Internal Energy and Thermodynamics

The internal energy of a system, often denoted by \(U\), is a fundamental concept in thermodynamics. It represents the total energy contained within the system, including the kinetic and potential energies of its molecules. When a system undergoes a change, its internal energy can change due to heat transfer and work done.

Thermodynamic Process and Internal Energy Change

For any infinitesimal process involving a system, the change in internal energy (\(dU\)) is related to the infinitesimal heat transferred to the system (\(dQ\)) and the infinitesimal work done *by* the system (\(dW\)). This relationship is expressed by the First Law of Thermodynamics:

\(dU = dQ - dW\)

This equation states that the change in internal energy is the heat added to the system minus the work done *by* the system.

Reversible Process and its Implications

The question specifies an "infinitesimal reversible process". A reversible process is an idealized process that proceeds infinitely slowly through a series of equilibrium states. For a reversible process, the heat transfer (\(dQ\)) and the work done (\(dW\)) can be expressed in terms of state variables. For an infinitesimal reversible process:

• The heat transfer \(dQ\) is given by \(TdS\), where \(T\) is the absolute temperature and \(dS\) is the change in entropy. This is derived from the definition of entropy, \(dS = dQ_{rev}/T\).
• The work done \(dW\) *by* the system is typically pressure-volume work, given by \(pdV\), where \(p\) is the pressure and \(dV\) is the change in volume.

Deriving the Change in Internal Energy

Now, we can substitute these expressions for \(dQ\) and \(dW\) into the First Law of Thermodynamics equation for the infinitesimal reversible process:

\(dU = dQ - dW\)

Substituting \(dQ = TdS\) and \(dW = pdV\):

\(dU = TdS - pdV\)

This equation gives the change in internal energy (\(dU\)) for an infinitesimal reversible process between two equilibrium states in terms of changes in entropy (\(dS\)) and volume (\(dV\)). This is a fundamental thermodynamic relation, sometimes called the first TdS equation.

Comparing with Options

Let's look at the given options and compare them with our derived equation:

• Option 1: \(dU = TdS + pdV\)
• Option 2: \(dU = TdS - pdV\)
• Option 3: \(dU = pdV - TdS\)
• Option 4: \(dU = -TdS - pdV\)

Our derived relation, \(dU = TdS - pdV\), matches Option 2. Therefore, the change in internal energy for a system that goes through an infinitesimal reversible process between two equilibrium states is \(dU = TdS - pdV\).