Titanium metal requires light with a maximum wavelength of 286 nm to emit electrons. (a) What is the minimum energy of the photons necessary to emit electrons from titanium via the photoelectric effect? (b) If titanium is irradiated with light of wavelength 276 nm, what is the maximum possible kinetic energy of the emitted electrons?

Verified step by step guidance

Step 1: Understand the photoelectric effect, which states that electrons are emitted from a material when it is exposed to light of a certain minimum frequency or maximum wavelength. The energy of the photons must be greater than or equal to the work function of the material.

Step 2: Use the equation for the energy of a photon, E = \frac{hc}{\lambda}, where h is Planck's constant (6.626 \times 10^{-34} \text{ J s}), c is the speed of light (3.00 \times 10^8 \text{ m/s}), and \lambda is the wavelength of the light. Calculate the minimum energy of the photons using the maximum wavelength of 286 nm.

Step 3: Convert the wavelength from nanometers to meters for calculation purposes. Remember that 1 nm = 1 \times 10^{-9} m.

Step 4: For part (b), calculate the energy of the photons with a wavelength of 276 nm using the same formula, E = \frac{hc}{\lambda}.

Step 5: Determine the maximum kinetic energy of the emitted electrons by using the equation KE_{max} = E_{photon} - \phi, where \phi is the work function of titanium, which is the energy calculated in Step 2.

Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Photoelectric Effect

The photoelectric effect is a phenomenon where electrons are emitted from a material when it absorbs light of sufficient energy. This effect demonstrates the particle nature of light, where photons must have a minimum energy, determined by the material's work function, to dislodge electrons. The relationship between the energy of the photons and the emitted electrons is crucial for understanding how light interacts with matter.

Photon Energy Calculation

The energy of a photon can be calculated using the equation E = hc/λ, where E is the energy, h is Planck's constant (6.626 x 10^-34 J·s), c is the speed of light (3.00 x 10^8 m/s), and λ is the wavelength of the light in meters. This relationship shows that shorter wavelengths correspond to higher energy photons, which is essential for determining whether the light can cause electron emission in the photoelectric effect.

Kinetic Energy of Emitted Electrons

The maximum kinetic energy (KE) of emitted electrons in the photoelectric effect can be calculated using the equation KE = E - φ, where E is the energy of the incoming photon and φ is the work function of the material. This concept is important for understanding how excess energy from the photon translates into the kinetic energy of the emitted electrons, allowing for the calculation of their maximum speed and behavior.

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