Chemical Thermodynamics

Pressure-volume work


Work done in isothermal expansion :
For isothermal process
ΔT = 0 , Δw = 0
q = − w

For reversible expansion :
dW = − (Pext − dP).dV
= − Pext.dV + dP dV
dW = - P.dv ; W = -\int_{V_{1}}^{V_{2}}PdV
W = -nRT \ ln \frac{V_{2}}{V_{1}}
⇒ W = -2.303 \ nRT \ log \frac{V_{2}}{V_{1}}
 W = -2.303 \ nRT \ log \frac{P_{1}}{P_{2}}

Work done in reversible isothermal compression :
W = 2.303 \ nRT \ log \frac{V_{2}}{V_{1}}
W = 2.303 \ nRT \ log \frac{P_{1}}{P_{2}}

Work done in irreversible isothermal expansion :
Case 1 : free expansion of gas
Work done = expansion × Vacuum
In this case Pext = 0
⇒ W = 0,   q = 0
  ΔE = 0 , ΔH = 0
Case 2 : Intermediate expansion
W = Pext (ΔV)

Work done in adiabatic process :
q = 0
first law ΔE = q + w
ΔE = ±W
ΔE (expansion) = −W ⇒ CV.ΔT = −PΔV
                       = −W = CV(T2 − T1)
Note : in Expansion T↓,   Compression T ↑

In reversible adiabatic expansion :
\frac{T_{2}}{T_{1}} = \left(\frac{V_{1}}{V_{2}}\right)^{r - 1} →\ 1
T_{1}V_1^{r - 1} = T_{2}V_2^{r - 1}
⇒ TVr − 1 = constant
PVr = constant

Irreversible adiabatic process :
w = 0, ΔE = 0, ΔH = 0, q = 0 (free against vacuum, i.e free vacuum)

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Expression for pressure - Volume Work :

1. Work of irreversible expansion against constant pressure p under isothermal conditions

                  q = - WpV = pext ΔV

2. Work of reversible expansion under isothermal conditions
           q = - Wrev = 2.303 nRT log \tt \left(\frac{V_{2}}{V_{1}}\right)

          or q = - Wrev = 2.303 nRT log \tt \frac{p_{1}}{p_{2}}