Question

The pressure of \(6.0 \mathrm{~L}\) of an ideal gas in a flexible container is decreased to one-third of its original pressure, and its absolute temperature is decreased by one-half. What is the final volume of the gas?

Short answer

The final volume of the gas is \(18.0 \mathrm{~L}\)

Step-by-step solution

Identify the Known Variables

The known variables are: Initial volume \(V1 = 6.0 \mathrm{~L}\), initial pressure \(P1\) (unknown), and initial temperature \(T1\) (unknown). We also know that the pressure falls to one-third of its original value, so the final pressure \(P2 = P1/3\), and the temperature decreases by one-half, so the final temperature \(T2 = T1/2\).

Apply The Ideal Gas Law

The Ideal Gas Law states that \(PV=nRT\), where \(P\) is the pressure, \(V\) is the volume, \(n\) is the number of moles, \(R\) is the universal gas constant, and \(T\) is the temperature. Since this exercise doesn't give the number of moles or the value for \(R\), both can be eliminated from the formula due to their remaining constant during the procedure. Thus, the equation for the initial state can be written as \(P1V1 = T1\) and for the final state as \(P2V2 = T2\). Now, since \(P2 = P1/3\) and \(T2 = T1/2\) we can substitute these values into our second equation giving us \(P1/3 * V2 = T1/2\). Now set these two equations equal to each other: \(P1V1 = P1/3 * V2\).

Solve For The Final Volume

Solve the equation for \(V2\) and we find that \(V2 = 3* V1\). Substituting our known initial volume, \(V2 = 3 * 6.0 L = 18.0 L\). Therefore, the final volume of the gas is 18.0 L.