Question

A system consists of N particles, interacting with each other (for example, protein molecules). Which one of the following statements is FALSE?

• A. The motion of the system can be split into translational, rotational, and vibrational motions
• B. Number of rotational degrees of freedom are 3
• C. Number of vibrational degrees of freedom are 3
• D. Number of translational degrees of freedom are 3
• E. The system, if isolated, will conserve both total energy and total angular momentum.

Hint:

When analyzing the degrees of freedom for a system of particles, consider the physical state and type of particles involved (e.g., atoms, molecules).

Answer 1

The Correct Option is C

Let's analyze each statement regarding a system of N interacting particles (like a protein molecule):

1. Statement (a): The motion of the system can be split into translational, rotational, and vibrational motions.
   • This is a standard approach in physics and chemistry for analyzing the dynamics of multi-particle systems. The overall motion can be described by the translation of the center of mass, the rotation of the system around the center of mass, and the internal vibrations of the particles relative to each other.
   • This statement is generally TRUE.
2. Statement (d): Number of vibrational degrees of freedom are 3.
   • The translational motion of the entire system is described by the movement
   • These 3N degrees of freedom are distributed among translation, rotation, and vibration.
   • Number of Vibrational Degrees of Freedom = Total DOF - Translational DOF - Rotational DOF
   • Number of Vibrational Degrees of Freedom = 3N - 3 - 3 = 3N - 6 (for a non-linear system).
   • The statement claims the number of vibrational degrees of freedom is 3. This is only true if 3N - 6 = 3, which means 3N = 9, or N = 3. For any system with N ≠ 3 particles, the number of vibrational degrees of freedom is not 3. For a protein molecule, N is large, so 3N-6 is much larger than 3.
   • Statement (d) is FALSE in general.
3. Evaluate Statement (e): Conservation Laws
   • An isolated system is one with no external forces or torques acting on it.
   • Conservation of Energy: In the absence of external work and non-conservative internal forces, the total energy (kinetic + potential) of an isolated system is conserved.
   • Conservation of Angular Momentum: In the absence of external torques, the total angular momentum of an isolated system is conserved.
   • These are fundamental principles of physics.
   • Statement (e) is TRUE.
4. Conclusion: Statement (c) makes a claim about the number of vibrational degrees of freedom that is generally incorrect for a system of N particles.

The FALSE statement is (c).

Answer 2

Concept:
A system of $N$ interacting particles (e.g., molecules) can move in the following ways:

• Translational motion: Whole system moves in space (3 degrees of freedom)
• Rotational motion: System can rotate around 3 axes (up to 3 degrees of freedom for 3D objects)
• Vibrational motion: Internal motions of the particles relative to each other

Total degrees of freedom:
For $N$ particles in 3D, total degrees of freedom = $3N$. Breakdown:

• Translational: 3 degrees
• Rotational: 3 degrees (for a rigid 3D body)
• Vibrational: remaining degrees of freedom = $3N - 6$ (if the body is non-linear)

Now, evaluating each statement:

• (a) TRUE – The motion splits into translational, rotational, and vibrational
• (b) TRUE – For a general 3D body, there are 3 rotational degrees
• (c) FALSE – Vibrational degrees are usually $3N - 6$ (not just 3) for a non-linear system
• (d) TRUE – Translational motion always contributes 3 degrees
• (e) TRUE – Isolated systems conserve total energy and angular momentum

Correct answer: (c) Number of vibrational degrees of freedom are 3