A system consists of N particles, interacting with each other (for example, protein molecule). 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 or translational degrees of freedom are 3 (d) Number of vibrational degrees of freedom are 3 (e) The system, if isolated, will conserve both total energy and total angular momentum . No links , give detailed answers , please
Dear Student,
Degree of freedom can be defined as the direction in which a particle can move freely provided that it does not have any constraints on it.( A vague definition but that will do for now)
Now suppose a particle is free to move in 3 dimensions ,then it can move in three axes (x,y,z) so it has 3 degree of freedom.
If I take 2 particles then DOF will be 6 , Three for each.
Thus for N no. of particle, Total degree of freedom= 3N.
The whole system can travel in all three dimensions. So, No. of translational degrees of freedom=3.
Similarly, the system can rotate about any of the axis in 3-D. So, No. of translational degrees of freedom=3.
As Total DOF=3N. Remaining DOF will go with vibrational degrees of freedom i.e. No. of vibrational degrees of freedom=3N-6.
Clearly, options a, b, c are correct. d is incorrect. Let us check for option e.
Law of equipartition of energy states that for a dynamical system in thermal equilibrium the total energy of the system is shared equally by all the degrees of freedom.
The energy associated with each degree of freedom per molecule is ½ kT, where k is the Boltzmann’s constant.
And No. of degrees of freedom =3N.
So, Energy of whole system = .
After isolating, DOF of one molecule=3 and energy associated with each degree of freedom per molecule is ½ kT.
So, Energy of one molecule= .
As there are N isolated particles, Energy of N particles after isolating= .
So, Energy is conserved if system of N particles is isolated as Energy before isolating and after isolating are equal. And as there is net external torque on the system of N particles = 0 because all the interactive forces between particles are internal to system. So, there is no external force to generate torque. Thus, Angular momentum will also conserve.
So, Only incorrect option is (d).
Regards
Degree of freedom can be defined as the direction in which a particle can move freely provided that it does not have any constraints on it.( A vague definition but that will do for now)
Now suppose a particle is free to move in 3 dimensions ,then it can move in three axes (x,y,z) so it has 3 degree of freedom.
If I take 2 particles then DOF will be 6 , Three for each.
Thus for N no. of particle, Total degree of freedom= 3N.
The whole system can travel in all three dimensions. So, No. of translational degrees of freedom=3.
Similarly, the system can rotate about any of the axis in 3-D. So, No. of translational degrees of freedom=3.
As Total DOF=3N. Remaining DOF will go with vibrational degrees of freedom i.e. No. of vibrational degrees of freedom=3N-6.
Clearly, options a, b, c are correct. d is incorrect. Let us check for option e.
Law of equipartition of energy states that for a dynamical system in thermal equilibrium the total energy of the system is shared equally by all the degrees of freedom.
The energy associated with each degree of freedom per molecule is ½ kT, where k is the Boltzmann’s constant.
And No. of degrees of freedom =3N.
So, Energy of whole system = .
After isolating, DOF of one molecule=3 and energy associated with each degree of freedom per molecule is ½ kT.
So, Energy of one molecule= .
As there are N isolated particles, Energy of N particles after isolating= .
So, Energy is conserved if system of N particles is isolated as Energy before isolating and after isolating are equal. And as there is net external torque on the system of N particles = 0 because all the interactive forces between particles are internal to system. So, there is no external force to generate torque. Thus, Angular momentum will also conserve.
So, Only incorrect option is (d).
Regards