The ideal gas law can be expressed in a number of ways. We can use one form of it to easily derive the laws of Boyle, Charles, and Gay-Lussac.

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### The Ideal Gas Law and Avogadro's Number

In terms of Avogadro's number N

_{A}, we can write the number of moles n as:1) n = N/N

_{A}where N represents the total number of molecules in a gas and N

_{A}= 6.02 x 10^{23}molecules/mole. We found in part one that the ideal gas law may be written as:2) PV = nRT

Substituting 1 into 2 yields:

3) PV = N/N

_{A}RTWe can simplify this equation even further with the use of

**Boltzmann's contant**. Boltzmann's constant is defined as:4) k = R/N

_{A}which is 1.38 x 10

^{-23}J/K in SI units. Then equation 3 becomes:5) PV = NkT

This is another standard way of writing the ideal gas equation.

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### Another Form

It is not always necessary to use the number of molecules or Boltzmann's constant. In this case, the ideal gas law is also commonly written as:

6) P

_{1}V_{1}/ T_{1}= P_{2}V_{2}/ T_{2}where P

_{1},V_{1}, and T_{1}are the original values of the gas, while P_{2},V_{2}, and T_{2}represent its final values.We can use equation 6 to derive Boyle's, Charles', and Gay-Lussac's Laws. We do this by considering isothermal, isobaric, and isochoric thermodynamic processes.

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### Derivation of Boyle's Law

**Isothermal**Isothermal means that the temperature is constant. When we do this, T1 = T2 = T, so:

7) P

_{1}V_{1}/ T = P_{2}V_{2 }/ TThe Ts cancel, and we are left with Boyle's Law P

_{1}V_{1}= P_{2}V_{2}. - slide 4 of 7
### Derivation of Charles' Law

**Isobaric**Isobaric means that the pressure is constant, so P1 = P2 = P, giving us:

8) PV

_{1 }/ T_{1}= PV_{2 }/ T_{2}With the Ps canceling, we are left with Charles' Law V

_{1}/T_{1}= V_{2}/T_{2}. - slide 5 of 7
### Derivation of Gay-Lussac's Law

**Isochoric**Finally, isochoric means the volume is constant, such that V1 = V2 = V, and thus we have:

9) P

_{1}V / T_{1 }= P_{2}V / T_{2}The Vs cancel, giving us Gay-Lussac's Law P

_{1}/T_{1}= P_{2}/T_{2}. - slide 6 of 7
### References

Physics for Scientists and Engineers by Douglas Giancoli

Fundamentals of Physics by Halliday, Resnick, and Walker

*Image Credits**Ideal gas law from www.EngineersEdge.com* - slide 7 of 7