What is Inexact differential: Definition and 14 Discussions
An inexact differential or imperfect differential is a type of differential used in thermodynamics to express changes in path dependent quantities. In contrast, an integral of an exact differential (a differential of a function) is always path independent since the integral acts to invert the differential operator. Consequently, a quantity with an inexact differential cannot be expressed as a function of only the variables within the differential; i.e. its value cannot be inferred just by looking at the initial and final states of a given system. Inexact differentials are primarily used in calculations involving heat and work because they are path functions, not state functions.
This is from Callen's thermodynamics. What does the differentiation with respect to T means for an inexact differential like dQ. Also why is T treated as a constant if we start by replacing dQ by TdS? Any references to the relevant mathematics will be much appreciated.
Today the inexact differential is usually denoted with δ, but in a text by a Russian author I found a dyet (D-with stroke, crossed-D) instead:
In response to my question to the author about this deviation from normal usage, he stated that this was a suggestion from von Neumann. (Which of course...
In Thermodynamics, I have seen that some equations are expressed in terms of inexact differentials, ##\delta##, instead of ##d##. I understand that this concept is introduced to point out that these differential forms are path-dependent, although I am not clear how they can be handled.
So, are...
Here, M = ##siny*cosy +xcos^{2}y ## and N = x
## M_y = (1/2)cos(2y) -xsin(2y)##
and ##N_x = 1##
Theorems:
If R = ## \frac{1}{N} (M_y - N_x) = f(x), then I.F. = e^{ \int f(x) dx} ##
If R = ## \frac{1}{M} (N_x - M_y) = g(y), then I.F. = e^{ \int g(x) dx} ##
Neither is holding true.
What should...
Homework Statement
The question is to solve the inexact equation by turning it into exact.the equation is ##( x + y + 4 ) d x + ( - x + y + 6 ) d y = 0##
Where "x" and "y" are variable.
2. Homework Equations [/B]
1.(x+y+4)=m and (-x+y+6)=n
2.Integrating Factor =##\frac { 1 } { x ^ { 2 } + y...
How is mass flow rate within an annular region of a pipe taken to be an inexact differential?
I read it in Fluid Mechanics textbook by Yunus A. Cengel and John M. Cimbala.
The mass flow rate through the annulus is given to be inexact differential. Why is mass flow through the annulus not equal...
Homework Statement
##dz=Mdx+Ndy## is an exact differential if ##(\frac{\partial M}{\partial y})_x=(\frac{\partial N}{\partial x})_y##.
By invoking the condition for an exact differential, demonstrate that the
reversible heat ##Q_R## is not a thermodynamic property.
Homework Equations...
I am looking for a general expression for an integrating factor μ(x,t) to solve the following diffential equation for x(t)
\frac{dx}{dt} = \frac{x - f}{x}
f = f(t) is an arbitrary function of t with f > 0 and df/dt < 0
Any ideas?
hello ... I wanted to demonstrate the solution of a differential equation, and as I do not yet know the latex language, then I take pictures of the leaves with my solutions ... the user who wants to give more solutions is well received and is welcome ...
att
jefferson alexander vitola (Smile)...
Hey,
Given the fundamental thermodynamic relation (dN=0) ; TdS=pdV+dU, I have to argue that pdV is inexact. I know that dS and dU are exact differentials and that the integral of an exact differential around a closed loop will give a zero result, not too sure where to start, the question is...
Homework Statement
I have another question. The following equation is inexact. Find an integrating factor u that makes the equation exact.
(-9/6)x4y6+3x8y7+x6)dx+(−1x5y5+(13/21)x9y6)dy=0
Homework Equations
Call the part before dx M, and the part before dy N.
The Attempt at a Solution
We did...
Homework Statement
Is (x2 -y)dx + xdy = dF an exact differential ?
dF = path dependant ( i.e. d has a line strikethrough it as h in Planck's constant.)
Homework Equations
(x2 -y)dx + xdy = dF
The Attempt at a Solution
I don't know what exact and inexact differentials are...