# Thermodynamic Equation of State

## Thermodynamic Equation of State

Thermodynamic equation of state is a thermodynamic equation relates state variables that characterize the state of matter under a given set of physical conditions. Equations of state are used to describe gases, fluids, fluid mixtures, solids, and the interior of stars.## First Thermodynamic Equation of State

The change in internal energy with respect to change in volume at constant temperature is called first thermodynamic equation of state. Mathematically and dimensionally its value is equal to internal pressure of a gas.We know that-

dU = TdS − PdV -----Equation-1

Differentiating this equation with respect to V at constant T, we get-

Equation-3 is called first thermodynamic equation of state.

For ideal gas-

PV = RT for one mole.

P = RT/V -----Equation-4

Differentiating this equation with respect to T at constant V, we get-

## Second Thermodynamic Equation of State

The change in enthalpy with respect to change in pressure at constant temperature is called second thermodynamic equation of state.We know that-

dH = TdS + VdP -----Equation-1

Differentiating this equation with respect to P at constant T, we get-

Equation-4 is called second thermodynamic equation of state.

For ideal gas-

PV = RT for one mole.

V = RT/P -----Equation-5

Differentiating this equation with respect to T at constant P, we get-

So, second thermodynamic equation of state for ideal gas is also zero.

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