Which law states that internal energy is a state function and that the increase in internal energy of a system equals the heat supplied to the system plus the work done on (or equivalently, minus the work done by) the system?

Introduction / Context:
Thermodynamics deals with heat, work and internal energy. One of its foundational principles connects these quantities and expresses the law of conservation of energy for thermodynamic systems. This principle states that any change in the internal energy of a system results from the heat added to the system and the work done on or by the system. This question asks you to identify the named law that encapsulates this relationship.

Given Data / Assumptions:

• A thermodynamic system exchanges heat and work with its surroundings.
• Internal energy is treated as a state function depending on variables like temperature and volume.
• The change in internal energy is related to heat supplied and work done.
• We use the sign convention that increase in internal energy equals heat supplied plus work done on the system (or heat supplied minus work done by the system).

Concept / Approach:
The first law of thermodynamics is essentially a statement of energy conservation in thermodynamic processes. In differential form, it can be expressed as dU = dQ - dW, where dU is the change in internal energy, dQ is the heat supplied to the system and dW is the work done by the system. Alternatively, with a different sign convention for work, it may be written as dU = dQ + dW_on. The crucial idea is that internal energy is a state function and its change depends only on initial and final states, while heat and work are path dependent.

Step-by-Step Solution:
Step 1: Identify the key statement: increase in internal energy equals heat supplied plus or minus work term, depending on sign convention. Step 2: Recognise that this is a direct expression of energy conservation applied to thermodynamic systems. Step 3: Recall that the law saying stress is proportional to strain within elastic limit is Hooke's law, not related to heat and work. Step 4: Recall that Coulomb's law describes the electrostatic force between point charges, again unrelated to internal energy changes. Step 5: Faraday's law deals with electromagnetic induction, relating induced emf to the rate of change of magnetic flux. Step 6: Therefore, the only option that matches the described heat work energy relationship is the first law of thermodynamics.

Verification / Alternative check:
Examples from thermodynamics courses show that for an isochoric process (constant volume) where no work is done, the change in internal energy equals the heat supplied. For isothermal processes of ideal gases, internal energy change is zero, so heat supplied equals work done by the gas. These applications all follow directly from the first law and confirm its role as the energy conservation rule in thermodynamic contexts.

Why Other Options Are Wrong:
Hooke's law: States that within the elastic limit of a material, stress is proportional to strain; it does not involve internal energy, heat or work.

Coulomb's law: Gives the magnitude of electrostatic force between two charges and depends on distance and charge values, not on thermodynamic variables.

Faraday's law: Relates induced emf in a circuit to the time rate of change of magnetic flux through it, unrelated to heat and internal energy in a thermodynamic sense.

Common Pitfalls:
Students sometimes mix up the numerical forms of gas laws and the first law. It is useful to remember that the first law always links three central quantities internal energy, heat and work and is a qualitative and quantitative statement of energy conservation. The presence of these three key words in the statement is a strong clue that the answer is the first law of thermodynamics.

Final Answer:
The statement describes the first law of thermodynamics.