Derive Laplace’s law for spherical membrane of bubble due to surface tension.

Derive Laplace’s law for a spherical membrane.

#### Solution 1

Consider a spherical liquid drop and let the outside pressure be P_{o} and inside pressure be P_{i}, such that the excess pressure is P_{i} − P_{o}.

Let the radius of the drop increase from r Δto r, where Δr is very small, so that the pressure inside the drop remains almost constant.

Initial surface area (A_{1}) = 4Πr^{2}

Final surface area (A_{2}) = 4Π(r + Δr)^{2}

^{ }= 4π(r^{2 }+ 2rΔr + Δr^{2})

= 4Πr^{2} + 8ΠrΔr + 4ΠΔr^{2}

As Δr is very small, Δr2 is neglected (i.e. 4πΔr^{2}≅0)

Increase in surface area (dA) =A_{2} - A_{1}= 4Πr^{2} + 8ΠrΔr - 4Πr^{2}

Increase in surface area (dA) =8ΠΔr

Work done to increase the surface area 8ΠrΔr is extra energy.

∴dW=TdA

∴dW=T*8πrΔr .......(Equ.1)

This work done is equal to the product of the force and the distance Δr.

dF=(P_{1} - P_{0})4πr^{2}

The increase in the radius of the bubble is Δr.

dW=dFΔr= (P_{1} - P_{0})4Πr^{2}*Δr ..........(Equ.2)

Comparing Equations 1 and 2, we get

(P_{1} - P_{0})4πr^{2}*Δr=T*8πrΔr

∴`(P_1 - P_0) = (2T)/R`

This is called the Laplace’s law of spherical membrane.

#### Solution 2

Let us consider a liquid drop which is spherical in shape with surface area $A$

$`therefore triangle "A" = 4pi xx [("r" + triangle "r")^2 - "r"^2]`$

$`therefore triangle "A" = 4 pi xx 2"r" triangle "r"` ...(∵ Δ r is very small Δr2 is still smaller and hence ignored and considered as zero)$

$Δ A = 8 π r Δ r$

$Work done in expanding the drop = gain in energy Δ W = T Δ A = T 8 π r Δ r ....(i)$

$Here T is surface tension.$

$Excess pressure = Pi - P0$

$Excess force = excess pressure times × area$

$Δ F = (Pi - P0) 4 πr2$

$Work done = ΔF × Δr = (Pi - P0) 4 πr2 Δ r .......(ii)$

$From 1 and 2 we get , T 8 π r Δ r = (Pi - P0) 4 πr2 Δ r$

$`therefore "2T"/"r" = "P"_"i" - "P"_0` ...... Lapalce's Law$

$For a bubble, there are 2 surface areas, internal and external, the Laplace's Law gets changed as `(4"T")/"r" = "P"_"i" - "P"_0`$