Chapter 5, Problem 5.98PAE

Interpretation Introduction

Interpretation:

It is required to calculate the number of initial moles of the reactants, identify the limiting and excess reactant, calculate the moles of the products and use the ideal gas equation to find the final pressure of the system.

Concept introduction:

• The ideal gas equation relates the pressure, volume, and temperature of a system.
• $\text{Pressure}\times \text{Volume = moles }\times \text{ Gas Constant }\times \text{Temperature}$
• The limiting reactant is the chemical species that will be consumed totally, stoichiometric calculations must be done based on the limiting reactant.
• The excess reactant is the chemical species that will not be consumed totally.
• To change units of a measured quantity a proper conversion factor is required.

Answer to Problem 5.98PAE

Solution:

The total pressure is $650.62\text{ torr}$.

Explanation of Solution

Step 1: Write the chemical equation

The ${\text{SO}}_{\text{2}}$ and ${\text{O}}_{\text{2}}$ react to produce ${\text{SO}}_{\text{3}}$ according to the equation:

${\text{2 SO}}_{\text{2}}+{\text{O}}_2\to 2{\text{ SO}}_{\text{3}}$

Step 2: Calculate the number of moles of ${\text{SO}}_{\text{2}}$ and ${\text{O}}_{\text{2}}$

It is possible to use the ideal gas equation to calculate the moles of ${\text{SO}}_{\text{2}}$ and ${\text{O}}_{\text{2}}$ that are available.

$\text{Pressure}\times \text{Volume = moles }\times \text{ Gas Constant }\times \text{Temperature}$

Rearrange the equation to find the number of moles:

$\frac{\text{Pressure }\times \text{Volume}}{\text{Gas Constant }\times \text{ Temperature}}=\text{moles}$

Pressure must be in atmospheres

Gas constant is $\text{0}\text{.0821 }\frac{\text{atm L}}{\text{K mole}}$

The temperature must be in Kelvin

The volume must be in liters

From the statement:

Pressure is 500 torr, change this value to atm:

$\text{1 atm = 760 torr}$ ;

$500\text{ torr x }\frac{\text{1atm}}{\text{760 torr}}=0.658\text{ atm}$

The temperature must be in kelvin, change Celsius to Kelvin by adding 273.15

$\text{Temperature = 27 + 273}\text{.15 = 300}\text{.15 Kelvin}$

Volume is 1 Liter

Calculate the number of moles using this information

$\frac{\text{0}\text{.658 atm }\times \text{1 Liter}}{\text{0}\text{.082}\frac{\text{atm L}}{\text{Kelvin mole}}\times \text{ 300}\text{.15 Kelvin}}=0.0268\text{ moles}$

Apply the same calculations to find the number of moles of ${\text{O}}_{\text{2}}$.

From the statement:

Pressure is 400 torr, change this value to atm:

$\text{1 atm = 760 torr}$ ;

$400\text{ torr x }\frac{\text{1atm}}{\text{760 torr}}=0.526\text{ atm}$

The temperature must be in kelvin, change Celsius to Kelvin by adding 273.15

$\text{Temperature = 27 + 273}\text{.15 = 300}\text{.15 Kelvin}$

Volume is 1 Liter

Calculate the number of moles using this information

$\frac{\text{0}\text{.526 atm }\times \text{ 1 Liter}}{\text{0}\text{.082}\frac{\text{atm L}}{\text{Kelvin mole}}\times \text{ 300}\text{.15 Kelvin}}=0.0214\text{ moles}$

Step 3: Calculate the ${\text{SO}}_{\text{3}}$ produced

Let’s look at the chemical equation:

${\text{2 SO}}_{\text{2}}+{\text{O}}_2\to 2{\text{ SO}}_{\text{3}}$

2 moles of ${\text{SO}}_{\text{2}}$ require 1 mole of ${\text{O}}_{\text{2}}$

There are 0.0214 moles of ${\text{O}}_{\text{2}}$ and 0.0267 moles of ${\text{SO}}_{\text{2}}$. It is required to identify the limiting reactant

2 moles of ${\text{SO}}_{\text{2}}$ require 1 mole of ${\text{O}}_{\text{2}}$, since there are 0.0267 moles of ${\text{SO}}_{\text{2}}$ then we require:

$0.0268{\text{ moles SO}}_2\text{ x }\frac{1{\text{ mole O}}_2}{2{\text{ moles SO}}_2}=0.0134{\text{ moles of O}}_2$

Since there is more oxygen than the required then it is true that Oxygen is the excess reactant and ${\text{SO}}_{\text{2}}$ is the limiting reactant.

Only 0.0134 moles of oxygen will react, so there will be:

$0.0214-0.0134=0.008{\text{ moles of unreacted O}}_2$

According to the chemical equation, 2 moles of ${\text{SO}}_{\text{2}}$ will produce 2 moles of ${\text{SO}}_{\text{3}}$, so there will be 0.0268 moles of ${\text{SO}}_{\text{3}}$

Step 4: Calculate the pressure

Count the moles at the end of the reaction:

moles of ${\text{SO}}_{\text{2}}$ = 0

moles of ${\text{O}}_{\text{2}}$ = 0.008

moles of ${\text{SO}}_{\text{3}}$ = 0.0268

Apply the ideal gas equation to calculate the pressure:

$\text{Pressure = }\frac{\text{moles }\times \text{ Gas Constant }\times \text{Temperature}}{\text{Volume}}$

Total moles available:

$0+0.008+0.0268=0.0348\text{ moles}$

Temperature is 600 Kelvin

Volume is 2 Liters

Calculate the pressure using these values:

$\text{Pressure = }\frac{\text{0}\text{.0348 moles }\times \text{ 0}\text{.082 }\frac{\text{atm L}}{\text{Kelvin mole}}\times \text{ 600 Kelvin}}{\text{2 Liters}}\text{=0}\text{.856 atm}$

Change atm to torr:

$\text{0}\text{.856 atm }\times \frac{\text{760 torr}}{1\text{ atm}}=650.62\text{ torr}$

Conclusion

The pressure of a system can be determined using the ideal gas equation.