Comprehensive Study Notes: Ideal and Combined Gas Laws Worksheet Part 1

The Ideal Gas Law is defined by the mathematical expression: $PV = nRT$ .
This equation is used to isolate and solve for specific variables affecting an ideal gas sample.
$P$ represents Pressure: Measured in units of atmospheres ( $\text{atm}$ ) or $\text{torr/mm Hg}$ .
$V$ represents Volume: In this equation, volume must be converted to cubic decimeters ( $dm^3$ ).
$n$ represents Amount of Substance: Measured as the number of moles ( $mol$ ).
$R$ represents the Universal Gas Constant: A numerical value that varies depending on the units used for pressure.
$T$ represents Temperature: This value must always be expressed in Kelvin ( $K$ ).

Standard Temperature and Pressure (STP) provides a standard set of conditions for gaseous comparisons.
Standard Pressure ( $P$ ): Defined as $1\,\text{atm}$ , which is equivalent to $760\,\text{torr}$ and $760\,\text{mm Hg}$ .
Standard Temperature ( $T$ ): Defined as $273.15\,K$ (commonly simplified to $273\,K$ ), which equals $0^\circ\text{C}$ and $32^\circ\text{F}$ .
Molar Volume at STP: Exactly $1\,\text{mol}$ of any ideal gas occupies a volume of $22.4\,dm^3$ at STP.
The designated amount ( $n$ ) for standard molar volume is $1\,\text{mole}$ .

The value of $R$ is derived by rearranging the ideal gas law: $R = \frac{P \times V}{n \times T}$ .
Calculation using Atmospheres: $R = \frac{1\,\text{atm} \times 22.4\,dm^3}{1\,\text{mol} \times 273\,K} = 0.0821\,dm^3\cdot\text{atm}\cdot mol^{-1}\cdot K^{-1}$ .
Note on Precision: Online resources and certain materials may list this constant as $0.08206\,dm^3\cdot\text{atm}\cdot mol^{-1}\cdot K^{-1}$ .
Calculation using Torr/mm Hg: $R = \frac{760\,\text{torr (mm Hg)} \times 22.4\,dm^3}{1\,\text{mol} \times 273\,K} = 62.4\,dm^3\cdot\text{torr}\cdot mol^{-1}\cdot K^{-1}$ .
Testing Requirement: Students are not required to commit the $R$ constant values to memory; they will be provided during in-class tests, specifically Test #4 and the Final Exam (FE).

Problem 1: Calculate the pressure, in $\text{atm}$ , exerted by $0.400\,\text{moles}$ of a gas contained in a $5.00\,dm^3$ vessel at a temperature of $17^\circ\text{C}$ .
Problem 2 (Part A): Determine the number of moles of oxygen gas ( $O_2$ ) contained in a $50.0\,dm^3$ tank at a temperature of $22^\circ\text{C}$ and a pressure of $136\,\text{atm}$ .
Problem 2 (Part B): Calculate the volume (in $dm^3$ ) of $75\,\text{moles}$ of ammonia gas ( $NH_3$ ) when maintained at $22^\circ\text{C}$ and a pressure of $729\,\text{torr}$ .
Problem 3: Identify the temperature required for $25.2\,\text{moles}$ of Xenon ( $Xe$ ) gas to occupy a volume of $645\,dm^3$ with a pressure of $732\,\text{torr}$ .
Problem 4: Determine the volume (in $dm^3$ ) of $105\,\text{moles}$ of methane ( $CH_4$ ) under conditions of $39^\circ\text{C}$ and $1.5\,\text{atm}$ .
Problem 5: Find the amount in moles of carbon dioxide ( $CO_2$ ) held in a volume of $9.55\,dm^3$ at $45^\circ\text{C}$ and $752\,\text{torr}$ .
Problem 6: Calculate the temperature, in degrees Celsius ( $^\circ\text{C}$ ), of a $37.5\,\text{mole}$ Argon ( $Ar$ ) sample occupying $725\,dm^3$ at a pressure of $675\,\text{mm Hg}$ .

The Combined Gas Law is applied when a gas sample is subjected to changing conditions (initial state vs. final state).
Mathematical Formula: $\frac{P_1 \times V_1}{T_1} = \frac{P_2 \times V_2}{T_2}$ .
Variables with subscript "1" represent the initial properties, while variables with subscript "2" represent the properties after the change.

Problem 1: A gas sample occupies a volume of $725\,cm^3$ at $825\,\text{mm Hg}$ . Determine the new pressure if the volume is modified to $283\,cm^3$ .
Problem 2: A gas sample occupies $125\,cm^3$ at $21^\circ\text{C}$ . Calculate the resulting volume if the temperature is raised to $822^\circ\text{C}$ .
Problem 3: Given an initial volume of $20.0\,dm^3$ for ammonia gas ( $NH_3$ ) at $5^\circ\text{C}$ and $730\,\text{torr}$ , solve for the new volume at $50^\circ\text{C}$ and $800\,\text{torr}$ .
Problem 4: A balloon filled with gas occupies $50.0\,dm^3$ at $20^\circ\text{C}$ and $742\,\text{torr}$ . Determine the balloon's volume if it is moved to STP conditions ( $1\,\text{atm}$ and $273\,K$ ).
Problem 5: A $125\,cm^3$ sample of gas at STP undergoes a change to a new temperature of $75^\circ\text{C}$ and a pressure of $11\,\text{atm}$ . Calculate the new volume.
Problem 6: Determine the temperature to which a $5.00\,dm^3$ gas sample (at $50^\circ\text{C}$ and $600\,\text{torr}$ ) must be heated to achieve a new volume of $10.0\,dm^3$ and a final pressure of $800\,\text{torr}$ .