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Derive Maxwell's equations.

Mumbai university > MECH > SEM 3 > THERMO

Marks: 4M

Year: May 2014

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  • A pure substance existing in a single phase has only two independent variables.

  • Of the eight quantities p, V, T, S, U, H, F (Helmholtz function), and G (Gibbs function) any one may be expressed as a function of any two others.

  • For a pure substance undergoing an infinitesimal reversible process

    a) dU = TdS - pdV

    b) dH = dU + pdV + Vdp = TdS + Vdp

    c) dF = dU - TdS - SdT = -pdV - SdT

    d) dG = dH - TdS - SdT = Vdp - SdT

  • Since U, H, F and G are thermodynamic properties and exact differentials of the type $$dz = Mdx + Ndy$$ then

    $(\frac{∂M}{∂y})_x = (\frac{∂N}{∂x})_y$

  • Applying this to the four equations we get,

    $(\frac{dT}{dv})_s = (\frac{∂P}{∂s})_v$

    $(\frac{dT}{dP})_s = (\frac{∂v}{∂s})_P$

    $(\frac{∂s}{∂v})_T = (\frac{∂P}{∂T})_v$

    $(\frac{∂s}{∂P})_T = (\frac{∂v}{∂T})_P$

  • These four equations are known as Maxwell’s equations.

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