Problem 134 - Chemistry The Molecular Science

Question

The energy of a photon needed to cause ejection of an electron from a photoemissive metal is expressed as the sum of the binding energy of the photon plus the kinetic energy of the emitted electron. Calculate the kinetic energy of an electron that is emitted from a strontium metal surface irradiated with photons of \(4.20 \times 10^{-7} \mathrm{~m}\) light. The binding energy of strontium is \(4.39 \times 10^{-19} \mathrm{~J}\).

Step-by-Step Solution

Answer

The kinetic energy of the electron is approximately \( 0.34 \times 10^{-19} \text{ J}.\)

Step 1: Calculate the Frequency of the Incident Light

First, we need to find the frequency \( u \) of the light. We use the speed of light equation: \( c = \lambda u \), where \( c = 3.00 \times 10^8 \text{ m/s} \) is the speed of light, and \( \lambda = 4.20 \times 10^{-7} \text{ m} \) is the wavelength of the light. Rearrange the formula to solve for \( u \): \[ u = \frac{c}{\lambda} = \frac{3.00 \times 10^8 \text{ m/s}}{4.20 \times 10^{-7} \text{ m}} \approx 7.14 \times 10^{14} \text{ Hz}. \]

Step 2: Calculate the Energy of the Photon

Using Planck's equation \( E = hu \), where \( h = 6.63 \times 10^{-34} \text{ J} \cdot \text{s} \) is Planck's constant, calculate the energy \( E \) of the photon: \[ E = hu = (6.63 \times 10^{-34} \text{ J} \cdot \text{s})(7.14 \times 10^{14} \text{ Hz}) \approx 4.73 \times 10^{-19} \text{ J}. \]

Step 3: Calculate the Kinetic Energy of the Emitted Electron

The kinetic energy (KE) of the emitted electron is obtained by subtracting the binding energy (BE) from the photon's energy (E): \[ \text{KE} = E - \text{BE} = 4.73 \times 10^{-19} \text{ J} - 4.39 \times 10^{-19} \text{ J} = 0.34 \times 10^{-19} \text{ J}. \] Thus, the kinetic energy of the electron is approximately \( 0.34 \times 10^{-19} \text{ J}. \)

Key Concepts

Photon Energy

Photon energy is the energy carried by a single photon and is an essential concept in understanding the photoelectric effect. A photon is essentially a packet of light energy, and its energy is directly proportional to its frequency. To calculate the energy of a photon, we use the formula:
\[ E = hu \]
Here, \( E \) represents the energy of the photon, \( h \) is Planck's constant \( (6.63 \times 10^{-34} \, \text{J} \cdot \text{s}) \), and \( u \) is the frequency of the photon.

• Higher frequency photons have more energy.
• Visible light, X-rays, and gamma rays are all made up of photons, differing only by their frequency and energy levels.

Photons with sufficient energy can cause electrons to be ejected from metal surfaces, a phenomenon explained by the photoelectric effect. Understanding photon energy helps us predict whether an electron will be released from a given material when light is shone upon it.

Electron Binding Energy

Electron binding energy, or work function, is the minimum energy required to remove an electron from a solid's surface. In the context of the photoelectric effect, this is the energy that must be overcome by the incoming photon's energy to eject an electron. If a photon has energy equal to or greater than the binding energy:

• The electron will be released.
• If the photon's energy exceeds the binding energy, the excess energy becomes kinetic energy.

For materials like metals, binding energy is a critical factor. Different metals have distinct binding energies, which is why certain metals are more efficient for the photoelectric effect. For instance, in the given problem, the binding energy of strontium metal is \( 4.39 \times 10^{-19} \) Joules. Only photons with energy greater than this value can cause electron ejection from strontium's surface.

Kinetic Energy Formula

The kinetic energy formula is central to the photoelectric effect as it describes the excess energy of the ejected electron after overcoming its binding energy. The kinetic energy \( (KE) \) of an emitted electron is calculated using the photon energy \( (E) \) and the electron's binding energy \((\text{BE})\) as follows:
\[ \text{KE} = E - \text{BE} \]

• \( KE \) is the leftover energy after subtracting the binding energy from the total photon energy.
• A higher kinetic energy implies a faster-moving electron.

In our exercise, the calculation showed that the kinetic energy of the emitted electron from strontium is \( 0.34 \times 10^{-19} \) Joules. This outcome arises because the photon energy \( (4.73 \times 10^{-19} \, \text{J}) \) exceeds the binding energy, and the difference translates into the kinetic energy, dictating the velocity of the emitted electron.