Deriving combined gas law from Boyle's and Charles' laws

I know that the combined gas law, $$\frac{PV}{T}=k$$ should be derivable from Boyle's Law and Charles' Law. Since these are very basic equations, I presumed that it would be a simple matter, so I tried it myself.

Charles' Law is $$\frac{V}{T}=k_1$$ and Boyle's Law is $$PV=k_2$$The subscripts are arbitrary. In the derivation on Wikipedia, they jump from this to $$PV=k_2T$$

I'm sure I'm just overlooking something silly, but I see no way of combining Charles' and Boyle's to achieve an equation in which we don't cancel at least one of $P$, $V$, or $T$.

What am I missing? Thanks.

You have to realize first that Charles' law is the change in volume with respect to temperature for constant pressure while Boyle's law is the change in volume with respect to pressure for constant temperature. So when you combine them, you need to account for these

If I take a gas of volume $V_1$, pressure $P_1$ and temperature $T_1$ and let it change have a state $(V_*,\,P_2,\,T_1)$, then via Boyle's law, $$P_1V_1=P_2V_*\tag{1}$$ Then keeping this constant pressure we move to state $(V_2,\,P_2,\,T_2)$ via Charles' law, $$\frac{V_*}{T_1}=\frac{V_2}{T_2}\tag{2}$$ Solving for $V_*$ in both (1) and (2) then equating them, we get $$\frac{P_1V_1}{T_1}=\frac{P_2V_2}{T_2}$$ We've changed the pressure, volume and temperature of the gas but still find their product equal, suggesting that the relation is constant: $$\frac{PV}{T}=k$$

• Thanks Kyle - exactly what I was looking for. Seems the tricky thing in Thermo is always knowing which values to keep constant. I'd upvote your answer, but I just joined and don't have the reputation yet :P – stett Feb 16 '14 at 0:58
• Can $k$ be expressed in terms of the constants of proportionality of Boyle's and Charle's experimental laws? – Geremia Jul 31 '16 at 1:52

Once you have the two laws for isothermic and isochoric processes for a perfect gas, you can deduce the state equation.

We assume that there exists a “set of possible configurations $(P,V,T)$”, where the two laws (isothermal, isochoric) are both satisfied: $$PV=\phi (T),\quad T=P\xi (V),$$ for some functions $\phi,\xi$. We can then show that $\phi$ is the multiplication by a constant and $\xi = \phi ^{-1}$ .

Proof. If $P=1$, $$V=\phi(T)=\phi(\xi (V)).$$ Since $\xi$ is surjective, $\phi = \xi ^{-1}$. With $V=1$ we have $$\xi(P)=\xi (\phi(T))=T=P\xi(1).$$ So $\xi$ is the multiplication by $\xi (1)$, q.e.d.

According to Boyle's law: At constant temperature a volume gas is inversely proportionally to applied pressure this is boyle's law.

• This doesn't really answer the question. OP knows what Boyle's law and Charles' laws are, he just didn't see how to connect them such that one obtains $PV=kT$. – Kyle Kanos Jul 6 '14 at 14:16

The volume of a fixed mass of gas is inversely proportional to its pressure at constant temperature . This is known as Boyle's law.

P1 V1=P2 V2
PV =a constant
P inversely proportional to 1/V at constant T.
PV = a constant T.

This is mathematical equation or algebraic law for Boyele's law.

protected by David Z♦May 7 at 8:59

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