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

Question

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.

Answer 1

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: $$ \frac{PV}{T}=k $$

Answer 2

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.

Answer 3

Charles' Law is $$\frac{V}{T}=k_1$$ and Boyle's Law is $$PV=k_2$$The subscripts are arbitrary.

Instead: \begin{align}\frac VT=f(P)\\ PV=g(T),\tag1\end{align} where the functions $f$ and $g$ are not identically zero. (When two proportional variables are varying against each other, every other relevant variable in the system is being fixed at a constant value.)

Thus, $$P\times f(P)=\frac{g(T)}T.$$ Since $P$ and $T$ are independent variables, $\big(P\times f(P)\big)$ and $\dfrac{g(T)}T$ must be a (nonzero) constant, say $k.$ Then, substituting $g(T)=kT$ into $(1)$ gives the combined gas law $$\frac{PV}{T}=k,$$ as required.

Answer 4

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

$$P_{1}V_{1}=P{2}V_{2}$$ $$PV= {\rm a\, constant}$$ $$P\, {\rm inversely\, proportional\, to}\, \frac{1}{V}\, {\rm at\, constant}\, T$$ $$PV = {\rm a\, constant} T$$

This is the mathematical equation or algebraic law for Boyle's law.