Other Forms of the Ideal Gas Law

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 10²³ 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_ART

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.

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) P₁V₁ / T₁ = P₂V₂ / T₂

where P₁,V₁, and T₁ are the original values of the gas, while P₂,V₂, and T₂ 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.

Isothermal

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

7) P₁V₁ / T = P₂V₂ / T

The Ts cancel, and we are left with Boyle's Law P₁V₁ = P₂V₂.

Isobaric

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

8) PV₁ / T₁ = PV₂ / T₂

With the Ps canceling, we are left with Charles' Law V₁/T₁ = V₂/T₂.

Isochoric

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

9) P₁V / T₁ = P₂V / T₂

The Vs cancel, giving us Gay-Lussac's Law P₁/T₁ = P₂/T₂.

Physics for Scientists and Engineers by Douglas Giancoli

Fundamentals of Physics by Halliday, Resnick, and Walker

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Ideal gas law from www.EngineersEdge.com

Comprehensive List of the Various Forms of the Ideal Gas Law