Combined Gas Law Calculator: Solve P1V1/T1 = P2V2/T2 Step by Step
A combined gas law calculator solves P₁V₁/T₁ = P₂V₂/T₂ for whichever variable is missing when a fixed amount of gas moves from one set of pressure, volume, and temperature conditions to another. Unlike the ideal gas law (PV = nRT), the combined gas law never needs the number of moles or the gas constant R — because the amount of gas doesn't change between the two states, both n and R cancel out of the equation entirely. That makes this approach the faster tool whenever a problem describes a sealed container, a balloon, or a cylinder being heated, cooled, compressed, or expanded, and asks what happens to one of the other variables as a result. This guide covers where the formula comes from, the exact algebra for isolating each variable, two fully worked numeric examples, the mistakes that most often break a combined gas law calculation, and practice problems with checked answers.
Inhalt
- 01What Is a Combined Gas Law Calculator and What Does P1V1/T1 = P2V2/T2 Mean?
- 02How Do Boyle's Law, Charles's Law, and Gay-Lussac's Law Combine into One Formula?
- 03How Do You Rearrange P1V1/T1 = P2V2/T2 to Solve for Any Variable?
- 04Worked Example 1: What Happens to Pressure When You Compress and Heat a Gas?
- 05Worked Example 2: How Does a Weather Balloon's Volume Change with Altitude?
- 06What Common Mistakes Break a Combined Gas Law Calculator Result?
- 07How Is the Combined Gas Law Different from the Ideal Gas Law?
- 08Practice Problems: Can You Solve These Combined Gas Law Problems?
- 09How Can Solvify's Combined Gas Law Calculator Help You Check Your Work?
What Is a Combined Gas Law Calculator and What Does P1V1/T1 = P2V2/T2 Mean?
A combined gas law calculator applies the equation P₁V₁/T₁ = P₂V₂/T₂ — read aloud as "P1V1 over T1 equals P2V2 over T2" — to a fixed amount of gas that changes from an initial state (subscript 1) to a final state (subscript 2). P is pressure, V is volume, and T is absolute temperature in kelvin. The subscript 1 values describe the gas before the change, and the subscript 2 values describe it after, whether that change is heating a sealed cylinder, compressing a piston, or letting a balloon rise into thinner, colder air. Because the same amount of gas is present in both states, moles (n) and the gas constant (R) are identical on both sides of the equation and cancel out algebraically, leaving only pressure, volume, and temperature to track. That's the entire appeal of this approach over the full ideal gas law: fewer variables to look up, fewer units to juggle, and a direct comparison between "before" and "after" instead of a full solve from scratch. As with every gas law, temperature must be in kelvin — never Celsius or Fahrenheit — because the relationship is built on absolute temperature, where 0 K represents the theoretical point at which gas particles stop moving. Pressure and volume can be in any consistent units (atm, kPa, or mmHg for pressure; L or mL for volume) as long as the same unit is used for both the initial and final state, since this comparison is really just two ratios set equal, rather than plugging into a formula with a fixed constant.
A combined gas law calculator compares two states of the same gas sample — no moles, no gas constant, just pressure, volume, and temperature before and after.
How Do Boyle's Law, Charles's Law, and Gay-Lussac's Law Combine into One Formula?
The combined gas law formula isn't a new discovery — it's three older gas laws merged into a single relationship, each describing what happens when one variable is held constant while the other two change.
1. Boyle's law (constant temperature)
P₁V₁ = P₂V₂. At constant temperature, pressure and volume are inversely related — compress a gas and its pressure rises. This is sometimes called the Boyle law relationship, and it's the P-V portion of the combined formula.
2. Charles's law (constant pressure)
V₁/T₁ = V₂/T₂. At constant pressure, volume and absolute temperature are directly proportional — heat a gas in a flexible container and it expands. This is the Charles law relationship contributing the V-T portion.
3. Gay-Lussac's law (constant volume)
P₁/T₁ = P₂/T₂. At constant volume, pressure and absolute temperature are directly proportional — heat a sealed rigid container and the pressure inside climbs. This is the Gay-Lussac law relationship contributing the P-T portion.
4. Merging the three into the combined gas law
Multiplying the P-V, V-T, and P-T relationships together and canceling the repeated terms produces P₁V₁/T₁ = P₂V₂/T₂ — a single formula that works whether pressure, volume, or temperature (or any two of the three at once) are changing simultaneously.
The combined gas law formula is Boyle's law, Charles's law, and Gay-Lussac's law folded into one equation, so it can be applied even when more than one variable changes at once.
How Do You Rearrange P1V1/T1 = P2V2/T2 to Solve for Any Variable?
Solving P₁V₁/T₁ = P₂V₂/T₂ comes down to isolating one of six variables — P₁, V₁, T₁, P₂, V₂, or T₂ — by cross-multiplying the two ratios and dividing.
1. Solving for final pressure
P₂ = P₁V₁T₂ / (T₁V₂)
2. Solving for final volume
V₂ = P₁V₁T₂ / (T₁P₂)
3. Solving for final temperature
T₂ = P₂V₂T₁ / (P₁V₁)
4. Solving for any initial-state variable
The same pattern works in reverse for P₁, V₁, or T₁ — just swap which side of the equation you're isolating from. For example, P₁ = P₂V₂T₁ / (T₂V₁).
Cross-multiply P₁V₁/T₁ = P₂V₂/T₂, then divide by whatever is left multiplying your unknown — that's every rearrangement the combined gas law requires.
Worked Example 1: What Happens to Pressure When You Compress and Heat a Gas?
A sealed 4.00 L container of gas is at 1.00 atm and 300 K. The gas is compressed to 2.00 L and heated to 350 K. What is the new pressure? Because the gas is sealed, moles stay constant, so this is a direct combined gas law problem using P₂ = P₁V₁T₂ / (T₁V₂).
1. Step 1 — List the known values
P₁ = 1.00 atm, V₁ = 4.00 L, T₁ = 300 K, V₂ = 2.00 L, T₂ = 350 K. All temperatures are already in kelvin, so no conversion is needed.
2. Step 2 — Substitute into P₂ = P₁V₁T₂ / (T₁V₂)
P₂ = (1.00 atm × 4.00 L × 350 K) / (300 K × 2.00 L)
3. Step 3 — Compute the numerator and denominator
Numerator: 1.00 × 4.00 × 350 = 1400 atm·L·K. Denominator: 300 × 2.00 = 600 L·K.
4. Step 4 — Divide
P₂ = 1400 / 600 = 2.333 atm, which rounds to 2.33 atm.
5. Step 5 — Check the answer
Both volume decreased (which alone would double the pressure, since V was halved) and temperature increased (which alone would raise pressure further). A final pressure more than double the original 1.00 atm is consistent with both effects acting in the same direction, confirming 2.33 atm is reasonable.
2.33 atm — when volume drops and temperature rises at the same time, the combined gas law captures both effects in a single step instead of solving them separately.
Worked Example 2: How Does a Weather Balloon's Volume Change with Altitude?
A weather balloon has a volume of 2.50 L at ground level, where pressure is 1.00 atm and temperature is 293 K. It rises to an altitude where pressure drops to 0.500 atm and temperature drops to 250 K. What is the balloon's new volume? This uses V₂ = P₁V₁T₂ / (T₁P₂).
1. Step 1 — List the known values
P₁ = 1.00 atm, V₁ = 2.50 L, T₁ = 293 K, P₂ = 0.500 atm, T₂ = 250 K.
2. Step 2 — Substitute into V₂ = P₁V₁T₂ / (T₁P₂)
V₂ = (1.00 atm × 2.50 L × 250 K) / (293 K × 0.500 atm)
3. Step 3 — Compute the numerator and denominator
Numerator: 1.00 × 2.50 × 250 = 625 atm·L·K. Denominator: 293 × 0.500 = 146.5 K·atm.
4. Step 4 — Divide
V₂ = 625 / 146.5 = 4.266 L, which rounds to 4.27 L.
5. Step 5 — Check the answer
Pressure dropped to half its original value, which alone would double the volume to 5.00 L. Temperature also dropped, which alone would shrink the volume somewhat. The net result, 4.27 L, sits below the pressure-only estimate of 5.00 L, exactly as expected once the cooling effect is factored in — confirming the answer is reasonable.
4.27 L — the combined gas law is what makes this kind of two-directional altitude problem solvable in one pass instead of two separate gas law calculations.
What Common Mistakes Break a Combined Gas Law Calculator Result?
Most incorrect combined gas law answers come from a handful of predictable setup errors rather than a misunderstanding of the algebra itself.
1. Forgetting to convert Celsius to kelvin
T must be in kelvin: K = °C + 273.15. Plugging a Celsius value directly into either T₁ or T₂ produces a badly wrong ratio, since the combined gas law depends on absolute temperature.
2. Mixing pressure or volume units between states
P₁ and P₂ must share the same unit (both atm, or both kPa) and V₁ and V₂ must share the same unit (both L, or both mL). The combined gas law compares ratios, so mismatched units between the initial and final state invalidate the whole calculation even if each individual number is correct.
3. Assuming moles stay constant when the container isn't sealed
The combined gas law only applies when the amount of gas doesn't change. If gas is added, removed, or reacts chemically between the two states, n is no longer constant and the full ideal gas law (PV = nRT) is required instead.
4. Swapping subscript 1 and subscript 2 values
Keep the "before" state consistently as subscript 1 and the "after" state as subscript 2 across P, V, and T. Mixing an initial pressure with a final volume in the same substitution is one of the most common setup errors.
5. Rounding intermediate results too early
Carrying only 2 significant figures through the numerator and denominator can shift the final answer. Keep at least 4 significant figures until the last step, then round to match the precision of the given data.
Matching units and matching subscripts, state by state, is what a good combined gas law calculator gets right automatically — and what a hand calculation most often gets wrong.
How Is the Combined Gas Law Different from the Ideal Gas Law?
The ideal gas law, PV = nRT, describes a single state of a gas sample — one specific pressure, volume, moles, and temperature at one moment — and requires knowing (or solving for) the number of moles and the gas constant R. The combined gas law, P₁V₁/T₁ = P₂V₂/T₂, instead compares the same fixed amount of gas between two different states, so n and R cancel out and never need to be looked up at all. A combined gas law calculator is the right tool when a problem gives you a full "before" state and a partial "after" state and asks for the missing piece; the ideal gas law calculator is the right tool when you're given a single, complete state and asked to find one of its four variables directly, or when moles themselves are part of the question.
1. When to use the combined gas law
Use P₁V₁/T₁ = P₂V₂/T₂ whenever a sealed, fixed amount of gas changes between two sets of conditions — heating, cooling, compressing, or expanding — and moles are not part of what's given or asked.
2. When to use the ideal gas law instead
Use PV = nRT when you need to find moles, when moles change during the problem (a reaction producing or consuming gas), or when you're given a single complete state rather than two states to compare.
3. Quick cross-check example
For the balloon problem above, since n is constant, plugging the same 2.50 L / 1.00 atm / 293 K state into PV = nRT would first require solving for n, then using that n again at the new pressure and temperature to find V₂ — arriving at the same 4.27 L, but through more steps. That's why the combined gas law, not the full ideal gas law, is the faster path whenever moles are fixed.
If the amount of gas doesn't change, the combined gas law gets you there in fewer steps than the full ideal gas law every time.
Practice Problems: Can You Solve These Combined Gas Law Problems?
Work through each problem by hand before checking the answer. All temperatures must be converted to kelvin before substitution if given in Celsius.
1. Problem 1 — Find final volume
A gas occupies 3.00 L at 2.00 atm and 280 K. It is allowed to expand until the pressure drops to 1.00 atm at a temperature of 310 K. Find the new volume. Answer: V₂ = P₁V₁T₂/(T₁P₂) = (2.00)(3.00)(310)/[(280)(1.00)] = 1860/280 = 6.64 L.
2. Problem 2 — Find final temperature
A rigid sealed tank holds gas at 4.00 atm, 6.00 L, and 295 K. The gas is heated at constant volume until the pressure reaches 5.50 atm. Find the new temperature. Answer: since V is constant, T₂ = P₂T₁/P₁ = (5.50)(295)/4.00 = 1622.5/4.00 = 405.6 K.
3. Problem 3 — Find final pressure with a Celsius conversion
A gas sample at 1.20 atm, 3.50 L, and 20°C is compressed to 1.75 L and heated to 80°C. Find the new pressure. First convert: T₁ = 20 + 273.15 = 293.15 K, T₂ = 80 + 273.15 = 353.15 K. Then P₂ = P₁V₁T₂/(T₁V₂) = (1.20)(3.50)(353.15)/[(293.15)(1.75)] = 1483.23/513.01 = 2.892 atm, which rounds to 2.89 atm.
4. Problem 4 — Find initial volume
A gas ends up at 1.00 atm, 8.00 L, and 320 K after starting at 2.50 atm and 260 K. Find the initial volume. Answer: V₁ = P₂V₂T₁/(T₂P₁) = (1.00)(8.00)(260)/[(320)(2.50)] = 2080/800 = 2.60 L.
If Problem 3 came out near 2.89 atm, the Celsius-to-kelvin conversion and the cross-multiplication were both handled correctly — that combination is exactly what trips up most first attempts at a combined gas law problem.
How Can Solvify's Combined Gas Law Calculator Help You Check Your Work?
Once you understand where P₁V₁/T₁ = P₂V₂/T₂ comes from and how to rearrange it, a combined gas law calculator becomes a way to verify your own reasoning rather than a shortcut around learning it. Solvify's step-by-step chemistry solver works like a gas pressure volume temperature calculator that shows the full cross-multiplication, unit handling, and arithmetic, so you can line up your handwritten work against the calculator's steps and find exactly where a mistake happened instead of just seeing a different final number. A practical workflow: solve the problem by hand first, keeping subscript 1 and subscript 2 values clearly separated and writing out any Celsius-to-kelvin conversions explicitly. Then compare your answer against the calculator, and if the numbers don't match, walk through the calculator's steps to find the exact line where your setup diverged — usually a missed kelvin conversion, a mismatched unit between P₁ and P₂, or a swapped subscript. That comparison builds real understanding of the combined gas law formula, because it forces you to locate your own specific error rather than just reading a solved answer.
The goal of a combined gas law calculator should be to confirm your reasoning, not replace it — work the cross-multiplication by hand first, then check.
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