Probability density function

Template:Orphan Template:Simplify Template:Continuum mechanics The combined gas law is a formula about ideal gases. It comes from putting together three different laws about the pressure, volume, and temperature of the gas. They explain what happens to two of the values of that gas while the third stays the same. The three laws are:

• Charles's law, which says that volume and temperature are directly proportional to each other as long as pressure stays the same.
• Boyle's law says that pressure and volume are inversely proportional to each other at the same temperature.
• Gay-Lussac's law says that temperature and pressure are directly proportional as long as the volume stays the same.

The combined gas law shows how the three variables are related to each other. It says that:

The formula of the combined gas law is:

[math]\displaystyle{ \qquad \frac {PV}{T}= k }[/math]

where:

Template:Math is the pressure

Template:Math is the volume

Template:Math is the temperature measured in kelvin

Template:Math is a constant (with units of energy divided by temperature).

To compare the same gas with two of these cases, the law can be written as:

[math]\displaystyle{ \qquad \frac {P_1V_1}{T_1}= \frac {P_2V_2}{T_2} }[/math]

By adding Avogadro's law to the combined gas law, we get what is called the ideal gas law.

Derivation from the gas laws

Boyle's Law states that the pressure-volume product is constant:

[math]\displaystyle{ PV = k_1 \qquad (1) }[/math]

Charles's Law shows that the volume is proportional to the absolute temperature:

[math]\displaystyle{ \frac{V}{T} = k_2 \qquad (2) }[/math]

Gay-Lussac's Law says that the pressure is proportional to the absolute temperature:

[math]\displaystyle{ P = k_3T \qquad (3) }[/math]

where P is the pressure, V the volume and T the absolute temperature of an ideal gas.

By combining (1) and either of (2) or (3), we can gain a new equation with P, V and T. If we divide equation (1) by temperature and multiply equation (2) by pressure we will get:

[math]\displaystyle{ \frac{PV}{T} = \frac{k_1(T)}{T} }[/math]

[math]\displaystyle{ \frac{PV}{T} = k_2(P)P }[/math].

As the left-hand side of both equations is the same, we arrive at

[math]\displaystyle{ \frac{k_1(T)}{T} = k_2(P)P }[/math],

which means that

[math]\displaystyle{ \frac{PV}{T} = \textrm{constant} }[/math].

Substituting in Avogadro's Law yields the ideal gas equation.

Physical derivation

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

[math]\displaystyle{ P = k_V\, T \,\! }[/math]Template:Spaces(1) Gay-Lussac's Law, volume assumed constant

[math]\displaystyle{ V = k_P T \,\! }[/math]Template:Spaces(2) Charles's Law, pressure assumed constant

[math]\displaystyle{ P V = k_T \,\! }[/math]Template:Spaces(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

[math]\displaystyle{ PVPV = k_V T k_P T k_T \,\! }[/math]

Taking the square root of both sides and dividing by T appears to produce of the desired result

[math]\displaystyle{ \frac {PV}{T} = \sqrt{k_P k_V k_T} \,\! }[/math]

However, if before applying the above procedure, one merely rearranges the terms in Boyle's Law, k_T = PV, then after canceling and rearranging, one obtains

[math]\displaystyle{ \frac{k_T}{k_V k_P} = T^2 \,\! }[/math]

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 = k₁T, while at constant volume V₂ it might appear P = k₂T. Denoting this "variable constant volume" by k_V(V), rewrite the law as

[math]\displaystyle{ P = k_V(V) \,T \,\! }[/math]Template:Spaces(4)

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

[math]\displaystyle{ V = k_P(P) \,T \,\! }[/math]Template:Spaces(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

[math]\displaystyle{ T_0 = \frac{P_0}{k_V(V_0)} \quad and \quad T_0 = \frac{V_0}{k_P(P_0)} }[/math]

Since these both describe what is happening at the same point on the surface, the two numeric expressions can be equated and rearranged

[math]\displaystyle{ \frac{k_V(V_0)}{k_P(P_0)} = \frac{P_0}{V_0}\,\! }[/math]Template:Spaces(6)

Note that Template:Sfrac and Template:Sfrac are the slopes of orthogonal lines parallel to the P-axis/V-axis and through that point on the surface above the PV plane. The ratio of the slopes of these two lines depends only on the value of Template:Sfrac 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

[math]\displaystyle{ \frac{k_V(V)}{k_P(P)} = \frac{P}{V} \quad\forall P, \forall V }[/math]Template:Spaces(7)

This says that each point on the surface has its 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.

[math]\displaystyle{ V\,k_V(V) = P\,k_P(P) }[/math]

Choose any pressure P₁. The right side evaluates to some arbitrary value, call it k_arb.

[math]\displaystyle{ V\,k_V(V) = k_{\text{arb}} \,\! }[/math]Template:Spaces(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

[math]\displaystyle{ k_V(V) = \frac{k_{\text{arb}}}{V} }[/math]Template:Spaces(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

[math]\displaystyle{ \frac{PV}{T} = k_{\text{arb}}\,\! }[/math]

Note that while Boyle's law was not used in this derivation, it 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.

Applications

The combined gas law can be used to explain the mechanics where pressure, temperature, and volume are affected. For example: air conditioners, refrigerators and the formation of clouds and also use in fluid mechanics and thermodynamics.

Related pages

• Dalton's law

Sources

• Raff, Lionel. Principles of Physical Chemistry. New Jersey: Prentice-Hall 2001

Other websites

• Interactive Java applet on the combined gas law by Wolfgang Bauer