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Internal energy, often denoted as \(U\), is a key concept in thermodynamics that refers to the total energy contained within a system. It encompasses all forms of energy, like kinetic and potential energy, that are contained within the system's particles.
One of the crucial aspects of internal energy is that it is a state function. This means that the internal energy depends only on the current state of the system, not on the path used to reach that state. Hence, if a system is taken from one state to another, the change in internal energy \(ΔU\) will remain the same irrespective of the process path taken. This is why in the given problem, the change \(ΔU\) for both paths iaf and ibf is the same, i.e., \(30\) cal.
The relationship between internal energy, heat, and work is governed by the first law of thermodynamics, which states:

• \( ΔU = Q - W \)

Therefore, changes in internal energy can occur due to heat transfer into or out of the system or work done by or on the system.

Heat transfer, symbolized as \(Q\), is a vital thermodynamic concept referring to the exchange of thermal energy between the system and its surroundings. It can occur through different means, such as conduction, convection, or radiation.
In the context of thermodynamics, heat is energy in transit and is a process-dependent function. This means it varies depending on the path taken to reach a particular state. Therefore, for the paths discussed in this textbook problem, the heat transferred varies:

• For path iaf, \(Q = 50\) cal
• For path ibf, \(Q = 36\) cal

The first law of thermodynamics incorporates heat transfer, enabling us to determine the internal energy changes. Hence, the application of this law in solving problems like this requires careful measurement of \(Q\) as a function of path.

The term "work done," denoted as \(W\), refers to energy transfer performed by a system on its surroundings or vice versa. Work is associated with a force causing a displacement, and it is also path-dependent. This means the amount of work done varies, depending on the route taken between two states.
For thermodynamic systems, work can often involve volume changes, like in the expansion or compression of gases. In the given exercise, we see two different values for work done along the paths:

• Along path iaf, \(W = 20\) cal
• The correct calculation for path ibf, given \(ΔU\) and \(Q\), yields \(W = 6\) cal

The first law of thermodynamics helps us calculate work, as we rearrange the law to solve for \(W\): \(W = Q - ΔU\).
Mastering how work is calculated in different thermodynamics scenarios is crucial for accurate energy system analysis.