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Deriving the Combined Gas Equation
Boyle's Law states that, at constant temperature, the volume (V) of a given amount of gas is inversely proportional to its pressure (P):
Charles' Law states that, at constant pressure, the volume (V) of the same quantity of gas is directly proportional to its temperature (T) in Kelvin:
V ∝ T (in Kelvin)
So, combining both Boyle's Law and Charles' Law we would say that the volume (V) of a given quantity of gas is directly proportional to its temperature (T) in kelvin and also inversely proportional to its pressure (P):
that is:
We could use a constant of proportionality (k) to write an equation for this expression:
and we could rearrange this equation by multipling both sides by P:
P × V 
= 
P × k × T(K)
P 
P × V 
= 
k × T(K) 
then divide both sides of the equation by T(K):
P × V T(K) 
= 
k × T(K)
T(K) 
P × V T(K) 
= 
k 
If we know the pressure (P), the volume (V) and the temperature in Kelvin (T) for a given amount of gas we can find the value of the constant (k).
For example, let's say that we have some cold gas in a syringe.
Initially this gas has a temperature of 250 K and a volume of 20 mL at a pressure of 100 kPa.
We can calculate the value of the constant (k) for this quantity of gas:
P_{1} × V_{1} T_{1}(K) 
= 
k 
because we know:
P_{1} = 100 kPa
V_{1} = 20 mL
T_{1} = 250 K
so
If we now heat the same gas in the syringe to 400 K and allow the gas in the syringe to expand, pushing the plunger (piston) up until the gas occupies a volume of 25 mL, we can calculate the pressure of the gas in the syringe because we have already calculated the value of k for this quantity of gas:
P_{2}(kPa) × V_{2}(mL) T_{2}(K) 
= 
8 
and now
V_{2} = 25 mL
T_{2} = 400 K
P_{2}(kPa) × 25 400 
= 
8 
P_{2}(kPa) × 0.0625 
= 
8 
P_{2}(kPa) × 0.0625
0.0625 
= 
8 0.0625 
P_{2}(kPa) 
= 
128 kPa 
So, under the new conditions of temperature and volume, our gas trapped in the syringe now exerts a pressure of 128 kPa.
Now, we could take a shortcut. Because the value of the constant (k) is the same during the whole of this experiment (because we are using a fixed amount of gas), we could have written:
P_{1}(kPa) × V_{1}(mL) T_{1}(K) 
= 
8 
= 
P_{2}(kPa) × V_{2}(mL) T_{2}(K) 
In other words:
P_{1}(kPa) × V_{1}(mL) T_{1}(K) 
= 
P_{2}(kPa) × V_{2}(mL) T_{2}(K) 
The Combined Gas Equation requires that the amount of gas remains constant, and is usually used in this general form:
P_{1} × V_{1} T_{1}(K) 
= 
P_{2} × V_{2} T_{2}(K) 
In which
P_{1} = initial pressure of gas 

P_{2} = final pressure of gas (same units as P_{1}) 
V_{1} = initial volume of gas 

V_{2} = final volume of gas (same units as V_{1}) 
T_{1} = initial temperature of gas (in Kelvin) 

T_{2} = final temperature of gas (in Kelvin) 
We need to know 5 of the 6 values in order to calculate the 6^{th} value:
To calculate initial pressure (P_{1}):
P_{1} = 
P_{2} × V_{2} × T_{1} V_{1} × T_{2}



To calculate final pressure (P_{2}):
P_{2} = 
P_{1} × V_{1} × T_{2} V_{2} × T_{1}


To calculate initial volume (V_{1}):
V_{1} = 
P_{2} × V_{2} × T_{1} P_{1} × T_{2}



To calculate final volume (V_{2}):
V_{2} = 
P_{1} × V_{1} × T_{2} P_{2} × T_{1}


To calculate initial temperature (T_{1}):
T_{1} = 
P_{1} × V_{1} × T_{2} P_{2} × V_{2}



To calculate final temperature (T_{2}):
T_{2} = 
P_{2} × V_{2} × T_{1} P_{1} × V_{1}

