Question

At the bottom of a lake where the temperature is $7^\circ {\text{C}}$ , the pressure is $2.8{\text{ atm}}$. An air bubble of radius $1{\text{ cm}}$ at the bottom rises to the surface. Where the temperature is $27^\circ {\text{C}}$. The radius of the air bubble at the surface is
A. ${3^{\dfrac{1}{3}}}$
B. ${4^{\dfrac{1}{3}}}$
C. ${5^{\dfrac{1}{3}}}$
D. ${6^{\dfrac{1}{3}}}$

Answer 1

We are given an air bubble at the bottom of a lake in which temperature and pressure are given and we are required to find the radius of the air bubble at the surface which is at a different temperature. The pressure of the air bubble at the surface will be equal to the atmospheric pressure.

Complete step by step answer:
We are given the following values at the bottom of the lake and surface. At the bottom,
${P_1} = 2.8{\text{ atm}}$
$\Rightarrow {V_1} = \dfrac{4}{3}\pi {r_1}^3$ ${T_1} = 7^\circ {\text{C = 280 K}}$
$\Rightarrow {r_1} = 1{\text{ cm}}$
At the surface,
${P_2} = 1{\text{ atm}}$
$\Rightarrow {V_2} = \dfrac{4}{3}\pi {r_2}^3$ ${T_2} = 27^\circ {\text{C = 300 K}}$
$\Rightarrow {r_2} = ?$
We need to find the value of the radius of the air bubble at the surface.Now we know according to the universal gas law $\dfrac{{PV}}{T} = {\text{constant}}$.

Therefore comparing the pressure, volume and temperature at the bottom of the lake to the surface of the lake we get
$\dfrac{{{P_1}{V_2}}}{{{T_1}}} = \dfrac{{{P_2}{V_2}}}{{{T_2}}}$
Substituting the values we get
$\dfrac{2.8 \times \dfrac{4}{3} \pi (1)^{3}}{280} = \dfrac{1 \times \dfrac{4}{3} \pi (r_2)^{3}}{300}$
${r_2} = {\left( {2.8 \times \dfrac{{30}}{{28}}} \right)^{\dfrac{1}{3}}}$
Hence, we get the final answer as,
$\therefore {r_2} = {3^{\dfrac{1}{3}}}$

Hence option A is correct.

Additional information: The universal gas law or the ideal gas law gives us the ideal gas equation according to which pressure is directly proportional to the number of molecules and temperature and is inversely proportional to the volume of the gas. Ideal gas law has two assumptions. One that the particles in a gaseous state do not have forces acting among them and second that these particles do not take up any space which means their atomic volume is being ignored.

Note: It should be noted that ideal gas laws are those which obeys the kinetic theory of gases and are not found in reality but to simplify the calculations we assume that the gases are in their ideal state where it is unaffected by the real-world conditions. And also the value of the atmospheric pressure at sea level is constant.