Solve For V2: Quick Rearrangement Of The Gas Law
To solve the combined gas law for $$V_2$$, rearrange it to $$V_2 = \frac{P_1V_1T_2}{P_2T_1}$$. If you start with $$\frac{P_1V_1}{T_1} = \frac{P_2V_2}{T_2}$$, the final volume is isolated by multiplying both sides by $$T_2$$ and then dividing by $$P_2$$; a standard worked form of the equation is also shown as $$V_2 = \frac{P_1 \times V_1 \times T_2}{P_2 \times T_1}$$.
Gas Law Trick: Isolate V2 in One Step
The combined gas law connects pressure, volume, and temperature for a fixed amount of gas, and the quickest way to solve for the final volume is to put $$V_2$$ alone on one side of the equation. In practical chemistry problems, that means using the rearranged form $$V_2 = \frac{P_1V_1T_2}{P_2T_1}$$, with temperature in kelvins and consistent pressure and volume units.
Core Formula
The starting point is the combined gas law, written as $$\frac{P_1V_1}{T_1} = \frac{P_2V_2}{T_2}$$, where the subscripts 1 and 2 represent the initial and final states of the same gas sample.
To isolate $$V_2$$, rearrange algebraically:
- Multiply both sides by $$T_2$$.
- Divide both sides by $$P_2$$.
- Leave $$V_2$$ by itself on the right side.
That gives the solved form $$V_2 = \frac{P_1V_1T_2}{P_2T_1}$$, which is the form most chemistry references use for final-volume problems.
Step-by-step setup
- Write the combined gas law as $$\frac{P_1V_1}{T_1} = \frac{P_2V_2}{T_2}$$.
- Identify the unknown as $$V_2$$, not $$V_1$$, because the final volume is what you want.
- Move $$T_2$$ to the left by multiplying both sides by $$T_2$$.
- Move $$P_2$$ to the denominator by dividing both sides by $$P_2$$.
- Substitute your known values and solve.
Worked example
A common example uses $$P_1 = 0.833$$ atm, $$V_1 = 2.00$$ L, $$T_1 = 308$$ K, $$P_2 = 1.00$$ atm, and $$T_2 = 273$$ K. Plugging those into the rearranged equation gives $$V_2 = \frac{0.833 \times 2.00 \times 273}{1.00 \times 308}$$, which evaluates to about 1.48 L.
This example shows why the formula is useful: it converts a multi-variable gas problem into a straightforward calculation once the algebra is done.
Unit rules
Use kelvin for temperature, because gas-law relationships require absolute temperature rather than Celsius. In many classroom problems, pressure is given in atm and volume in liters, but the exact units matter less than consistency, as long as the same unit system is used on both sides of the equation.
| Variable | Meaning | Common unit | Role in $$V_2$$ formula |
|---|---|---|---|
| $$P_1$$ | Initial pressure | atm | Multiplies the numerator |
| $$V_1$$ | Initial volume | L | Multiplies the numerator |
| $$T_1$$ | Initial temperature | K | Appears in the denominator |
| $$P_2$$ | Final pressure | atm | Appears in the denominator |
| $$T_2$$ | Final temperature | K | Multiplies the numerator |
| $$V_2$$ | Final volume | L | The unknown being isolated |
Why the rearrangement works
The algebra works because the combined gas law is just a ratio relationship that stays constant for the same gas sample when the amount of gas does not change. By clearing the fractions carefully, you can isolate any one variable, but $$V_2$$ is especially easy because it ends up in a simple product-over-product form.
This is also why many textbooks encourage students to memorize the rearranged version rather than re-deriving it every time. The formula is efficient, direct, and less error-prone when you are working under time pressure.
Common mistakes
- Using Celsius instead of kelvin.
- Putting $$P_2$$ in the numerator instead of the denominator.
- Mixing up initial and final values.
- Forgetting that the gas amount must stay constant.
- Dropping units too early and losing track of consistency.
Those errors are common because the equation looks simple, but the variable placement matters a lot. A quick dimensional check can help: the result for $$V_2$$ should come out in volume units such as liters.
Historical context
The combined gas law is a classroom-level synthesis of older gas relationships, bundling Boyle's law, Charles's law, and Gay-Lussac's law into one equation for a fixed amount of gas. That makes it especially useful in introductory chemistry, where students need one formula that handles changing pressure, volume, and temperature at the same time.
Modern teaching resources still emphasize the same rearrangement because it is the shortest path from the general law to a usable final-volume expression. In practice, the algebraic isolation of $$V_2$$ is one of the first gas-law manipulations students learn because it builds confidence for later ideal-gas work.
Fast answer summary
If the combined gas law is $$\frac{P_1V_1}{T_1} = \frac{P_2V_2}{T_2}$$, then solving for $$V_2$$ gives $$V_2 = \frac{P_1V_1T_2}{P_2T_1}$$.
That is the final form to use whenever the question asks for the new volume of a gas after pressure and temperature change, assuming the amount of gas stays the same.
Everything you need to know about Solve For V2 Quick Rearrangement Of The Gas Law
What is the combined gas law for?
It relates pressure, volume, and temperature of a fixed amount of gas, letting you find an unknown variable when the others are known.
Why is temperature in kelvin?
Kelvin is required because gas-law equations depend on absolute temperature, and Celsius would distort the proportional relationship.
How do I solve for V2 quickly?
Use $$V_2 = \frac{P_1V_1T_2}{P_2T_1}$$, then substitute the values directly and calculate.
Can the combined gas law solve for other variables?
Yes. The same equation can be rearranged to solve for $$P_1$$, $$P_2$$, $$V_1$$, $$T_1$$, or $$T_2$$, depending on which value is missing.