11.7: The Combined Gas Law: Pressure, Volume, and Temperature
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Learning Objectives
 Learn and apply the combined gas law.
One thing we notice about all the gas laws is that, collectively, volume and pressure are always in the numerator, and temperature is always in the denominator. This suggests that we can propose a gas law that combines pressure, volume, and temperature. This gas law is known as the combined gas law, and its mathematical form is
\[\frac{P_{1}V_{1}}{T_{1}}=\dfrac{P_{2}V_{2}}{T_{2}}\; at\; constant\; n\]
This allows us to follow changes in all three major properties of a gas. Again, the usual warnings apply about how to solve for an unknown algebraically (isolate it on one side of the equation in the numerator), units (they must be the same for the two similar variables of each type), and units of temperature must be in Kelvin.
Example \(\PageIndex{1}\):
A sample of gas at an initial volume of 8.33 L, an initial pressure of 1.82 atm, and an initial temperature of 286 K simultaneously changes its temperature to 355 K and its volume to 5.72 L. What is the final pressure of the gas?
SOLUTION
Steps for Problem Solving 


Identify the "given"information and what the problem is asking you to "find." 
Given: V_{1} = 8.33 L, P_{1} = 1.82 atm, and T_{1} = 286 K V2 = 5.72 L and T_{2} = 355 K Find: P_{2} = ? atm 
List other known quantities 
none 
Plan the problem 
First, rearrange the equation algebraically to solve for \(V_2\). \(P_2 = \frac{P_1 V_1 T_2 }{T_1V_2}\) 
Calculate 
Now substitute the known quantities into the equation and solve. \[P_2 = \frac{(1.82\, atm)(8.33\, \cancel{L})(355\, \cancel{K})}{(286\, \cancel{K})(5.72\, \cancel{L})}=3.22 atm\] 
Think about your result.  Ultimately, the pressure increased, which would have been difficult to predict because two properties of the gas were changing. 
Exercise \(\PageIndex{1}\)
If P_{1} = 662 torr, V_{1} = 46.7 mL, T_{1} = 266 K, P_{2} = 409 torr, and T_{2} = 371 K, what is V_{2}?
 Answer

105 mL
As with other gas laws, if you need to determine the value of a variable in the denominator of the combined gas law, you can either crossmultiply all the terms or just take the reciprocal of the combined gas law. Remember, the variable you are solving for must be in the numerator and all by itself on one side of the equation.
 The combined gas law relates pressure, volume, and temperature of a gas.
Contributors
Henry Agnew (UC Davis)