Question 184: Properties of Matter

Question

A soap bubble, blown by a mechanical pump at the mouth of a tube, increases in volume, with time, at a constant rate. The graph that correctly depicts the time dependence of pressure inside the bubble is given by :

Accepted Answer

When soap bubble is being inflated and its temperature remains constant, then it follows Boyle's law, so
pV = constant (k)
$$ \Rightarrow p = {k \over V}$$
Differentiating above equation with time, we get
$${{dp} \over {dt}} = k.\,{d \over {dt}}\left( {{1 \over V}} \right)$$
$$ \Rightarrow {{dp} \over {dt}} = k\left( {{{ - 1} \over {{V^2}}}} \right).\,{{dV} \over {dt}}$$
It is given that, $${{dV} \over {dt}} = c$$ (a constant)
So, $${{dp} \over {dt}} = {{ - kc} \over {{V^2}}}$$ ..... (i)
Now, from $${{dV} \over {dt}} = c$$, we get
$$dV = cdt$$
or, $$\int {dV = \int {cdt} } $$ or, $$V = ct$$ ..... (ii)
From Eqs. (i) and (ii), we get
$${{dp} \over {dt}} = {{ - kc} \over {{c^2}{t^2}}}$$
or, $${{dp} \over {dt}} =  - {\left( {{k \over c}} \right)^{{t^{ - 2}}}}$$
$$ \Rightarrow dp =  - {k \over c}.\,{t^{ - 2}}dt$$
Integrating both sides, we get
$$\int {dp =  - {k \over c}\int {{t^{ - 2}}dt} } $$
$$p =  - {k \over c}.\,\left( {{{{t^{ - 2 + 1}}} \over { - 2 + 1}}} \right)$$
$$ =  - {k \over c}.\,{{ - 1} \over t} = {k \over {ct}}$$
or, $$p \propto {1 \over t}$$
Hence, p versus $${1 \over t}$$ graph is a straight line, which is correctly represented in option (b).

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Properties of Matter

A soap bubble, blown by a mechanical pump at the mouth of a tube, increases in volume, with time, at a constant rate. The graph that correctly depicts the time dependence of pressure inside the bubble is given by :

Choose the correct answer: