To solve PV = nRT for one variable, rearrange the equation to isolate it: P = nRT/V, V = nRT/P, n = PV/RT, or T = PV/nR. Convert temperature to kelvin and match your pressure and volume units to R (0.08206 L atm / (mol K)), substitute the three known values, and compute.
The Four Rearrangements at a Glance
You never need to memorize four separate formulas. Learn one move, isolating the unknown by doing the opposite operation to both sides, and you can rebuild any of them in seconds. Here is the complete set:
| Unknown | Formula | In words |
|---|---|---|
| Pressure (P) | P = nRT / V | Moles times R times temperature, divided by volume |
| Volume (V) | V = nRT / P | Moles times R times temperature, divided by pressure |
| Moles (n) | n = PV / RT | Pressure times volume, divided by R times temperature |
| Temperature (T) | T = PV / nR | Pressure times volume, divided by moles times R |
Notice the pattern: P and V live on opposite sides of the division from n and T. If you can remember that P = nRT / V, you can flip it to get the other three. Below, each form gets a fully worked example with verified numbers.
Example 1: Solving for Pressure
Problem: What pressure does 2.00 mol of gas exert at 25 C (298 K) in a 48.9 L tank?
Step 1, isolate P: P = nRT / V.
Step 2, check units: temperature is already converted to kelvin, pressure will come out in atm because we use R = 0.08206 L atm / (mol K) with volume in liters.
Step 3, substitute and compute:
P = (2.00 mol x 0.08206 L atm / (mol K) x 298 K) / 48.9 L = 48.91 / 48.9 = 1.00 atm
The answer lands exactly on 1 atmosphere, which makes sense: this is essentially two molar volumes (2 x 24.45 L = 48.9 L) at room temperature, so the pressure must be 1 atm. Rather than working through the algebra by hand every time, an ideal gas law calculator runs these same steps instantly and shows the substitution it used.
Example 2: Solving for Volume
Problem: What volume does 1.00 mol of gas occupy at 0 C (273.15 K) and 1.00 atm?
Step 1, isolate V: V = nRT / P.
Step 2, check units: T = 273.15 K, P = 1.00 atm, R = 0.08206 L atm / (mol K). Everything matches.
Step 3, substitute and compute:
V = (1.00 mol x 0.08206 L atm / (mol K) x 273.15 K) / 1.00 atm = 22.41 L
This is the famous molar volume at standard temperature and pressure. Whenever a volume answer for one mole near 0 C comes out far from 22.4 L, recheck the temperature conversion first.
Example 3: Solving for Moles
Problem: How many moles are in a 22.4 L sample at 1.00 atm and 0 C (273.15 K)?
Step 1, isolate n: n = PV / RT.
Step 2, check units: atm and liters pair with R = 0.08206; T is in kelvin.
Step 3, substitute and compute:
n = (1.00 atm x 22.4 L) / (0.08206 L atm / (mol K) x 273.15 K) = 22.4 / 22.41 = 1.00 mol
The division 22.4 / 22.41 equals 0.9993, which rounds to 1.00 mol. This is the reverse of Example 2, and the pair shows how the equation stays consistent in both directions.
Example 4: Solving for Temperature
Problem: One mole of gas at 1.00 atm occupies 24.5 L. What is its temperature?
Step 1, isolate T: T = PV / nR.
Step 2, check units: atm, liters, and R = 0.08206 line up; the answer will be in kelvin automatically.
Step 3, substitute and compute:
T = (1.00 atm x 24.5 L) / (1.00 mol x 0.08206 L atm / (mol K)) = 24.5 / 0.08206 = 298.6 K
That is 25.4 C, just above room temperature, which is exactly what you would expect for a molar volume a touch larger than the 24.45 L that 1 atm and 25 C produce.
Three Mistakes That Break the Answer
- Plugging in Celsius. The number one error in gas problems. 25 C must become 298 K before it touches the equation. A quick habit: add 273.15 the moment you read a temperature.
- Mixing R with the wrong units. R = 0.08206 belongs to atm and liters. If your pressure is in kPa, either convert to atm (divide by 101.325) or switch to R = 8.314. Never mix 0.08206 with kilopascals.
- Solving for the wrong variable. Read the question twice. “What volume” means isolate V, even if the pressure value looks like the interesting number. Circling the unknown before touching the algebra prevents most slips.
Frequently Asked Questions
How do I solve PV = nRT for pressure?
Divide both sides by V to get P = nRT / V. Multiply moles by 0.08206 and by the temperature in kelvin, then divide by the volume in liters. The answer comes out in atmospheres.
How do I solve PV = nRT for volume?
Divide both sides by P to get V = nRT / P. Multiply moles by 0.08206 and by the kelvin temperature, then divide by the pressure in atm. The result is in liters.
How do I find moles with the ideal gas law?
Use n = PV / RT. Multiply pressure by volume, then divide by R times the kelvin temperature. Make sure pressure and volume units match your R value.
How do I find temperature with the ideal gas law?
Use T = PV / nR. The answer is always in kelvin. Subtract 273.15 if you want Celsius afterward, but never subtract before calculating.
Can I use mmHg or psi in the ideal gas law?
Yes, as long as you convert first. Divide mmHg by 760 or psi by 14.6959 to get atm, then use R = 0.08206. Or use the R value built for your units, such as 62.36 L torr / (mol K) for mmHg, since 1 mmHg equals 1 torr.
What if I am given mass instead of moles?
Convert mass to moles first: moles = mass / molar mass. For example, 8.0 g of helium (molar mass 4.0 g/mol) is 2.0 mol. Then use PV = nRT normally.
