First TdS Equation:
Entropy (S) can be defined in terms of any two of P, V, and T. Let us define entropy S as the function of V and T, i.e. S (V, T). Derivative of entropy S is:
dS = (∂S/∂V)TdV + (∂S/∂T)VdT
Multiply equation by T to get:
TdS = T(∂S/∂V)TdV + T(∂S/∂T)VdT
(∂S/∂V)T = (∂P/∂T)V and also at constant volume T(∂S/∂T)V = (∂U/∂T)V = CV, thus equation becomes
TdS = T(∂P/∂T)VdV + CVdT
This equation is first TdS equation.
Second TdS:
Let us define entropy S in terms P and T i.e. S (P, T), its derivative is
dS = (∂S/∂p)TdP + (∂S/∂T)PdT
Multiplying equation by T provides
TdS = T(∂S/∂p)TdP + T(∂S/∂T)PdT
From Maxwell's relation (∂S/∂p)T = - (∂V/∂T)P, and also at constant pressure T(∂S/∂T)P = (∂H/∂T)P = CP. Thus Equation becomes TdS = -T(∂V/∂T)PdP + CPdT
This equation is second TdS equation.
Third TdS:
Then last option is to define entropy S as the function of P and V, i.e. S(P, V), we have
dS = (∂S/∂p)VdP + (∂S/∂V)PdV
Multiplying equation by T to get
TdS = T(∂S/∂p)VdP + T(∂S/∂V)PdV
In the constant volume process, it can be proved that:
T(∂S/∂p)V = CV(∂T/∂P)V
And also constant pressure process, it can be proved that
T(∂S/∂V)P = CP(∂T/∂V)P
As a result equation becomes:
TdS = CV(∂T/∂P)VdP + CP(∂T/∂V)PdV
This equation is third of TdS equations.
These equations allow one to compute heat flow (TdS) in the reversible process. Also, change in entropy between two states of the system can be estimated from these equations given that equations of state are known. This is due to all partial derivatives in these equations can be attained from equation of state.
Expansion, Compression and TdS Equations:
Compression:
The way the volume of the material decreases with pressure at constant temperature is explained by isothermal compressibility k
k = -1/V(∂V/∂P)T
Isothermal compressibility is different from another parameter known as adiabatic compressibility kad.
kad = -1/V(∂V/∂P)s
Reciprocal of k (i.e. 1/k) is known as isothermal bulk modulus or isothermal incompressibility.
Expansion:
The way the volume of the material increases with temperature at constant pressure is explained by coefficient of volume expansion or expansivity β.
β = 1/V(∂V/∂T)P
Unit of expansivity is K-1
Using equations, one can demonstrate that
(∂P/∂T)V = β/k and (∂V/∂T)P = βV and reciprocal of equation provides
(∂T/∂P)V = k/β and (∂T/∂V)P = 1/βV
TdS equations in terms of k and β:
Replacing partial derivatives with these, three TdS equations become
dS = β/kdV + (CV/T)dT
dS = -βVdP + (CP/T)dT
ds = (CVk/Tβ)dP + (CP/TβV)dV
These Equations give change in entropy between two states in terms of P, V, T, k, β and heat capacities as the function of temperature and pressure of specific volume. Implication of these equations is that one doesn't even require equation of state to compute change in entropy.
Tutorsglobe: A way to secure high grade in your curriculum (Online Tutoring)
Expand your confidence, grow study skills and improve your grades.
Since 2009, Tutorsglobe has proactively helped millions of students to get better grades in school, college or university and score well in competitive tests with live, one-on-one online tutoring.
Using an advanced developed tutoring system providing little or no wait time, the students are connected on-demand with a tutor at www.tutorsglobe.com. Students work one-on-one, in real-time with a tutor, communicating and studying using a virtual whiteboard technology. Scientific and mathematical notation, symbols, geometric figures, graphing and freehand drawing can be rendered quickly and easily in the advanced whiteboard.
Free to know our price and packages for online physics tutoring. Chat with us or submit request at [email protected]
Oligosaccharides tutorial all along with the key concepts of Chemistry of oligosaccharides, Kinds of oligosaccharides, Maltose, Lactose, Sucrose, Trehalose and Properties of oligosaccharides
tutorsglobe.com extraction of silver assignment help-homework help by online occurrence and principles of extraction of silver tutors
tutorsglobe.com practical application of genetic transformation assignment help-homework help by online transgenic plants tutors
tutorsglobe.com isoquants assignment help-homework help by online production function tutors
Theory and lecture notes of Labor Market all along with the key concepts of labor market, MPL used for Cobb-Douglas production function, marginal product of labor, MPL, typical firm's demand, labor demand. Tutorsglobe offers homework help, assignment help and tutor’s assistance on Labor Market.
Alpha-Beta Unsaturated Carbonyl Compounds tutorial all along with the key concepts of Structures and properties of unsaturated carbonyl compounds, Preparation of a, ß - unsaturated compounds, Nucleophilic and Electrophilic Centers, Michael Addition, Diels-Alder reaction and Quinones
tutorsglobe.com in vitro fertilization assignment help-homework help by online fertilization tutors
Top load - involves a wider variety of available models, colors and features like they have been on the market longer, Front load - people are employed to seeing front load washers in Laundromats, several brands are now available for home use.
tutorsglobe.com ureotelism assignment help-homework help by online excretion tutors
inherent limitations in accounting, homework help, assignment help by online tutors. get solved accounting questions by tutors.
tutorsglobe.com reproduction in angiosperms assignment help-homework help by online reproduction biology tutors
mammalia tutorial all along with the key concepts of features of the mammal, external feature of the rabbit and adaptation to ecological environment
tutorsglobe.com braggs equation assignment help-homework help by online solid state chemistry tutors
Human genetics tutorial all along with the key concepts of The human karyotype, Human chromosomal abnormalities, Types of chromosomal abnormality, Human Allelic Disorders, Sex-linked Traits, Diagnosis of Human Genetic Diseases
tutorsglobe.com strain improvement assignment help-homework help by online industrial microbiology tutors
1933612
Questions Asked
3689
Tutors
1461315
Questions Answered
Start Excelling in your courses, Ask an Expert and get answers for your homework and assignments!!