## 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}RT

We 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.

## 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.

## 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 }/ T

The Ts cancel, and we are left with Boyle’s Law P_{1}V_{1} = P_{2}V_{2}.

## 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}.

## 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}.

## 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*

## Resources

Comprehensive List of the Various Forms of the Ideal Gas Law

## This post is part of the series: Introduction to the Ideal Gas Law

- What is The Ideal Gas Law?
- The Ideal Gas Law and the Mass of Air Inside Your Bedroom
- Other Forms of the Ideal Gas Law