The Ideal Gas Law Derivations: Study with Video Lessons, Practice Problems & Examples

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

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.

The Ideal Gas Law Derivations Example 1

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. 0.65 atm
• B. 0.83 atm
• C. 0.92 atm
• D. 1.6 atm
• E. 2.7 atm

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?

• A. 0.176 L
• B. 0.217 L
• C. 0.246 L
• D. 0.361 L
• E. 1.28 L

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?

• A. 750 torr
• B. 1800 torr
• C. 2190 torr
• D. 2850 torr
• E. 3300 torr

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·atm/(K·mol)\; 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 $P_{}{V}_{}=n_{}{R}_{}{T}_{}.\; Its\; value\; depends\; on\; the\; units\; used\; for\; pressure,\; volume,\; and\; temperature.\; Commonly,\; R\; is\; 0.0821\; L·atm/(K·mol)\; when\; using\; liters\; for\; volume,\; atmospheres\; for\; pressure,\; and\; Kelvin\; for\; temperature.\; R\; ensures\; that\; the\; relationship\; between\; pressure,\; volume,\; temperature,\; and\; moles\; of\; gas\; is\; consistent\; across\; different\; conditions.$

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.$$