# Combination of Gas Laws?

• uestions
In summary: We can see that the equations can be combined to produce the ideal gas law by taking the partial results and multiplying them.

#### uestions

What do textbooks mean when the gas laws are "combined" to make the ideal gas law?
I think that if the equations were combined, the result would look something like this:

P = k(T)T P = k(V)/V P = k(n)n
P^3 = (k(T)T * k(V)*k(n)n)/V

or

P/T = k(T) PV = k(V) P/n = k(n)
(P^3*V)/nT = k(T)*K(V)*K(n)

(k(n), k(T), k(V) = constants)
I get pressure raised to the third power instead of plain pressure. Any pointers or insights into what I am not understanding? I find it hard to believe the very top three equations can just be mushed together without multiplying their respective pressures to get pressure to the third, as what PV = nRT suggests in my mind.

We can write the relevant gas laws as:

$PV = k_{1}(N,T)$ (Boyle's law)
$\frac{V}{T} = k_{2}(N,P)$ (Charles's Law)
$\frac{P}{T} = k_{3}(N,V)$ (Gay-Lussac's Law)

If we solve the second and third equations for $T$, we get

$V k_{3}(N,V) = P k_{2}(N,P)$

Since this equation must be true for all values of $(P,V,N)$, each side must equal a constant; the same constant.

In order for $V k_{3}(N,V) =const$, for all values of $V$, we require that
$k_{3}(N,V) =\frac{k_{a}(N)}{V}$

The same thing is true for $P k_{2}(N,P)=const$, so that
$k_{2}(N,P) =\frac{k_{b}(N)}{P}$

Substituting $k_{2}(N,P)$ into our second equation, or $k_{3}(N,V)$ into our third equation, we find that

$\frac{PV}{T} =k_{a}(N) =k_{b}(N)\equiv k(N)$ (combined law)

This gets us almost all the way. the last law we need is Avogadro's law
$\frac{V}{N} = k_{4}(P,T)$ (Avogadro's law)

Solving both the combined law and Avogadro's law for the volume, we find

$\frac{k(N)}{N} = \frac{P}{T}k_{4}(P,T)$

Again, each side must be independently constant.

Where
$\frac{k(N)}{N} = const$
we find that
$k(N) = k_{c} N$
which gives us (from combined law)
$\frac{PV}{NT} =k_{c}$

Where
$\frac{P}{T}k_{4}(P,T)=const$
we find that
$k_{4}(P,T)=k_{d}\frac{T}{P}$
which gives us (from Avogadro's law)
$\frac{PV}{NT} = k_{d}=k_{c}\equiv k$

In either case, we arrive at the final result:
$PV = k NT$
where k is a constant of proportionality independent of $P,V,N$, or $T$ (i.e. a universal gas constant). More rigorous theoretical investigations show that k is Boltzmann's constant.

Working in units of mole number $n = \frac{N}{N_{A}}$, we have the more common version of the ideal gas equation $PV = n RT$, where $R=k N_{A}$.

In the simplest form, combined gas law is

$$\frac {P_1V_1}{T_1} = \frac {P_2V_2}{T_2}$$

It nicely follows as a generalization (although not in a strict way) from partial results

$${P_1V_1} = {P_2V_2}$$

$$\frac {P_1}{T_1} = \frac {P_2}{T_2}$$

$$\frac {V_1}{T_1} = \frac {V_2}{T_2}$$

## What is the combination of gas laws?

The combination of gas laws refers to the relationship between pressure, volume, and temperature of a gas. It is based on the combined laws of Boyle, Charles, and Gay-Lussac, which describe the behavior of gases under different conditions.

## What are the three gas laws involved in the combination of gas laws?

The three gas laws involved in the combination of gas laws are Boyle's law, which states that the pressure of a gas is inversely proportional to its volume at a constant temperature; Charles's law, which states that the volume of a gas is directly proportional to its temperature at a constant pressure; and Gay-Lussac's law, which states that the pressure of a gas is directly proportional to its temperature at a constant volume.

## How are the gas laws combined?

The gas laws are combined using the ideal gas law, which states that the product of pressure and volume is directly proportional to the product of the number of moles and the temperature of a gas. This is represented by the equation PV = nRT, where P is pressure, V is volume, n is the number of moles, R is the gas constant, and T is temperature.

## What is the significance of the combination of gas laws?

The combination of gas laws allows scientists to predict and explain the behavior of gases in various conditions. It is also used in various applications, such as in the design of engines and the production of industrial gases.

## What are some real-life examples of the combination of gas laws?

Some real-life examples of the combination of gas laws include the functioning of a propane tank, where Boyle's law is used to keep the gas under pressure, and the inflation of a balloon, where Charles's law is used to increase the volume of the gas inside the balloon as it is heated. The combination of gas laws is also used in the production of bottled carbonated drinks.