## Chemical Thermodynamics

# Pressure-volume work

**Work done in isothermal expansion :**

For isothermal process

ΔT = 0 , Δw = 0

q = − w

**For reversible expansion :**

dW = − (P_{ext} − dP).dV

= − P_{ext}.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 P_{ext} = 0

⇒ W = 0, q = 0

ΔE = 0 , ΔH = 0

**Case 2 :** Intermediate expansion

W = P_{ext} (ΔV)

**Work done in adiabatic process :**

q = 0

first law ΔE = q + w

ΔE = ±W

ΔE (expansion) = −W ⇒ C_{V}.ΔT = −PΔV

= −W = C_{V}(T_{2} − T_{1})**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}

⇒ TV^{r − 1} = constant

PV^{r} = 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* = - *W _{pV}* =

*p*

_{ext}Δ

*V*

2. Work of reversible expansion under isothermal conditions

*q* = - *W*_{rev} = 2.303 *nRT* log \tt \left(\frac{V_{2}}{V_{1}}\right)

or *q* = - *W*_{rev} = 2.303 *nRT* log \tt \frac{p_{1}}{p_{2}}