Internal energy and the first law of thermodynamics

Internal energy describes the state

Internal energy U accounts for microscopic energy associated with the selected material, including relevant motions and interactions. A thermodynamic state description determines its value relative to a chosen reference. The change ΔU = U_final − U_initial compares endpoints; a reference offset cancels from the difference.

Here the system is closed and stationary, with no separately changing bulk kinetic or gravitational potential energy. That scope lets the simple internal-energy balance focus on heat and work. Matter flow or other explicitly tracked energy stores require a broader ledger.

Add the transfers with their signs

Using the chemistry convention, the first law for this setting is ΔU = q + w. Receiving 500 J of heat and doing 200 J of work gives q = +500 J, w = −200 J and ΔU = +300 J. The internal-energy increase is smaller than the received heat because some energy leaves through work.

If the surroundings instead do work on the system, that contribution is positive. Do not attach a negative sign simply because the word work appears. First identify its direction relative to the chosen system, then add the signed transfers.

Endpoint changes and paths answer different questions

U is a state function: the same initial and final states have the same ΔU. Heat q and work w depend on how the process connects those states. A useful fictional accounting comparison stipulates two paths between the same states, one with +500 J heat and −200 J work, another with +450 J heat and −150 J work.

Both paths give ΔU = +300 J, although their separate heat and work differ. The table is an original ledger example, not a physical engine simulation. A consistent balance is necessary, while a real apparatus also needs a physically possible path and complete transfer accounting.

Zero net change does not imply no transfers

A system can have ΔU = 0 while receiving heat and doing an equal magnitude of work. Likewise, a closed cycle returns to its initial state and therefore has zero net change in U, even though heat and work can have crossed its boundary along the way.

For an isolated system in the stated setting, q and w are both zero and U remains constant. For an adiabatic system, only q is constrained to zero; its ΔU can equal nonzero work. These conclusions follow from the distinct boundary conditions rather than from interchangeable names.

Worked examples

Try it yourself

1. Must q have the same value for every path between the same endpoints?

   Show the answer and reasoning

   No.

   Heat is path dependent, while ΔU is determined by the endpoint states.

2. For q = +450 J and w = −150 J, what is ΔU?

   Show the answer and reasoning

   +300 J.

   Add the signed transfers.

3. If ΔU = 0, must heat and work each be zero?

   Show the answer and reasoning

   No.

   Nonzero transfers can cancel in the net ledger.

4. In this setting, an adiabatic system receives +100 J of work. What is ΔU?

   Show the answer and reasoning

   +100 J.

   q = 0, so ΔU = w.

What to carry forward

• Internal-energy change is an endpoint property.
• Heat and work depend on the path.
• Add transfers using the system’s sign convention.
• A zero net change can coexist with nonzero transfers.