Combined First and Second Laws of Thermodynamics:
First law of Thermodynamics is
dQ = dU + dW
And second law of thermodynamics is:
dQ = TdS
But differential work
dW = PdV
Combination of the three equations in any infinitesimal reversible process, for PVT system provides
TdS = dU + PdV
It is the combined form of first and second laws of thermodynamics for PVT system. Few useful thermodynamics relations can be derived from by selecting T and V, T and P, P and V as independent variables. It is significant to note that state of the pure substance can be stated or specified by any two of its properties.
T and V Independent:
Using equation TdS = dU + PdV we can write:
dS = 1/T (dU + PdV)
Consider internal energy U as the function of T and V, i.e. U(T,V), then derivative of U is
dU = (∂U/∂T) VdT + (∂U/∂V) TdV
Further we get:
1/T(∂U/∂T)V = (∂S/∂T)V
1/T[(∂U/∂V)T + P](∂S/∂V)T
If differentiate equation partially with respect to V at constant T and differentiate partially with respect to T and constant V, equating mixed second-order partial derivatives of S provides
(∂U/∂V)T = T(∂P/∂T)V - P
It can be shown that (∂P/∂T)V = β/k
Thus equation becomes:
(∂U/∂V)T = Tβ/k - P
Equations show dependence of internal energy of the system on volume, at constant temperature and this can be computed from equation of state, or from values of β, κ, T and P.
By making use of other equations
CP - CV = T(∂P/∂T)V(∂V/∂T)P = β2TV/k
Therefore difference (CP - CV) can be computed for any substance, from equation of state or from values of T, V, β and κ.
T and P Independent:
In terms of enthalpy H = U + PV, equation can be rewritten as:
dS = 1/T(dH - VdP)
Consider enthalpy H as the function of T and P, i.e. H(T, P), then derivative of H is:
dH = (∂H/∂T)PdT + (∂H/∂P)TdT
Also consider entropy S as function of two independent variables T and P, i.e. S(T, P), then derivative of S is:
dS = (∂S/∂T)pdT + (∂S/∂P)TdP
By comparing equation, we get:
(∂S/∂T)p = 1/T(∂H/∂T)p and (∂S/∂P)T = 1/T[(∂H/∂P)T - V]
By further solving
(∂H/∂P)T = -βVT + V
This equation illustrates dependence of enthalpy on pressure, at constant temperature and this can be computed from equation of state or from values of β, T and V.
Also as we know that (∂H/∂T)p = CP,
Thus we get:
dh = CPdT - [T(∂V/∂T)P - V]dP
P and V Independent:
Consider entropy S as the function of P and V, i.e. S(P, V), derivative of S is
dS = (∂S/∂P)VdP + (∂S/∂V)PdV
Also consider internal energy U as the function of P and V, i.e. U(P, V), derivative of U is:
dU = (∂U/∂P)VdP + (∂U/∂V)PdV
Further solving we get:
(∂S/∂P)V = 1/T (∂U/∂P) V and (∂S/∂V) P = 1/T (∂U/∂V) P + P/T
(∂U/∂P)V can be written as (∂U/∂T) V (∂T/∂P) V then equation becomes:
(∂S/∂P)V = 1/T (∂U/∂T) V (∂T/∂P) V
But (∂U/∂T) V = CV, and (∂T/∂P) V = k/β
Solving equation we get:
(∂S/∂P)V = (CV/T) (k/β)
Equation provides change in entropy with respect to pressure at constant volume (∂S/∂P) V in terms of measurable quantities CV, k, β, and T.
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]
tutorsglobe.com transmitting antenna types assignment help-homework help by online antenna tutors
Other properties of group 14 tutorial all along with the key concepts of Chemical Properties of group 14, Carbides, Halides, Fluorocarbons, Hydrides, Oxides and Complex Formation
www.tutorsglobe.com offers preparation of 1º amines homework help, preparation of 1º amines assignment help, online tutoring assistance, organic chemistry solutions by online qualified tutor's help.
keynes theory of income and employment & factors influencing the level of investment, www.tutorsglobe.com offers keynes theory assignment help - keynes theory homework help by economics tutor's help.
tutorsglobe.com functions of an entrepreneur assignment help-homework help by online entrepreneurship tutors
Capacity utilisation cost audit program - the idle ability in any production shop or of transport make easy for distribution is not excessive.
Inter firm comparison can be described as, "a management method through the use of which it is made feasible for an organization to compare its performance with that of the other units occupied in similar activity."
medical equipment is intended to assist in the analysis, checking or treatment of medical conditions. medical equipment devices are generally designed with precise safety standards.
tutorsglobe.com deadlocks assignment help-homework help by online operating system tutors
Viscosity tutorial all along with the key concepts of Velocity Gradient in Moving Fluids, Coefficient of Viscosity, Variationof with Temperature, Poiseuille's Formula, Stokes' Law and Terminal Velocity, Measuring by Falling Sphere
what is Incentive Plans? To eliminate the defects of both piece-wage and time wages systems, incentive plans are employed. Within these plans, the benefits of time and piece wage systems are joint and incentives are offered to workers to work hard.
tutorsglobe.com fiscal policy and capital formation assignment help-homework help by online objectives of fiscal policy tutors
tutorsglobe.com structure of a prokaryotic cell assignment help-homework help by online prokaryotes tutors
gaseous state of matter tutorial all along with the key concepts of physical characteristics of gases, ideal gas law, boyle's law, charles' law, combined gas law equation, gay lussac's law, avogadro's law, dalton's law of partial pressure, graham's law of diffusion
tutorsglobe.com criticism of marshall statement assignment help-homework help by online alfred marshall’s definition tutors
1954220
Questions Asked
3689
Tutors
1450513
Questions Answered
Start Excelling in your courses, Ask an Expert and get answers for your homework and assignments!!