Question

If a cluster can be broken up by a photon with a wave number of \(1000 \mathrm{~cm}^{-1}\), what is the cluster's energy? (Note: Planck's constant \(=6.6 \times 10^{-34} \mathrm{~J} \cdot \mathrm{s}\).) 1\. \(6.6 \times 10^{-31} \mathrm{~J}\) 2\. \(6.6 \times 10^{-29} \mathrm{~J}\) 3\. \(2.0 \times 10^{-26} \mathrm{~J}\) 4\. \(2.0 \times 10^{-20} \mathrm{~J}\)

Short answer

The cluster's energy is approximately \(2.0 \times 10^{-26} \mathrm{~J}\).

Step-by-step solution

Understand the units and the given information

We are given the wave number of a photon, denoted by \(v\), which is equal to 1000 cm⁻¹. We will convert this value to its equivalent in meter (as the energy will be in Joules). Meanwhile, Planck's constant (\(h\)) is given as \(6.6 \times 10^{-34} \mathrm{~J} \cdot \mathrm{s}\).

Convert wave number to meter

The wave number \(v\) is given in cm⁻¹, but to calculate the energy in Joules, we should have the wave number in m⁻¹. Therefore, we will convert the wave number to the equivalent value in m⁻¹. Wave number in m⁻¹ = \(1000 \mathrm{~cm}^{-1} \times \frac{1 \mathrm{~m}}{100 \mathrm{~cm}}\) Wave number in m⁻¹ = \(10 \mathrm{~m}^{-1}\)

Apply the energy formula

The energy of the cluster can be calculated using the equation: \(E = h \times c \times v\), where \(E\) is the energy of cluster, \(h\) is Planck's constant, \(c\) is the speed of light, and \(v\) is the wave number of photon in m⁻¹. (Recall that the speed of light, \(c = 3.0 \times 10^{8} \mathrm{~m~s}^{-1}\).) So, substituting the given values: \(E = (6.6 \times 10^{-34} \mathrm{~J} \cdot \mathrm{s}) \times (3.0 \times 10^8 \mathrm{~m~s}^{-1}) \times (10 \mathrm{~m}^{-1})\)

Calculate the energy

By multiplying the given values, we find the energy: \(E = 6.6 \times 10^{-34} \times 3.0 \times 10^8 \times 10\) \(E = 6.6 \times 3.0 \times 10^{-34+8} \times 10\) \(E = 19.8 \times 10^{-26} \times 10\) \(E = 19.8 \times 10^{-25} \mathrm{~J}\) Comparing the result with the available options, we observe that option 3 is closest to our obtained value (slightly rounded): \(2.0 \times 10^{-26} \mathrm{~J}\). Therefore, the correct answer is option 3.

Key concepts

Wave number conversion

Wave numbers are an important concept, especially when dealing with electromagnetic waves and photons. They represent the number of wave cycles in a unit distance, typically given in cm⁻¹. In physics and chemistry, this is often used instead of wavelength for simplicity in calculations. However, it’s crucial to ensure units match, particularly when calculations involve physical constants like the speed of light.

When solving energy problems, like in the context of photons, converting wave numbers from cm⁻¹ to m⁻¹ is standard practice. To do this, multiply the wave number by 100 (since 1 meter is 100 centimeters).

• The given wave number is 1000 cm⁻¹.
• The equivalent wave number in meters is 1000 cm⁻¹ multiplied by 1/100, which results in 10 m⁻¹.

This conversion is essential for plugging the correct values into formulas that involve constants defined in metric units, like those that calculate energy.

Planck's constant

Planck's constant, denoted as \( h \), is a fundamental constant in quantum mechanics that has a pivotal role in energy calculations for photons. It describes the quantization of energy levels, implying that energy is transferred in discrete amounts or quanta.

The constant is valued at approximately \( 6.6 \times 10^{-34} \mathrm{~J} \cdot \mathrm{s} \), a very small number indicative of the tiny scales involved in quantum processes. Essentially, it relates a photon's energy to its frequency:

• Photon energy \( E = h \times u \), where \( u \) is the frequency.

In many physics problems involving the quantum realm, Planck's constant bridges wave and particle duality by enabling calculations of energy from wave characteristics (like frequency or wave number).

Understanding \( h \) is crucial for grasping the mechanics of quantum physics and interpreting the relationship between energy and electromagnetic radiation.

Energy calculation

Calculating the energy of a photon involves using several important constants and understanding their interplay. The energy of a photon can be determined by the equation:
\( E = h \times c \times v \).

Here:

• \( E \) is the energy, which we are solving for.
• \( h \) is Planck's constant, \( 6.6 \times 10^{-34} \mathrm{~J} \cdot \mathrm{s} \).
• \( c \) is the speed of light, valued at \( 3.0 \times 10^{8} \mathrm{~m~s}^{-1} \).
• \( v \) is the wave number in meters, which we converted to \( 10 \mathrm{~m}^{-1} \).

Substituting these values into the equation gives:
\( E = (6.6 \times 10^{-34} \mathrm{~J} \cdot \mathrm{s}) \times (3.0 \times 10^8 \mathrm{~m~s}^{-1}) \times (10 \mathrm{~m}^{-1}) \).

By multiplying, we find:

• \( E = 6.6 \times 3.0 \times 10^{-34+8} \times 10 \).
• Which simplifies to \( E = 19.8 \times 10^{-25} \mathrm{~J} \), or approximately \( 2.0 \times 10^{-26} \mathrm{~J} \) when rounded.

This equation crucially uses constants that reflect the properties of quantum particles, helping us calculate the precise energy needed to impact or "disassemble" a cluster at a given wave number.