# What's wrong with this derivation of pressure-temperature for an ideal gas?

The pressure temperature relationship for an ideal gas, known as Amontons law or Gay Lussac's law should be obtainable from Boyle's and Charles's laws.

I did the following.

For a given sample of an ideal gas, by Boyle's law,

$$pV=a$$

And by Charle's law

$$V=bT$$

Where $b,a$ are constants for the given sample of gas. Substituting for $V$ in the first equation from the second,

$$pT=\frac{a}{b}$$

Which is not the correct relationship. Pressure should be directly proportional to temperature. Where did I go wrong?

• I believe different things are implicitly held constant in Boyle's and Charles' laws. – Ryan Unger Feb 2 '15 at 4:29
• @0celo7: Yes, they are. But for a given sample of gas, all things are constant. – Gerard Feb 2 '15 at 4:31

## 1 Answer

Note your first equation says that pV is constant implying you do not change temperature and you define a. The second equation says keep p constant and you define b. Your final equation contains a and b implying constant T and p hence pT is a constant. Therefore there is nothing wrong.