Deriving the Combined Gas Law

Question

Just finished learning about Gas Laws like Charle's, Boyle's, and Gay- Lussac's gas laws

We were then given this problem.

P= Pressure

V= Volume

T= Temperature (kelvin)

Equation 1:

P x V = k₁

First equation says that pressure and volume are related in such a way that the product (of P x V) is a constant (same number). This is an inverse proportionality. For example, as pressure goes up, the volume goes down. 

Equation 2:

V x k₂ = T

Second equation says that volume and temperature are related in such a way that the volume multiplied by a constant (another number) is equal to the temperature. This is a direct proportionality. For example, as volume goes up, the temperature goes up. 

Equation 3:

P x k₃ = T

Third equation says that pressure and temperature are related in such a way that the pressure multiplied by a constant (another number) is equal to the temperature. This is also a direct proportionality. For example, as pressure goes up, the temperature goes up. 

These equations are then combined to make the combined gas law:

P₁ x V₁ / T₁ = P₂ x V₂ / T₂

And the problem is to show how one would derive the combined gas law from equations 1-3

Answer 1

The equation that ALL of the above are derived from is the Ideal Gas Law:

PV = nRT where n is the number of moles of the gas and R is the Ideal Gas Constant.

The Ideal Gas Law is not derived from the others but visa versa,

We can take the Ideal Gas Law (PV = nRT) and solve it for "nR" making it:

PV/T = nR or PV/T = k₄ (I used K₄ because you used k₁ thru k₃ in your equations)

In looking at your equation 1 (PV = k₁), this is ONLY true if temperature is held constant. So, if I algebraically manipulate my ideal gas law PV/T = k₄ by multiplying each side by T (a constant), I get your equation 1. (PV = Tk₄ = k₁)

The same holds true for your equation 2 (which is ONLY true if pressure is held constant).If P is a constant, I can algebraically manipulate my ideal gas law PV/T = k₄ by dividing both sides by P to get V/T = k₄/P = k₂. So if V/T = k₂, then V•k₂ = T.

The same holds true for your equation 3 (which ONLY holds true if volume is held constant). If V is constant, I can algebraically manipulate my ideal gas law PV/T = k₄ by dividing both sides by V to get P/T = k₄/V = k₃ and if P/T = k₃ then P•k₃ = T

All of these equations are derived from the Ideal Gas Law, PV = nRT and deal with specialized cases, where one of the 3 variables is held constant. The Ideal Gas Law can also be written as PV/T = nR

From this Ideal Gas Law, IF YOU HAVE A CLOSED SYSTEM, n is a fixed amount (a constant) and, since "R" is a constant, so is nR. That means that at time 1 (when P, V, and T are some values depending on the conditions), and at time 2 (when P, V, and T are some OTHER values as the conditions change), the value of PV/T will equal the same constant at both times meaning that P₁V₁/T₁ = P₂V₂/T₂ This is the most general case for a closed gas system (what you called the "combined gas law")

When you have specialized conditions, you get equations 1 thru 3.