2.2 Find the numerical value of using , and . · OlympiadBench #1013

Physics · thermodynamics

Problem

2.2 Find the numerical value of

\frac{\rho_{i} T_{i}}{\rho_{a} T_{a}}-1

using

\gamma=0.0250 \mathrm{Nm}^{-1}, R_{0}=1.00 \mathrm{~cm}

, and

P_{a}=1.013 \times 10^{5} \mathrm{Nm}^{-2}

. Given constants: An Electrified Soap Bubble A spherical soap bubble with internal air density

\rho_{i}

, temperature

T_{i}

and radius

R_{0}

is surrounded by air with density

\rho_{a}

, atmospheric pressure

P_{a}

and temperature

T_{a}

. The soap film has surface tension

\gamma

, density

\rho_{s}

and thickness

t

. The mass and the surface tension of the soap do not change with the temperature. Assume that

R_{0} \gg t

. The increase in energy,

d E

, that is needed to increase the surface area of a soap-air interface by

d A

, is given by

d E=\gamma d A

where

\gamma

is the surface tension of the film. Context question: 2.1 Find the ratio

\frac{\rho_{i} T_{i}}{\rho_{a} T_{a}}

in terms of

\gamma, P_{a}

and

R_{0}

. Context answer:

1+\frac{4 \gamma}{R_{0} P_{a}}