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A cylinder with a movable piston records a volume of 11.6 L when 3.2 mol of oxygen is added. The gas in the cylinder has a pressure of 5.2 atm. The cylinder develops a leak and the volume of gas is now recorded to be 10.5 L at the same pressure. How many moles of oxygen are lost?

This question aims to find the moles of oxygen gas in a cylinder after a leak. The moles of oxygen gas need to be determined at the same pressure inside the cylinder.

The question is based on the concepts of Ideal Gas Law and Avogadro’s Law. Ideal gas law states that the volume of any gas is directly proportional to the number of moles of oxygen gas when the temperature and pressure of the gas remain constant. Ideal gas law is given as:

                                                             PV = nRT 

Avogadro’s law states that the two gasses with the same temperature and pressure will have the same number of molecules if their volume is the same. Avogadro’s law is given as:

\[ \dfrac{ V_1 }{ n_1 } = \dfrac{ V_2 }{ n_2 } \]

Expert Answer

We can use Avogadro’s law to solve this problem considering the oxygen gas to be a separate gas after the leak. The information given in this problem is as follows:

\[ Volume\ of\ Oxygen\ V_1 = 11.6\ L \]

\[ Moles\ of\ Oxygen\ n_1 = 3.2\ mol \]

\[ Pressure\ of\ Oxygen\ P = 5.2\ atm \]

\[ Volume\ of\ Oxygen\ after\ Leak\ V_2 = 10.5\ L \]

We need to determine the moles of oxygen remaining after the leak first and then we can deduct that amount from the original amount to determine the lost gas.

We can use Avogadro’s Law as:

\[ \dfrac{ V_1 }{ n_1 } = \dfrac{ V_2 }{ n_2 } \]

\[ \dfrac{ 11.6 }{ 3.2 } = \dfrac{ 10.5 }{ n_2 } \]

\[ n_2 = \dfrac{ 3.2 \times 10.5 }{ 11.6 } \]

\[ n_2 = 2.9\ mol \]

Now that we know, how much moles of oxygen are remaining, we can deduct it from the original amount. The amount of oxygen lost during leak is:

\[ Moles\ of\ Lost\ = n_1\ -\ n_2 \]

\[ Moles\ of\ Lost\ = 3.2\ -\ 2.9 \]

\[ Moles\ of\ Lost\ = 0.3\ mol \]

Numerical Result

The moles of oxygen lost during the leak while the pressure in the cylinder remained the same is calculated to be:

\[ Moles\ of\ Oxygen\ Lost\ = 0.3\ mol \]

Example

A cylinder containing 5 L of hydrogen gas containing 1.8 moles develops a leak . Find the amount of hydrogen gas remaining in the cylinder if the volume of hydrogen gas is now recorded to be 3.5 L while the pressure of 3 atm remained the same.

The information given in this problem are as follows:

\[ Volume\ of\ Hydrogen\ V_1 = 5\ L \]

\[ Moles\ of\ Hydrogen\ n_1 = 1.8\ mol \]

\[ Pressure\ of\ Hydrogen\ P = 3\ atm \]

\[ Volume\ of\ Hydrogen\ after\ Leak\ V_2 = 3.5\ L \]

Using the Avogadro’s Law, we can determine the number of moles remaining in the cylinder after the leak.

\[ \dfrac{ V_1 }{ n_1 } = \dfrac{ V_2 }{ n_2 } \]

\[ \dfrac{ 5 }{ 1.8 } = \dfrac{ 3.5 }{ n_2 } \]

\[ n_2 = \dfrac{ 1.8 \times 3.5 }{ 5 } \]

\[ n_2 = 1.26\ mol \]

The remaining amount of hydrogen gas is 1.26 moles.

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