Deriving the Combined Gas Law

[PFuser1232]

How does one derive the combined gas law rigorously from Boyle's Law, Charle's Law and Gay-Lussac's law? I started out as follows (I will be using $P$ for pressure, $T$ for absolute temperature, and $V$ for volume - for constants I will use subscripts to denote the parameter on which the constant is dependent):

$PV = k_T = f(T)$
$P = k_V T = g(V)$
$V = k_P T = h(P)$

Multiplying those three equations together, one obtains:

$(PV)^2 = k_T k_V k_P T^2$
$\frac{PV}{T} = \sqrt{k_T k_V k_P}$

Now, where exactly do I go from here? I know it should equal $Nk_B$ (or $nR$) but how do I show that?
In other words, how can I prove that the square root of the product of three functions of pressure, volume and temperature equals a function independent of pressure, volume, and temperature?

 

[Philip Wood]

Two remarks, neither of which, I think, will satisfy you…
(1) I'd say 'clearly', or if you can't stand the dishonesty, 'intuitively', pV =nRT. Then I'd show that Boyle's law and so on are included in this equation. For example, if we hold n and T constant, then pV = constant.
(2) That pV is proportional to T is not an experimental law. It is true by definition of the ideal gas temperature scale, or the thermodynamic temperature scale, when the gas density approaches zero. Experiments in which gas pressure or volume is plotted against temperature as measured with (say) a mercury thermometer do give roughly straight lines, but that is not the real basis for the equation pV = nRT, which is exact as gas density approaches zero.

 

[256bits]

In other words, how can I prove that the square root of the product of three functions of pressure, volume and temperature equals a function independent of pressure, volume, and temperature?

I do not think you can. Pressure, volume and temperature are not independent upon one another, but are related by the ideal gas law, which also by the way, includes another fourth variable, n, the number of moles of substance.

As Philip stated, the ideal gas law is PV = nRT, and considering the variables P,V, T, with constant n, the setting any 2 will determine the third. This you can see from Boyle's Law, Charle's Law and Gay-Lussac's law, where your constants kT, kV, and kP implicitly include T, V, and P, as applicable.

Unless of course I am misinterpreting what you mean by the word "independent".

Perhaps it is a mathematical proof that you are asking of, for an equation such as xyz = C.

 

[DrClaude]

That pV is proportional to T is not an experimental law.

Could you please clarify what you mean here?

 

[Philip Wood]

Hello DrC. pv = constant when n and T are held constant is indeed an experimental law. [It holds fairly well for most gases at 'ordinary' densities. As the gas density approaches zero it holds more and more accurately. Hence the idea of an ideal gas as the low density limit of a real gas.]

But how would you check experimentally that p is proportional to T at constant V and n? How would you measure T? It's usual when teaching about gases at an elementary level to use a thermometer like a mercury thermometer to monitor the temperature of a water bath which is heated gradually, while reading the pressure of a the gas in a constant volume gas thermometer, immersed in the bath. This does give, roughly, a straight line, which extrapolates back to zero pressure at about -273°C.

But how has the thermometer been calibrated? Either on the mercury-in-glass empirical Celsius scale defined by $\frac{\theta}{100} = \frac{L_{Hg, \theta} - L_{Hg, 0}}{L_{Hg, 100} - L_{Hg, 0}}$ (or the Fahrenheit equivalent), in which case your straight line relates gas pressure merely to a property of mercury in glass or according to the thermodynamic scale, which is equivalent to the ideal gas scale, in which case claiming that your experiment shows that p is proportional to T is a circular argument.

A sounder approach in more advanced work is to admit that T is defined as proportional to ideal gas pressure. One traditional approach is to define thermodynamic temperature in terms of Carnot cycle heat input and output, then to 'use' an ideal gas (defined as obeying Boyle's law and having internal energy dependent only on temperature) as the working substance. It then emerges that the ideal gas obeys pV = nRT. So T has been defined in such a way that the equation is obeyed.

 

[PFuser1232]

I made a mistake in my first post.
It should be: $P = g(V)T$ and $V = h(P)T$ (I forgot to include temperature on the RHS).

 

[PFuser1232]

Hello DrC. pv = constant when n and T are held constant is indeed an experimental law. [It holds fairly well for most gases at 'ordinary' densities. As the gas density approaches zero it holds more and more accurately. Hence the idea of an ideal gas as the low density limit of a real gas.]

But how would you check experimentally that p is proportional to T at constant V and n? How would you measure T? It's usual when teaching about gases at an elementary level to use a thermometer like a mercury thermometer to monitor the temperature of a water bath which is heated gradually, while reading the pressure of a the gas in a constant volume gas thermometer, immersed in the bath. This does give, roughly, a straight line, which extrapolates back to zero pressure at about -273°C.

But how has the thermometer been calibrated? Either on the mercury empirical Celsius scale defined by $\frac{\theta}{100} = \frac{L_{Hg, \theta} - L_{Hg, 0}}{L_{Hg, 100} - L_{Hg, 0}}$ (or the Fahrenheit equivalent), in which case your straight line relates gas pressure merely to a property of mercury or according to the thermodynamic scale, which is equivalent to the ideal gas scale, in which case claiming that your experiment shows that p is proportional to T is a circular argument.

A sounder approach in more advanced work is to admit that T is defined as proportional to ideal gas pressure. One traditional approach is to define thermodynamic temperature in terms of Carnot cycle heat input and output, then to 'use' an ideal gas (defined as obeying Boyle's law and having internal energy dependent only on temperature) as the working substance. It then emerges that the ideal gas obeys pV = nRT. So T has been defined in such a way that the equation is obeyed.

But your argument seems to suggest that Charle's Law and Gay-Lussac's Law are not experimental laws.

 

[PFuser1232]

I do not think you can. Pressure, volume and temperature are not independent upon one another, but are related by the ideal gas law, which also by the way, includes another fourth variable, n, the number of moles of substance.

As Philip stated, the ideal gas law is PV = nRT, and considering the variables P,V, T, with constant n, the setting any 2 will determine the third. This you can see from Boyle's Law, Charle's Law and Gay-Lussac's law, where your constants kT, kV, and kP implicitly include T, V, and P, as applicable.

Unless of course I am misinterpreting what you mean by the word "independent".

Perhaps it is a mathematical proof that you are asking of, for an equation such as xyz = C.

This is basically deriving the three gas laws from the ideal gas equation. I wanted the opposite.

 

[Philip Wood]

MR97. Post 8 I know what you want. I used to want to do it. But now I think it's not worth trying. It's perfectly valid - and much easier - to see intuitively that pV = nRT fits all the laws, and then to show that it does.

MR97. Post 7 Yes. C's law and GL's law were certainly once regarded as experimental. And they still are, to the extent that we accept what a thermometer reads as being 'temperature'. But when we formulate an equation like pV = nRT, which is an exact equation for an ideal gas, T is not the reading on any real thermometer - except a thermometer that has been calibrated on the ideal gas scale.

 

[Philip Wood]

I know that my assertion that pV = nRT is not simply the fusion of three experimental laws is unpalatable. It spoils a nice story. But I'm sure that if you examine the foundations (especially the matter of thermometer calibration) carefully, you'll find that things aren't as simple.