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Thermodynamic equation of state:
(1) First thermodynamic equation of state: The charge in internal energy with respect to volume at constant temperature is known as first thermodynamic equation of state. i.e.
Dimensionally it is equal to pressure. Thus it is also called internal pressure (π).
We know that
dU = TdS – PdV
∴
(2) Second thermodynamic equation of state: The change in enthalpy w.r.t. pressure at constant temperature is known as second thermodynamic equation of state.
dH = TdS + VdP
⇒
⇒
Comparison of Isothermal and Adiabatic Expansions: Let us consider isothermal and adiabatic expansio ns of an ideal gas from initial volume V_{i} and pressure P_{i} to a commo n final vo lume V_{f}. If P_{iso} and P_{adia} are final pressure, then
P_{i}V_{i} = P_{iso} V_{f} (for isothermal expansio n)
and P_{i}V_{i} ^{r} = P_{adia} V_{f} ^{r} (for adiabat ic expansio n)
Accordingly,
and
Since for expansio n V_{f} > V_{i} and for all r > 1, hence
From graph (a) & (b), it is clear that work done in isothermal expansion (shown by area ABCD) is greater than the work done in adiabatic expansion (shown by area AECD).
Consider the expansions in which the final pressure P_{f} is the same in both cases. If V_{iso} and V_{adia} are the final volume is isothermal and adiabatic expansion then
P_{i}V_{i} = P_{f} V_{iso} (for isothermal expansion)
and P_{i} V_{i}^{r} = P_{f} V_{adia} (for adiabatic expansion)
then
⇒
or [∵ r > 1 then]
or V_{adia }< V_{iso}
Reversible Isothermal expansion of a Real gas Using vendor walls equation we find the expression for W, ΔV, ΔH and q for reversible isothermal expansion of real gas.
Work of expansion
For the vendor walls gas,
= nRT so that
Hence,
Internal energy change
We know that,
∴ (at constant temperature)
Enthalpy change
Heat change. We know that ΔU = q + w or q = ΔU  w
Subst ituting the value of w & ΔU in this equation we get
Comparison of Work of Expansion of an ideal gas and a real gas.
We know that
and
If V >> nb, than
Hence,
Since for the expansio n of a gas, V_{2} > V_{1} then
W_{ideal} > W_{real}
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