By rearranging the ideal gas law, we can derive new equations connected to pressure, volume, moles, and temperature. We're going to say these derivations are required when we have variables with 2 sets of different values. So basically, we'll be dealing with a question where the ideal gas law is in play. And within the question, they may give you 2 pressures and 2 temperatures, or 2 volumes and 2 moles. That's when we have to do one of these types of derivations. So just remember, we're still utilizing the ideal gas law; we're just changing it a bit when we're dealing with 2 pressures or 2 volumes, 2 moles, or 2 temperatures within any given question. Now that we've seen this, let's go on to our example question in the next video.

- 1. Matter and Measurements4h 29m
- What is Chemistry?5m
- The Scientific Method9m
- Classification of Matter16m
- States of Matter8m
- Physical & Chemical Changes19m
- Chemical Properties8m
- Physical Properties5m
- Intensive vs. Extensive Properties13m
- Temperature (Simplified)9m
- Scientific Notation13m
- SI Units (Simplified)5m
- Metric Prefixes24m
- Significant Figures (Simplified)11m
- Significant Figures: Precision in Measurements7m
- Significant Figures: In Calculations19m
- Conversion Factors (Simplified)15m
- Dimensional Analysis22m
- Density12m
- Specific Gravity9m
- Density of Geometric Objects19m
- Density of Non-Geometric Objects9m

- 2. Atoms and the Periodic Table5h 23m
- The Atom (Simplified)9m
- Subatomic Particles (Simplified)12m
- Isotopes17m
- Ions (Simplified)22m
- Atomic Mass (Simplified)17m
- Atomic Mass (Conceptual)12m
- Periodic Table: Element Symbols6m
- Periodic Table: Classifications11m
- Periodic Table: Group Names8m
- Periodic Table: Representative Elements & Transition Metals7m
- Periodic Table: Elemental Forms (Simplified)6m
- Periodic Table: Phases (Simplified)8m
- Law of Definite Proportions9m
- Atomic Theory9m
- Rutherford Gold Foil Experiment9m
- Wavelength and Frequency (Simplified)5m
- Electromagnetic Spectrum (Simplified)11m
- Bohr Model (Simplified)9m
- Emission Spectrum (Simplified)3m
- Electronic Structure4m
- Electronic Structure: Shells5m
- Electronic Structure: Subshells4m
- Electronic Structure: Orbitals11m
- Electronic Structure: Electron Spin3m
- Electronic Structure: Number of Electrons4m
- The Electron Configuration (Simplified)22m
- Electron Arrangements5m
- The Electron Configuration: Condensed4m
- The Electron Configuration: Exceptions (Simplified)12m
- Ions and the Octet Rule9m
- Ions and the Octet Rule (Simplified)8m
- Valence Electrons of Elements (Simplified)5m
- Lewis Dot Symbols (Simplified)7m
- Periodic Trend: Metallic Character4m
- Periodic Trend: Atomic Radius (Simplified)7m

- 3. Ionic Compounds2h 18m
- Periodic Table: Main Group Element Charges12m
- Periodic Table: Transition Metal Charges6m
- Periodic Trend: Ionic Radius (Simplified)5m
- Periodic Trend: Ranking Ionic Radii8m
- Periodic Trend: Ionization Energy (Simplified)9m
- Periodic Trend: Electron Affinity (Simplified)8m
- Ionic Bonding6m
- Naming Monoatomic Cations6m
- Naming Monoatomic Anions5m
- Polyatomic Ions25m
- Naming Ionic Compounds11m
- Writing Formula Units of Ionic Compounds7m
- Naming Ionic Hydrates6m
- Naming Acids18m

- 4. Molecular Compounds2h 18m
- Covalent Bonds6m
- Naming Binary Molecular Compounds6m
- Molecular Models4m
- Bonding Preferences6m
- Lewis Dot Structures: Neutral Compounds (Simplified)8m
- Multiple Bonds4m
- Multiple Bonds (Simplified)6m
- Lewis Dot Structures: Multiple Bonds10m
- Lewis Dot Structures: Ions (Simplified)8m
- Lewis Dot Structures: Exceptions (Simplified)12m
- Resonance Structures (Simplified)5m
- Valence Shell Electron Pair Repulsion Theory (Simplified)4m
- Electron Geometry (Simplified)8m
- Molecular Geometry (Simplified)11m
- Bond Angles (Simplified)11m
- Dipole Moment (Simplified)15m
- Molecular Polarity (Simplified)7m

- 5. Classification & Balancing of Chemical Reactions3h 17m
- Chemical Reaction: Chemical Change5m
- Law of Conservation of Mass5m
- Balancing Chemical Equations (Simplified)13m
- Solubility Rules16m
- Molecular Equations18m
- Types of Chemical Reactions12m
- Complete Ionic Equations18m
- Calculate Oxidation Numbers15m
- Redox Reactions17m
- Spontaneous Redox Reactions8m
- Balancing Redox Reactions: Acidic Solutions17m
- Balancing Redox Reactions: Basic Solutions17m
- Balancing Redox Reactions (Simplified)13m
- Galvanic Cell (Simplified)16m

- 6. Chemical Reactions & Quantities2h 35m
- 7. Energy, Rate and Equilibrium3h 46m
- Nature of Energy6m
- First Law of Thermodynamics7m
- Endothermic & Exothermic Reactions7m
- Bond Energy14m
- Thermochemical Equations12m
- Heat Capacity19m
- Thermal Equilibrium (Simplified)8m
- Hess's Law23m
- Rate of Reaction11m
- Energy Diagrams12m
- Chemical Equilibrium7m
- The Equilibrium Constant14m
- Le Chatelier's Principle23m
- Solubility Product Constant (Ksp)17m
- Spontaneous Reaction10m
- Entropy (Simplified)9m
- Gibbs Free Energy (Simplified)18m

- 8. Gases, Liquids and Solids3h 25m
- Pressure Units6m
- Kinetic Molecular Theory14m
- The Ideal Gas Law18m
- The Ideal Gas Law Derivations13m
- The Ideal Gas Law Applications6m
- Chemistry Gas Laws16m
- Chemistry Gas Laws: Combined Gas Law12m
- Standard Temperature and Pressure14m
- Dalton's Law: Partial Pressure (Simplified)13m
- Gas Stoichiometry18m
- Intermolecular Forces (Simplified)19m
- Intermolecular Forces and Physical Properties11m
- Atomic, Ionic and Molecular Solids10m
- Heating and Cooling Curves30m

- 9. Solutions4h 10m
- Solutions6m
- Solubility and Intermolecular Forces18m
- Solutions: Mass Percent6m
- Percent Concentrations10m
- Molarity18m
- Osmolarity15m
- Parts per Million (ppm)13m
- Solubility: Temperature Effect8m
- Intro to Henry's Law4m
- Henry's Law Calculations12m
- Dilutions12m
- Solution Stoichiometry14m
- Electrolytes (Simplified)13m
- Equivalents11m
- Molality15m
- The Colligative Properties15m
- Boiling Point Elevation16m
- Freezing Point Depression9m
- Osmosis16m
- Osmotic Pressure9m

- 10. Acids and Bases3h 29m
- Acid-Base Introduction11m
- Arrhenius Acid and Base6m
- Bronsted Lowry Acid and Base18m
- Acid and Base Strength17m
- Ka and Kb12m
- The pH Scale19m
- Auto-Ionization9m
- pH of Strong Acids and Bases9m
- Acid-Base Equivalents14m
- Acid-Base Reactions7m
- Gas Evolution Equations (Simplified)6m
- Ionic Salts (Simplified)23m
- Buffers25m
- Henderson-Hasselbalch Equation16m
- Strong Acid Strong Base Titrations (Simplified)10m

- 11. Nuclear Chemistry56m
- BONUS: Lab Techniques and Procedures1h 38m
- BONUS: Mathematical Operations and Functions47m
- 12. Introduction to Organic Chemistry1h 34m
- 13. Alkenes, Alkynes, and Aromatic Compounds2h 12m
- 14. Compounds with Oxygen or Sulfur1h 6m
- 15. Aldehydes and Ketones1h 1m
- 16. Carboxylic Acids and Their Derivatives1h 11m
- 17. Amines38m
- 18. Amino Acids and Proteins1h 51m
- 19. Enzymes1h 37m
- 20. Carbohydrates1h 46m
- Intro to Carbohydrates4m
- Classification of Carbohydrates4m
- Fischer Projections4m
- Enantiomers vs Diastereomers8m
- D vs L Enantiomers8m
- Cyclic Hemiacetals8m
- Intro to Haworth Projections4m
- Cyclic Structures of Monosaccharides11m
- Mutarotation4m
- Reduction of Monosaccharides10m
- Oxidation of Monosaccharides7m
- Glycosidic Linkage14m
- Disaccharides7m
- Polysaccharides7m

- 21. The Generation of Biochemical Energy2h 8m
- 22. Carbohydrate Metabolism2h 22m
- 23. Lipids2h 26m
- Intro to Lipids6m
- Fatty Acids25m
- Physical Properties of Fatty Acids6m
- Waxes4m
- Triacylglycerols12m
- Triacylglycerol Reactions: Hydrogenation8m
- Triacylglycerol Reactions: Hydrolysis13m
- Triacylglycerol Reactions: Oxidation7m
- Glycerophospholipids15m
- Sphingomyelins13m
- Steroids15m
- Cell Membranes7m
- Membrane Transport10m

- 24. Lipid Metabolism1h 45m
- 25. Protein and Amino Acid Metabolism1h 37m
- 26. Nucleic Acids and Protein Synthesis2h 54m
- Intro to Nucleic Acids4m
- Nitrogenous Bases16m
- Nucleoside and Nucleotide Formation9m
- Naming Nucleosides and Nucleotides13m
- Phosphodiester Bond Formation7m
- Primary Structure of Nucleic Acids11m
- Base Pairing10m
- DNA Double Helix6m
- Intro to DNA Replication20m
- Steps of DNA Replication11m
- Types of RNA10m
- Overview of Protein Synthesis4m
- Transcription: mRNA Synthesis9m
- Processing of pre-mRNA5m
- The Genetic Code6m
- Introduction to Translation7m
- Translation: Protein Synthesis18m

# The Ideal Gas Law Derivations - Online Tutor, Practice Problems & Exam Prep

Rearranging the ideal gas law allows for the derivation of equations related to pressure, volume, moles, and temperature, particularly when dealing with two sets of values. This is essential for solving problems involving different pressures, volumes, or temperatures. The ideal gas law, expressed as $PV=nRT$, serves as the foundation for these calculations, emphasizing the relationships between these variables in various scenarios.

The** Ideal Gas Law Derivations** are a convenient way to solve gas calculations involving 2 sets of the same variables.

## Ideal Gas Law Derivations

### The Ideal Gas Law Derivations

#### Video transcript

### The Ideal Gas Law Derivations Example 1

#### Video transcript

Hey everyone. So now that we've talked about how we're able to derive different types of formulas from the ideal gas law formula, let's put it into action with the following example question. Here it says, a sample of sulfur hexachloride gas occupies 8.30 liters at 202 degrees Celsius. Assuming that the pressure remains constant, what temperature in degrees Celsius is needed to decrease the volume to 5.25 liters? Alright. So, step 1 tells us that we need to begin by writing out the ideal gas formula. Remember that is PV=nRT. Step 2, circle the variables in the ideal gas law formula that have 2 sets of different values. So in the question, we have 2 volumes, and let's see. Pressure is being held constant. That means it's not changing. There wouldn't be 2 values. They're giving us one temperature and asking for another.

Alright. So our 2 sets of values deal with volume and temperature. Now, step 3, cross out the variables in the ideal gas law formula that are not discussed or are remaining the same or constant. Since the R constant will be the same value you can also ignore it. So pressure is being held constant, you never mention moles because it's being held constant, and R is our constant. Next step 4, algebraically move all the circled variables to the left side of the ideal gas law. So now we have V/T. Make these circled variables equal to the second set of identical variables in order to derive a new formula. So it becomes V/T=V/T. And since we're dealing with 2 sets of data, it's Vt1/Tt1=Vt2/Tt2. If temperature is involved in the calculation, it must be in the units in the SI unit of Kelvin. So they want the answer in degrees Celsius, but don't fall for that. When we're doing our calculations, temperatures still need to be done in Kelvin.

Once we've gotten our final answer in Kelvin, then we change it to degrees Celsius. Alright. So, in the question, they're giving us 202 degrees Celsius so you're going to add 273.15 to this. Doing that is gonna give us our Kelvin which is 475.15 Kelvin. Okay. So that is gonna be T1. So let's come over here. We're told that our initial volume is 8.30 liters. T1 is 475.15 Kelvin. We know our second volume is 5.25 liters. And we don't know what our T2 is. That's what we need to find. To solve this, just cross multiply. So we're going to cross multiply these 2. Cross multiply these 2. So we're gonna get 8.30liters×t2=5.25liters×475.15Kelvin. Divide both sides now by 8.30 liters. Liters cancel out. I'll get T2 in Kelvin which comes out to 300.55 Kelvin. But, again, we want the answer in degrees Celsius. So, subtract 273.15 from this. When I do that, I get T2 equals 27.40 Kelvin. So, this would be my final answer.

Again, as long as you know the Ideal Gas Law, that's the first step. Next, look and see what are the 2 sets of data in relation to those variables of the ideal gas law formula. Focus on them. Once you've determined which ones are changing, ignore all the other ones that are being held constant. Move everything over to the left side and then it becomes a simple math problem to isolate the one missing variable. Doing that will ensure that you get the correct answer at the end. And always remember, when dealing with calculations of temperature, make sure that you're doing it in units of Kelvin. Even if they ask for degrees Celsius at the end, still change everything to Kelvin as you're doing your calculations. And change that Kelvin to Celsius at the very end. Right? So just keep that in mind. You'll be able to tackle any type of ideal gas law, derivation type of question.

A sample of nitrogen dioxide gas at 130 ºC and 315 torr occupies a volume of 500 mL. What will the gas pressure be if the volume is reduced to 320 mL at 130 ºC?

A cylinder with a movable piston contains 0.615 moles of gas and has a volume of 295 mL. What will its volume be if 0.103 moles of gas escaped?

On most spray cans it is advised to never expose them to fire. A spray can is used until all that remains is the propellant gas, which has a pressure of 1350 torr at 25 ºC. If the can is then thrown into a fire at 455 ºC, what will be the pressure (in torr) in the can?

750 torr

1800 torr

2190 torr

2850 torr

3300 torr

## Do you want more practice?

### Here’s what students ask on this topic:

How do you derive the combined gas law from the ideal gas law?

The combined gas law is derived from the ideal gas law, which is expressed as ${P}_{}{V}_{}={n}_{}{R}_{}{T}_{}.\; When\; dealing\; with\; two\; sets\; of\; conditions,\; we\; use$ {P}_{1}{V}_{1}/{T}_{1}={P}_{2}{V}_{2}/{T}_{2}.\; This\; equation\; combines\; Boyle\text{'}s\; Law,\; Charles\text{'}s\; Law,\; and\; Gay-Lussac\text{'}s\; Law,\; showing\; the\; relationship\; between\; pressure,\; volume,\; and\; temperature\; for\; a\; fixed\; amount\; of\; gas.$$

What is the relationship between pressure and volume in the ideal gas law?

In the ideal gas law, the relationship between pressure (P) and volume (V) is inversely proportional when temperature (T) and the number of moles (n) are constant. This is expressed as ${P}_{}{V}_{}={n}_{}{R}_{}{T}_{}.\; If\; the\; temperature\; and\; moles\; of\; gas\; remain\; constant,\; increasing\; the\; volume\; will\; decrease\; the\; pressure\; and\; vice\; versa.\; This\; is\; also\; known\; as\; Boyle\text{'}s\; Law.$

How do you use the ideal gas law to find the number of moles of a gas?

To find the number of moles (n) of a gas using the ideal gas law, rearrange the equation ${P}_{}{V}_{}={n}_{}{R}_{}{T}_{}to\; solve\; for\; n:$ {n}_{}={P}_{}{V}_{}/\; ({R}_{}{T}_{}).\; You\; need\; to\; know\; the\; pressure\; (P),\; volume\; (V),\; and\; temperature\; (T)\; of\; the\; gas,\; and\; use\; the\; ideal\; gas\; constant\; (R),\; which\; is\; 0.0821\; L\xb7atm/(K\xb7mol)\; for\; these\; units.$$

What is the significance of the ideal gas constant (R) in the ideal gas law?

The ideal gas constant (R) is a proportionality constant in the ideal gas law equation ${P}_{}{V}_{}={n}_{}{R}_{}{T}_{}.\; Its\; value\; depends\; on\; the\; units\; used\; for\; pressure,\; volume,\; and\; temperature.\; Commonly,\; R\; is\; 0.0821\; L\xb7atm/(K\xb7mol)\; when\; using\; liters\; for\; volume,\; atmospheres\; for\; pressure,\; and\; Kelvin\; for\; temperature.\; R\; ensures\; that\; the\; relationship\; between\; pressure,\; volume,\; temperature,\; and\; moles\; of\; gas\; is\; consistent\; across\; different\; conditions.$

How do you derive the ideal gas law from the combined gas law?

The ideal gas law can be derived from the combined gas law by considering a scenario where the number of moles (n) and the gas constant (R) are constant. The combined gas law is ${P}_{1}{V}_{1}/{T}_{1}={P}_{2}{V}_{2}/{T}_{2}.\; By\; setting\; the\; initial\; and\; final\; states\; to\; be\; the\; same,\; we\; get$ {P}_{}{V}_{}={n}_{}{R}_{}{T}_{},\; which\; is\; the\; ideal\; gas\; law.$$