the Energy Difference

The problem states that the energy difference between the 5d and 6s sublevels in gold is about 2.7 eV. Convert this energy difference to joules using the given conversion factor.

Convert Energy to Joules

Using the conversion factor provided, multiply 2.7 eV by the value of 1 eV in joules: \[ E = 2.7 \text{ eV} \times 1.602 \times 10^{-19} \text{ J/eV} = 4.324 \times 10^{-19} \text{ J} \]

Calculate the Wavelength of Light

Use the energy-wavelength relationship from quantum mechanics. The formula is given by: \[ E = \frac{hc}{\text{λ}} \] Here, - \( E \) is the energy in joules (\( 4.324 \times 10^{-19} \text{ J} \)) - \( h \) is Planck's constant (\( 6.626 \times 10^{-34} \text{ Js} \)) - \( c \) is the speed of light in a vacuum (\( 3.00 \times 10^{8} \text{ m/s} \))- \( \text{λ} \) is the wavelength. Rearrange to solve for wavelength (\( \text{λ} \)), we get: \( \text{λ} = \frac{hc}{E} = \frac{6.626 \times 10^{-34} \text{ Js} \times 3.00 \times 10^{8} \text{ m/s}}{4.324 \times 10^{-19} \text{ J} } = 4.59384 \times 10^{-7} \text{ m} \)

Convert Wavelength to Nanometers

Convert the wavelength from meters to nanometers as visible light wavelengths are usually represented in nanometers (1 meter = 10⁹ nanometers): \( \text{λ} \times 10^9 = 4.59384 \times 10^{-7} \text{ m} \times 10^9 \text{ nm/m} = 459.384 \text{ nm} \)

Compare to the Visible Spectrum

The wavelength of 459.384 nm falls within the visible light spectrum. Reference the visible light wavelength chart (approximately 390-700 nm) and note that around 459 nm corresponds to blue-violet light, not yellow. But remember, gold has a warm yellow color because the blue part of the spectrum is absorbed and the complementary color (yellow) is reflected.

Key concepts

Energy Levels

Energy levels in atoms are like steps electrons can stand on. These steps have specific energies.
When electrons jump between levels, they absorb or release energy.
This energy can be understood and calculated to predict properties like color.
Gold's unique warm yellow color is due to the energy difference between its 5d and 6s sublevels.

Wavelength Calculation

To find the wavelength of light related to an energy difference, we use a formula from quantum mechanics:
\ E = \frac{hc}{\lambda} \
Here's what these symbols mean:

• \( E \) is the energy in joules
• \( h \) is Planck's constant, \( 6.626 \times 10^{-34} \text{ Js} \)
• \( c \) is the speed of light, \( 3.00 \times 10^{8} \text{ m/s} \)
• \( \lambda \) is the wavelength

To solve for the wavelength, we rearrange the formula:
\ \lambda = \frac{hc}{E} \
After substituting the values, we get the wavelength of the light emitted when electrons in gold transition between the 5d and 6s sublevels.

Visible Light Spectrum

The visible light spectrum consists of the range of light wavelengths that humans can see.
This range is roughly between 390 and 700 nanometers (nm).
Different wavelengths correspond to different colors:

• Below 400 nm: Violet
• 450-495 nm: Blue
• 495-570 nm: Green
• 570-590 nm: Yellow
• 590-620 nm: Orange
• 620-750 nm: Red

In our case, we calculated a wavelength of 459.384 nm for gold's energy difference, which is in the blue-violet region.
However, gold appears yellow because it absorbs blue light and reflects yellow light, its complementary color.

Electron Volts Conversion

An electron volt (eV) is a unit of energy used in atomic and particle physics.
One electron volt is defined as the energy gained or lost by an electron when it moves through an electric potential difference of one volt.
To convert electron volts to joules, we use the conversion factor:
\( 1 \text{ eV} = 1.602 \times 10^{-19} \text{ J} \)
For example, the energy difference given was 2.7 eV:

• Multiply 2.7 eV by \( 1.602 \times 10^{-19} \)
• \( 2.7 \text{ eV} \times 1.602 \times 10^{-19} \text{ J/eV} = 4.324 \times 10^{-19} \text{ J} \)

This value is used in further calculations to determine the wavelength of light.