Ideal Gas Law

Read Section 5.3 of the textbook.

The Ideal Gas Law can be thought of as a combination of Avagadro's Law and the Combined Gas Law. Avagadro's Law states that equal volumes of gases at the same temperature and pressure contain the same number of molecules (or moles of molecules).

Mathematically this can be stated as  V = constant · n,
or V₁ / n₁  =   V₂ / n₂, where n = # of moles.

The Combined Gas Law is  P₁V₁ / T₁  =  P₂V₂ / T₂.

Putting the two laws together yields  P₁V₁ / (n₁T₁)  =  P₂V₂ / (n₂T₂).

This means that PV / nT equals a constant, which is represented by R. This constant is evaluated using this experimental data: One mole of any gas at STP occupies 22.4 L. So,

R = PV / (nT)

R = 0.0821 L atm/(mole K)

R is called the universal gas constant. It is true for any gas. The equation is usually written

PV = nRT, and this is called the Ideal Gas Law. It is true for any gas, within certain limits that we will discuss later.

R is a constant, but it may be written using different units - and so the numerical value changes accordingly. For example, if we choose to have R in terms of torr, we multiply the above value by 760 torr / atm. Write this out on paper, verify that the unit of atm cancels and that the answer is  R = 62.4 L torr / (moleK).

Example 1) A cylinder contains 2000.0 mL of O₂ gas at 1500.0 psi and 25°C. How many moles of O₂ gas are in the cylinder? What is the mass of O₂ gas in the cylinder?

PV = nRT, so

n = PV /(RT)

n = 8.34 mole

To convert moles to mass we use the molar mass of O₂, as we learned in Chapter 3:

8.34 mole x 32.0 g/mole = 267 g

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There are several different forms of the Ideal Gas Law that are important. In these forms the way to calculate n, the number of moles, has been re-expressed. On paper, verify for yourself using dimensional analysis that number of moles is calculated by dividing the mass of a sample by its molar mass. Symbolically this is n = m / M, where m = mass of sample and M = molar mass. Substituting this expression into the Ideal Gas Law gives PV = mRT/M.

Then, dividing both side by V and regrouping:

Notice that m/V is density, D. So,

P = DRT/M, where M is the molar mass of the gas.

Example 2) Calculate the density of N₂ gas at 1.00 atm and 25°C.

Use the Ideal Gas Law in the form P = DRT/M, but rearrange to solve for D.

D = PM / (RT)

Substitute in the values, and verify both the units & numerical value of the answer.

D = 1.14 g/L