A metal surface is exposed to 500 nm radiation. The threshold frequency of the metal for photoelectric current is \( 4.3 \times 10^{14} \text{ Hz} \). The velocity of ejected electron is \( x \times 10^5 \text{ m s}^{-1} \). (Nearest integer)
Use the following constants:
- Planck's constant (\(h\)) = \( 6.63 \times 10^{-34} \text{ J s} \)
- Mass of electron (\(m_e\)) = \( 9.0 \times 10^{-31} \text{ kg} \)
- Speed of light (\(c\)) = \( 3 \times 10^8 \text{ m s}^{-1} \)
Detailed Step-by-Step Solution
We will use Einstein's Photoelectric Equation: \( E = W_0 + KE_{\text{max}} \), where \( E \) is the energy of the incident photon, \( W_0 \) is the work function, and \( KE_{\text{max}} \) is the kinetic energy of the ejected electron.
Step 1: Calculate the Energy of Incident Photon (\(E\))
We are given the wavelength \( \lambda = 500 \text{ nm} = 500 \times 10^{-9} \text{ m} \).
\( E = \frac{19.89 \times 10^{-26}}{5 \times 10^{-7}} \)
\( E = 3.978 \times 10^{-19} \text{ J} \)
Step 2: Calculate the Work Function (\(W_0\))
We are given the threshold frequency \( \nu_0 = 4.3 \times 10^{14} \text{ Hz} \).
\( W_0 = 28.509 \times 10^{-20} \text{ J} \)
To easily subtract it from \(E\), write it in terms of \(10^{-19}\):
Step 3: Calculate Kinetic Energy and Velocity
Now, substitute \(E\) and \(W_0\) into the photoelectric equation to find \( KE_{\text{max}} \):
\( KE_{\text{max}} = (3.978 - 2.8509) \times 10^{-19} \text{ J} \)
\( KE_{\text{max}} = 1.1271 \times 10^{-19} \text{ J} \)
We know \( KE = \frac{1}{2}m_e v^2 \). Using \( m_e = 9.0 \times 10^{-31} \text{ kg} \), solve for \( v^2 \):
\( v^2 = \frac{2 \times 1.1271 \times 10^{-19}}{9.0 \times 10^{-31}} \)
\( v^2 = \frac{2.2542 \times 10^{-19}}{9.0 \times 10^{-31}} = 0.25046 \times 10^{12} \)
To find \(v\), take the square root. First, rewrite \(v^2\) to make the math easier:
\( v = \sqrt{25.046} \times 10^5 \approx 5.004 \times 10^5 \text{ m/s} \)
Conclusion: We found the velocity \( v \approx 5 \times 10^5 \text{ m/s} \). Comparing this to the given format \( x \times 10^5 \text{ m/s} \), the nearest integer value for \( x \) is 5.
Conquer Photoelectric Effect Numericals
This is a classic JEE Main question that heavily tests your ability to manipulate scientific notation and perform square roots of decimals quickly. Aligning the powers of 10 (e.g., matching \(10^{-20}\) to \(10^{-19}\) before subtracting) is a critical step to avoid careless calculation errors.
To master Einstein's Photoelectric Equation, work functions, and threshold energies, be sure to revise our in-depth notes on the Structure of Atom Class 11 Chemistry.
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