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## 杂谈 |

# Combined gas law

Continuum mechanics |
---|

FluidsFluid statics Viscosity: Newtonian LiquidsLiquid pressure GasesAtmosphere Combined gas
law Plasma |

The **combined gas law** is a gas law
which combines Charles's law,
Boyle's law,and
Gay-Lussac's
law. These laws each relate one
thermodynamic variable to another mathematically while holding
everything else constant. Charles's law states that volume and
temperature are directly proportional to each other as long as
pressure is held constant. Boyle's law asserts that pressure and
volume are inversely proportional to each other at fixed
temperature. Finally, Gay-Lussac's law introduces a direct
proportionality between temperature and pressure as long as it is
at a constant volume. The inter-dependence of these variables is
shown in the combined gas law, which clearly states that:

“ | The ratio between the pressure-volume product and the temperature of a system remains constant. | ” |

This can be stated mathematically as

where:

*p*is the pressure*V*is the volume*T*is the temperature measured in kelvins*k*is a constant (with units of energy divided by temperature).

For comparing the same substance under two different sets of conditions, the law can be written as:

The addition of Avogadro's law to the combined gas law yields the ideal gas law.

## Contents |

## [edit] Derivation from the Gas Laws

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

Charles's Law shows that the

## [edit] Physical Derivation

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

- ............(1) Gay-Lussac's Law, volume assumed constant
- ............(2) Charles's Law, pressure assumed constant
- ............(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

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

However, if before applying the above procedure, one merely
rearranges the terms in Boyle's Law, k_{t} = P V, then
after canceling and rearranging, one obtains

which is not very helpful if not misleading.

- ............(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
V_{0} and P_{0} define a T_{0}, a point on
that surface. Substituting these values in (4) and (5), and
rearranging yields

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

- ............(6)

The k_{v}(V_{0}) and
k_{p}(P_{0})are the slopes of orthogonal lines
through that surface point. Their ratio depends only on
P_{0} / V_{0} 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

- ............(7)

This says each point on the surface has it 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.

Choose any pressure P_{1}. the right side evaluates to
some arbitrary value, call it k_{arb}.

- ............(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

- ............(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

Note that Boyle's law was not used in this derivation but 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.^{[1]}

## [edit] 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.