# 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)

Disclaimer: Compete.etutor.co may from time to time provide links to third party Internet sites under their respective fair use policy and it may from time to time provide materials from such third parties on this website. These third party sites and any third party materials are provided for viewers convenience and for non-commercial educational purpose only. Compete does not operate or control in any respect any information, products or services available on these third party sites. Compete.etutor.co makes no representations whatsoever concerning the content of these sites and the fact that compete.etutor.co has provided a link to such sites is NOT an endorsement, authorization, sponsorship, or affiliation by compete.etutor.co with respect to such sites, its services, the products displayed, its owners, or its providers.

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}}