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The first law of thermodynamics is$$ \Delta U~=~Q-W \tag{1} \,,$$where:

  • $\Delta U$ is change in internal energy;

  • $Q$ is the amount of heat supplied to the system;

  • $W$ is the amount of work done by the system to the environment.

For a conservative force $F,$ work done by it is equal to

$$W = -\Delta U \,. \tag{2}$$

$\operatorname{Eq.}\left(2\right)$ is also the first law of thermodynamics (a version of the conservation of energy) with an adiabatic change $\left(Q=0\right),$ where positive work is done by the system.

Questions:

  1. Why isn't $\operatorname{Eq.}\left(2\right)$ applicable to a non-conservative force even though every object in this world follows conservation of energy?

  2. Is there any special case or condition where a non-conservative force, such as an external force by me on an object, follows the above equation?

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The above form is not the most general form of the 1st law of thermodynamics, which is: $$ dU = \delta Q - \delta W $$ Here $\delta Q$ is the change in heat, i.e., the heat flowing into or out of the system. This is a statement of energy conservation.

Now, if the heat flow is zero, it reduces to your above equation. Non-conservative forces are usually forces, e.g., friction, which cause some kind of heat dissipation, and thus the formula is not applicable.

There can be no special case/condition where $dU = - \delta W$ for non-conservative forces because the internal energy $U$ is a state function. Any process starting and ending at the same state must have the same internal energy. But, the definition of a non-conservative force is one where going "in a closed loop" from one state back to itself, does not preserve energy - thus there must be processes in that cycle where there is dissipation in the form of heat (otherwise the force would not be non-conservative). For these, $dU=-\delta W$ does not apply.

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  • $\begingroup$ OK in a general situation of electrostatics when we need to find electric potential at a point A of a +ve source charge we placed a unit +ve test charge which experiences a force F We moved it a little bit to B with same force F' but in opposite direction(towards the source charge) with constant velocity. F is a conservative force so the formula is applicable but for F' is it applicable. Or for F' it will be W=∆U Please reply $\endgroup$ Aug 21 '16 at 16:30
  • $\begingroup$ It is applicable if and only if $F'$ is a conservative force. So it really depends on how we move the charge to that point B. $\endgroup$
    – Sanya
    Aug 21 '16 at 17:42
  • $\begingroup$ So @Sanya u are saying that if F' is a conservative force or there is no dissipation of heat in any form then for F' work done W'=-∆U. Let it be. If u see that F(conservative force) is opposite to F' then work done W' by F' will be negative to work done W by F. Hence W' = -W Also W' = -∆U so -∆U = -W . Thus W = ∆U. How could this be possible if F is conservative force which is actually the electrostatic repulsion between source charge and test charge? $\endgroup$ Aug 21 '16 at 18:35
  • $\begingroup$ Either $W'=-\Delta U$ or $W=-\Delta U$, the other one is positive. The sign convention is yours to choose. $\endgroup$
    – Sanya
    Aug 21 '16 at 19:23
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A conservative force is one which acts against displacement, and stores all work energy for later full release back into work. Example: pressure on a piston in a cylinder of ideal gas, with frictionless movement between piston and cylinder, and perfectly insulating walls. Such a system produces a change in the internal energy of the system equal to the work done on the system. With no heat transfer, the first law equation $\Delta U = Q-W$ reduces to: $\Delta U=-W$

Question 1: Is the full first law equation applicable when non-conservative forces act in a non-adiabatic process? Specifically, is energy conserved in the above system when only a portion of the energy of work-energy is converted into displacement (reduced volume and increased pressure with a corresponding increased temperature), a portion lost to thermal energy by friction, and a portion of heat lost to the environment?

Answer 1: Yes. All processes conserve energy when considering the system and its environment. The sum of the recoverable energy stored in the compressed gas, plus the unrecoverable energies lost environment, equal the energy supplied to the system as work energy.

Question 2: Is the reduced first law $U=-W$ applicable in processes involving non-conservative forces?

Answer 2: No, the reduced first law is applicable only to adiabatic processes.

*The generalized form of the first law $\Delta U=Q-W$ is applicable to all processes, even if they involve non-conservative forces, irreversible processes, friction, non-ideal gasses, inelastic collisions, chemical bond breakage/formation, or convective-conductive-radiative heat loss. * The kinetic and potential energies associated with heat and work may convert into any other energy forms, but the total energy of the system and environment remains constant.

  • Example of Energy Conservation: Work in an adiabatic system, compressing a gas, increases its pressure. The kinetic energy of the moving piston transfers all its kinetic energy to the gas molecules. Work increases the Temperature proportional to the Internal Energy, as seen in the Internal Energy equation: for gas with n moles, at Temperature T Kelvin: $U=\frac{3}{2} nRT$.

  • In an adiabatic system, all the Internal Energy change is reflected in the temperature change, and in the case of a piston-cylinder-gas system, is due to compressive work.

  • But, there are two modes of energy transfer in the first law: heat and work. Obviously, if we only apply Work to the system, all of the increase in Temperature/Internal Energy necessarily comes from the applied Work.
  • But, in the real world, systems are only approximately adiabatic. Work-energy done on the system may not be conserved as useful energy, and non-conservative forces (such as friction) may convert work energy into thermal energy (instead of being stored as potential energy), and pass out of the system as heat.

Question 2: If a non-conservative force acts within a non-adiabatic system, can the system return to the same state at the end of a cycle?

Answer 2: No. The energy lost to the environment will not allow the system, to return to its original state - it does not have enough potential energy to restore the system to its original Internal energy.

  • A non-conservative force implies path dependence.

  • Example: friction as a non-conservative force: slide an eraser up the wall. The eraser gains $U=mgh$ joules of gravitational potential energy going from State A to State B. Gravity, a conservative force, stores energy as gravitational potential, but the sliding friction of the eraser requires additional energy which cannot be returned to the environment as work. The longer the path, the more energy is lost to friction - thus, path dependence.

  • Conservative Force Definition: A conservative force is a force with the property that the total work done in moving a particle between two points is independent of the taken path. ... When an object moves from one location to another, (a conservative) force changes the potential energy of the object by an amount that does not depend on the path taken.

  • In the real world, a gas-cylinder-piston system, is only approximately adiabatic and has minimal friction with good lubrication. Thus, compressing the gas will generate some thermal energy due to friction, and some heat will be lost to the environment.

  • The thermal energy lost to friction, and then to the environment, will not be available to return the piston to its original state. It is this loss of work energy to thermal energy, and that to the environment, which is the defining characteristic of the non-conservative force. When force and movement create heat, which is then lost from the system, and cannot be used to restore the original state by work, the force is non-conservative.
  • Example: In the case of the piston-cylinder-gas system, State A will have V1, P1, and T1. The work done will compress it to state B, with V2, P2, T2, while losing Qloss to the environment. Upon release of the compressive force on the piston, the system will return to a V3, P1, and T3. V3 will be smaller than V1 because energy was lost.
  • Friction is a non-conservative force, which makes the amount of work added and returned in a work cycle dependent upon the path taken. When there are non-conservative forces acting in a system, a cycle between State A and B, and back towards A, the system will not be able to fully return to State A without the addition of the heat lost to the environment.
  • The First Law still holds, it is merely that the system has generated thermal energy by friction, and lost heat to the environment in a method other than by returning work. The entire total system-environment energy has not changed, only the energy available to do work on the return cycle has been altered by the presence of the non-conservative (friction) force.
  • Note - regarding the storage of work-energy as thermal energy in a compressed gas: The increased gas pressure reduces the average intermolecular distance due to the increased kinetic energy of gas molecules. The increased kinetic energy allows and requires closer approach between electron clouds. The reduced average distance is necessary to produce enough repulsive force for rebound/recoil after the collision.
  • Nonconservative forces produce "losses" due to work-energy conversion into thermal energy through many methods, such as, inelastic deformation, radiation, vibratory and rotational modes of internal motion, and chemical bond formation.
  • To the extent that these processes increase Temperature, this increase in thermal energy content increases the Internal Energy. But, if the internal energy becomes locked (such as in metal deformation, or heat loss through conduction, convection, radiation to the environment), the system is non-conservative, and it will not be able to return to the original state without the addition of energy.
  • A system which produces non-useful, "parasitic heat loss" appears "non-conservative" from the perspective of using force to produce intended/desired work. But, non-conservation of energy (when including system + environment) is always illusory - just look for the hole where energy is leaking in or out - it's always there.
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