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]
Explain Methods of Wage Payment, and various factors to be considered in Methods of Wage Payment. Labour is one of the important issues of production. The achievement of a concern relies on the efficiency of labour to a huge extent. Low wages do not essentially result in low cost of production.
tutorsglobe.com shapes of orbitals assignment help-homework help by online wave nature of electrons tutors
Theory and lecture notes of Chi-square goodness-of-fit test all along with the key concepts of chi-square goodness-of-fit test, Interpreting the Claim. Tutorsglobe offers homework help, assignment help and tutor’s assistance on Chi-square goodness-of-fit test.
Photometry tutorial all along with the key concepts of Photometry and eye, Photometric quantities, Photometric versus radiometric quantities, Photometric measurement methods
Navigation in Birds tutorial all along with the key concepts of Compass Mechanisms, Sun compass, Star compass, Geomagnetic compass, Map-like or Navigational Mechanisms and Neural Representation of Space in Birds
tutorsglobe.com association of kp and kc assignment help-homework help by online attainment of equilibrium in chemical reactions tutors
cell phones are also termed as wireless phones or mobile phones are hand-held phones with built- in antennas. these are also called transceivers.
tutorsglobe.com reproduction in algae assignment help-homework help by online algae tutors
www.tutorsglobe.com offers answering questions to price elasticity of demand, perfectly inelastic demand, elastic demand, economics assignment help - homework help by tutor's help.
www.tutorsglobe.com offers related carbonyl derivatives homework help, related carbonyl derivatives assignment help, online tutoring assistance, organic chemistry solutions by online qualified tutor's help.
www.tutorsglobe.com offer theory and concepts of the journal, journal entries assignment help, journal entries homework help and preparing journal entries based questions answers.
tutorsglobe.com sorosis assignment help-homework help by online multiple or composite fruit tutors
As per the classification according to nature costs can be categorized into Direct and Indirect.
tutorsglobe.com limitations of the law assignment help-homework help by online equi-marginal utility tutors
tutorsglobe.com thallus organization assignment help-homework help by online algae tutors
1932901
Questions Asked
3689
Tutors
1467317
Questions Answered
Start Excelling in your courses, Ask an Expert and get answers for your homework and assignments!!