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Consider a system having spherical symmetry filled with gas. If it expands or compress, there is some work done by system or surroundings, respectively, according to the equation $W=nRT\log(v_2/v_1)$. But the coordinates of centre of mass of system is same at the centre of sphere, so according to original definition of work ($W$ equal to force times displacement) it should be zero. I can't understand where is the mistake.

AccidentalFourierTransform
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Deepanshu
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    The displacement is calculated at the point where the force is applied—in this case, the boundary. – Chemomechanics Jul 28 '18 at 16:36
  • I respectfully request that consideration be given to re-opening the question. I agree it is essentially the same question as the cited one, but feel that none of the previous 8 answers pointed out the difference between boundary work (which is involved in the two questions and involve internal energy) and work that changes the the KE (and possibly PE) of the system as a whole by changing the velocity or elevation of the center of mass of the system. Force x displacement of the center of mass equal the change in KE (work-energy principle) assuming no change in PE. – Bob D Jul 31 '18 at 13:53

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You need to understand the difference between boundary work on a closed system (expanding or compressing the sphere) vs work on or by an external force on the system as a whole (changing the velocity or position of the sphere). The former does not change the coordinates of the center of mass, but the latter does. It’s the difference between the change in internal energy of the system related to expansion or contraction of the boundary of the system, and the change in external energy of the system related to the change in velocity and/or position of the center of mass of the system with respect to an external frame of reference.

The long version of the first law for a closed system is:

Q – W = ΔE = ΔU + ΔKE + ΔPE

Where

ΔE = Total energy change of the system, which is the sum of change in internal and external energy of the system.

ΔKE = Change in kinetic energy of the system as a whole. This relates to a change in the velocity of the center of mass. By the work energy principle:

F x d = ΔKE

ΔPE = Change in potential energy of the system as a whole, such as a change in elevation of the center of mass (change in gravitational potential energy).

Q and ΔU are as always.

W now includes both boundary work and work done on or by the system a whole.

Bob D
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