Question

The two bulbs of volume $5litre$ and $10litre$ containing an ideal gas at $9atm$ and $6atm$ respectively are connected. What is the final pressure in the two bulbs if the temperature remains constant?
A. $15atm$
B. $7atm$
C. $12atm$
D. $21atm$

Answer 1

All systems when connected to each other, try to attain an equilibrium state. So, when two compartments with different pressures are connected, they tend to attain a common pressure after some time. As the temperature is kept constant, we can make use of Boyle's Law.
Formula used:
${P_1}{V_1} + {P_2}{V_2} = P({V_1} + {V_2})$
Where the symbols $P$ and $V$ are the pressure and volume of the ideal gas, and the subscripts $1,2$ indicate the first bulb and second bulb respectively.

Complete step by step answer:
As the temperature remains constant throughout the process, we can make use of Boyle's Law, which states that at constant temperature for the same amount of gas taken, pressure will be inversely proportional to the volume occupied by the gas. That is,
$P \propto \frac{1}{V} \Rightarrow PV =$ constant
As the two bulbs are connected to each other, and the gas is free to move around, the system tends to attain an equilibrium pressure which would be common across both the bulbs.
Let $P$ denote the final equilibrium pressure attained by the ideal gas. As the product of pressure and volume is a constant before and after the process, we can equate both the stages as:
${P_1}{V_1} + {P_2}{V_2} = P{V_1} + P{V_2} = P({V_1} + {V_2})$
Where the symbols $P$ and $V$ are the pressure and volume of the ideal gas, and the subscripts $1,2$ indicate the first bulb and second bulb respectively.
Here we have ${P_1} = 9atm,{V_1} = 5L,{P_2} = 6atm{\text{ and }}{V_2} = 10L$
Substituting these values into our equation, we get:
$(9 \times 5) + (6 \times 10) = P(5 + 10)$
On solving this, we get:
$105 = 15P \Rightarrow P = \dfrac{{105}}{{15}} = 7atm$
Hence the correct option is B.

Note:
Boyle’s Law, unlike the three other gas laws (Charle’s Law, Dalton’s Law and Avogadro’s Law) denote an inverse proportionality, while the other three laws have a direct proportionality. Make a note to remember this point so as to not use $P \propto V$.
Also note that the final equilibrium pressure was attained only because the bulbs were connected and the gas was free to move around. If they were not connected, the pressures would remain the same.