The combined gas law is a gas law which combines Charles's law, Boyle's law, and GayLussac's law. These laws each relate one thermodynamic variable to another mathematically while holding everything else constant. Charles' law states that volume and temperature are directly proportional to each other as long as pressure is held constant. Boyle's law asserts that pressure and volume are inversely proportional to each other at fixed temperature. Finally, GayLussac's law introduces a direct proportionality between temperature and pressure as long as it is at a constant volume. The interdependence of these variables is shown in the combined gas law, which clearly states that:
“  The ratio between the pressurevolume product and the temperature of a system remains constant.  ” 
This can be stated mathematically as
where:
For comparing the same substance under two different sets of conditions, the law can be written as:
The addition of Avogadro's law to the combined gas law yields the ideal gas law.
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Boyle's Law states that the pressurevolume product is constant:
Charles' Law shows that the volume is proportional to absolute temperature:
GayLussac's Law says that the pressure is proportional to the absolute temperature:
where P is the pressure, V the volume and T the absolute temperature and of an ideal gas.
By combining (1) and either of (2) or (3) we can gain a new
equation with P, V and T. Equation (2) is used in this example, and
the arbitrary subscript on the constant is dropped so that k =
k_{2}.
Substituting in Avogadro's Law yields the ideal gas equation.
A derivation of the combined gas law using only elementary algebra can contain surprises. For example, starting from the three empirical laws
where k_{v}, k_{p}, and k_{t} are the constants, one can multiply the three together to obtain
Taking the square root of both sides and dividing by T appears to produce the desired result
However, if before applying the above procedure, one merely rearranges the terms in Boyle's Law, k_{t} = P V, then after canceling and rearranging, one obtains
which is not very helpful if not misleading.
A physical derivation, longer but more reliable, begins by realizing that the constant volume parameter in GayLussac's law will change as the system volume changes. At constant volume V_{1} the law might appear P = k_{1} T while at constant volume V_{2} it might appear P= k_{2} T . Denoting this "variable constant volume" by k_{v}(V), rewrite the law as
The same consideration applies to the constant in Charles' law which may rewritten
In seeking to find k_{v}(V), one should not unthinkingly eliminate T between (4) and (5) since P is varying in the former while it is assumed constant in the latter. Rather it should first be determined in what sense these equations are compatible with one another. To gain insight into this, recall that any two variables determine the third. Choosing P and V to be independent we picture the T values forming a surface above the PV plane. A definite V_{0} and P_{0} define a T_{0}, a point on that surface. Substituting these values in (4) and (5), and rearranging yields
Since these both describe what is happening at the same point on the surface the two numeric expressions can be equated and rearranged
The k_{v}(V_{0}) and k_{p}(P_{0})are the slopes of orthogonal lines through that surface point. Their ratio depends only on P_{0} / V_{0} at that point.
Note that the functional form of (6) did not depend on the particular point chosen. The same formula would have arisen for any other combination of P and V values. Therefore one can write
This says each point on the surface has it own pair of orthogonal lines through it, with their slope ratio depending only on that point. Whereas (6) is a relation between specific slopes and variable values, (7) is a relation beween slope functions and function variables. It holds true for any point on the surface, i.e. for any and all combinations of P and V values. To solve this equation for the function k_{v}(V) first separate the variables, V on the left and P on the right.
Choose any volume P_{1}. the right side evaluates to some arbitrary value, call it k_{arb}.
This particular equation must now hold true, not just for one value of V but for all values of V. The only definition of k_{v}(V) that guarantees this for all V and arbitrary k_{arb} is
which may be verified by substitution in (8).
Finally substituting (9) in GayLussac's law (4) and rearranging produces the combined gas law
Note that Boyle's law was not used in this derivation but is easily deduced from the result. Generally any two of the three starting laws are all that is needed in this type of derivation  all starting pairs lead to the same combined gas law. ^{[1]}.
The combined gas law can be used to explain the mechanics where pressure, temperature, and volume are affected. For example:air conditioners, refrigerators and the formation of clouds.
