Introducing δQint =T d S into Eq. 6.10 gives Using this expression together with the test for ex

step 1 Okay to solve the next question we have that the simple relation is u equals to you into you in

Introducing δQint =T d S into Eq. 6.10 gives Using this...

Key Concepts

Exact Differential

An exact differential is a differential form that corresponds to the total change of a state function. In thermodynamics, if a differential form satisfies the condition ?M/?y = ?N/?x (for a form expressed as M(x, y)dx + N(x, y)dy), then it is exact, meaning its integral between two states is independent of the path taken.

State Function Versus Path Function

A state function is a property of a system that is determined solely by its current state, irrespective of the path by which the system arrived at that state. In contrast, a path function depends on the particular process. Since the integrated quantity of heat (Q_int) is path dependent, it cannot be regarded as a state function.

Inexact Differential

An inexact differential represents a quantity whose integral depends on the process or path taken between two states. In thermodynamics, heat and work are classic examples of quantities described by inexact differentials, which means that they are not determined solely by the state of the system but by the manner in which the system transits between states.

Test for Exactness

The test for exactness involves verifying whether the mixed partial derivatives of the components of a differential form are equal. When a differential form does not satisfy this condition, it is deemed inexact, indicating that the quantity defined by the differential (such as Q_int) is a path function and not a thermodynamic property that is solely dependent on the state.

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