Combined Gas Law - Physical Derivation

Physical Derivation

A derivation of the combined gas law using only elementary algebra can contain surprises. For example, starting from the three empirical laws

............(1) Gay-Lussac's Law, volume assumed constant

............(2) Charles's Law, pressure assumed constant

............(3) Boyle's Law, temperature assumed constant

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 of 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 Gay-Lussac's law will change as the system volume changes. At constant volume V₁ the law might appear P = k₁ T while at constant volume V₂ it might appear P= k₂ T . Denoting this "variable constant volume" by k_v(V), rewrite the law as

............(4)

The same consideration applies to the constant in Charles's law which may rewritten

............(5)

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 V₀ and P₀ define a T₀, 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

............(6)

The k_v(V₀) and k_p(P₀)are the slopes of orthogonal lines through that surface point. Their ratio depends only on P₀ / V₀ 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

............(7)

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 between 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 pressure P₁. the right side evaluates to some arbitrary value, call it k_arb.

............(8)

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

............(9)

which may be verified by substitution in (8).

Finally substituting (9) in Gay-Lussac'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.