9.4 Ideal Gas Law

The ideal gas law holds for all ideal gases at any temperature, pressure, and volume. But the only gases we have around us in the real world are real gases. Real gases behave most like ideal gases at low pressures ( atm or less) and high temperatures ( K or higher). It combines all the gas laws variable into one condition form.

Based on Boyle’s law: P/V = constant

Based on Charles’s law: V/T = constant

Based on Avogadro’s law: V/n= constant

Combining all, PV/n= constant. This constant is symbolized as R and called universal gas constant.

Figure 9.45 Ideal gas Law

Ref: common.wikimedia.org/

The ideal gas law has the form PV=nRT, where R= Ideal gas Constant (0.08206 L-atm/mol-K). With this equation, any one of the characteristics gas properties (P, V, T or n) can be calculated given the other three.

Gases

Ideal gas law follows al the assumptions of kinetic molecular theory. That is why it is called ideal gas law. In real world gas molecules behave differently. Below is the graph that shows how real molecules deviate from ideal gas law. Ideal gas law indicates that plot and PV vs. P should be a constant because PV= constant when temperature is a constant. Real molecules only behave ideally at low pressure and high temperature.

The ideal gas equation of state

If the variables P, V, T and n (the number of moles) have known values, then a gas is said to be in a definite state, meaning that all other physical properties of the gas are also defined. The relation between these state variables is known as an equation of state. By combining the expressions of Boyle’s, Charles’, and Avogadro’s laws (you should be able to do this!) we can write the very important ideal gas equation of state

where the proportionality constant R is known as the gas constant. This is one of the few equations you must commit to memory in this course; you should also know the common value and units of R.

An ideal gas is defined as a hypothetical substance that obeys the ideal gas equation of state.

Take note of the word “hypothetical” here. No real gas (whose molecules occupy space and interact with each other) can behave in a truly ideal manner. But we will see in the last lesson of this series that all gases behave more and more like an ideal gas as the pressure approaches zero. A pressure of only 1 atm is sufficiently close to zero to make this relation useful for most gases at this pressure.

Many textbooks show formulas, such as P₁V₁ = P₂V₂ for Boyle’s law. Don’t bother memorizing them; if you really understand the meanings of these laws as stated above, you can easily derive them on the rare occasions when they are needed. The ideal gas equation is the only one you need to know.

PVT surface for an ideal gas

In order to depict the relations between the three variables P, V and T we need a three-dimensional graph.

Each point on the curved surface represents a possible combination of (P,V,T) for an arbitrary quantity of an ideal gas. The three sets of lines inscribed on the surface correspond to states in which one of these three variables is held constant.

The red curved lines, being lines of constant temperature, or isotherms, are plots of Boyle’s law. These isotherms are also seen projected onto the P-V plane at the top right.

The yellow lines are isochors and represent changes of the pressure with temperature at constant volume.

The green lines, known

Figure 9.46 PVT Surface for Ideal gas

as isobars, and projected onto the V-T plane at the bottom, show how the volumes contract to zero as the absolute temperature approaches zero, in accordance with the law of Charles and Gay-Lussac.

Problem Example 1

A biscuit made with baking powder has a volume of 20 mL, of which one-fourth consists of empty space created by gas bubbles produced when the baking powder decomposed to CO₂. What weight of NaHCO₃ was present in the baking powder in the biscuit? Assume that the gas reached its final volume during the baking process when the temperature was 400°C.

(Baking powder consists of sodium bicarbonate mixed with some other solid that produces an acidic solution on addition of water, initiating the reaction
NaHCO₃(s) + H⁺ → Na⁺ + H₂O + CO₂

Solution: Use the ideal gas equation to find the number of moles of CO₂ gas; this will be the same as the number of moles of NaHCO₃ (84 g mol^–1) consumed :

9.1E–6 mol × 84 g mol^–1 = 0.0076 g

Here is another video to solve problems using ideal gas laws:

How to Use the Ideal Gas Law in Two Easy Steps

Example: How many moles of gas are in a typical human breath that takes in 0.50L of air at 1.0 atm pressure and 35⁰C?

Number of moles n is unknown. T= 35+ 273= 308K

n= PV/RT i.e. 1.0 * 0.50/(0.0821* 308)  n= 0.0198 mol or 0.020 mol

We can determine the mass of a gas from moles if the identity or molar mass of the gas is known. Moles= mass/molar mass.

What you should be able to do

Make sure you thoroughly understand the following essential ideas which have been presented above, and be able to state them in your own words. It is especially important that you know the precise meanings of all the green-highlighted terms in the context of this topic.

Boyle’s Law – The PV product for any gas at a fixed temperature has a constant value. Understand how this implies an inverse relationship between the pressure and the volume.

Charles’ Law – The volume of a gas confined by a fixed pressure varies directly with the absolute temperature. The same is true of the pressure of a gas confined to a fixed volume.

Avogadro’s Law – This is quite intuitive: the volume of a gas confined by a fixed pressure varies directly with the quantity of gas.

The E.V.E.N. principle – this is just another way of expressing Avogadro’s Law.

Gay-Lussac’s Law of Combining Volumes – you should be able to explain how this principle, that follows from the E.V.E.N. principle and the Law of Combining Weights,

The ideal gas equation of state – this is one of the very few mathematical relations you must know. Not only does it define the properties of the hypothetical substance known as an ideal gas, but it’s importance extends quite beyond the subject of gases

Volume, temperature, pressure and number of moles are interrelated in gaseous system.