Mole and Molar Mass in Thermodynamics
Summary:
The class introduces the mole and molar mass in thermodynamics, highlighting the importance of the relationships between particles in an object. It explains the need for statistical approaches for systems with many particles. The mole is defined with Avogadro’s Number, and it shows how to calculate the molar mass using the periodic table.
Learning Objectives
By the end of this class, the student will be able to
- Define what a mole is and its importance for representing large quantities of particles in substances.
- Learn the value and meaning of Avogadro’s Number as the number of particles in one mole.
- Identify the relationship between molar mass, the mole, and Avogadro’s Number.
- Calculate the molar mass of different substances using the periodic table.
TABLE OF CONTENTS
Introduction
Large Numbers in Thermodynamics
What is a Mole?
What is Molar Mass?
Introduction
Our study begins with understanding what a mole and molar mass are, but first we need to review another idea: that “Every object is more than the sum of its parts”. Because the constitution of an object comes from its parts and “their way of being in relation to each other”. This “way of being” goes beyond the qualities of each part individually, because it is about the relationship of all with all and not just with themselves. As we will see, this is why different objects exhibit different qualities even though they are made of the same thing. For example, changing the temperature turns water into ice without adding, removing, or changing anything inherent to the water. Thus, “THERMODYNAMICS” is born, where the behavior of systems with many particles is studied, so many that a statistical approach is necessary.
Large Numbers in Thermodynamics
But… What do we mean by “many particles”? We often see large numbers: the population of the Earth is around (6-7)\cdot 10^9 PEOPLE, and the public debt of the United States is about 23 \cdot 10^{12} USD. But even these numbers pale in comparison to the magnitudes involved in thermal physics. For example, any object within your reach easily has more than 10^{20} PARTICLES, and this sets serious limits on the calculations we can perform to understand them.
| Example |
A kilogram of Nitrogen gas contains approximately 2\cdot 10^{25} MOLECULES OF N_2. Suppose we have a personal computer with a 3GHz processor, and assuming it will use all its power solely to count molecules, let’s see how long it will take to count all the molecules in the kilogram of Nitrogen. |
Solution: Since the computer will use all its power to count molecules, it will count one molecule for each processor cycle. Therefore, the counting time will be: t = \dfrac{2\cdot 10^{25}}{3\cdot 10^9 \left[\dfrac{1}{s}\right]} \approx 6,\overline{6} \cdot 10^{15} [s] Now, we know that each year has 356 days, each day 24 hours, and each hour 3600 seconds. So, if we convert the seconds into years, we get the not insignificant amount of 211,398,613.2 years. We are talking about more than 200 million years. In this example, we only talk about counting molecules and the time it would take to do so, but we haven’t said anything about calculating the interaction between the particles. If something so simple takes so long, then calculating the combined interaction of all of them is unattainable. |
Thus, to advance in the study of thermodynamics, it is necessary to review some statistical issues, the thermodynamic limit and the concept of the mole. We will start by reviewing the latter first.
What is a Mole?
A mole is a name used to represent a certain number of things. Its function is similar to that of the word “dozen” when you buy eggs (1 dozen eggs is 12 eggs). The mole, however, is designed to allow us to deal with numbers as grotesque as the number of atoms in a certain substance. Its definition is as follows:
| Definition |
A mole is the amount of matter that contains as many objects as the number of atoms in exactly 12[g] of ^{12}C |
The mole is also roughly defined as the amount of matter that contains as many objects as the number of atoms in exactly 1[g] of ^{1}H, but Carbon is preferred for the definition because, being in a solid state, it is much easier to measure precisely.
One mole of atoms is equivalent to one Avogadro’s Number N_A of atoms. Avogadro’s Number, expressed with four significant figures, is:
\boxed{N_A = 6.022 \cdot 10^{23}}
Avogadro’s Number is also often written with “units” as N_A = 6.022 \cdot 10^{23} \left[\dfrac{1}{mol}\right] as a reminder of its definition, although it is a dimensionless quantity (like [mol]).
| Example |
|
What is Molar Mass?
The molar mass of a substance is the mass contained in one mole of that substance. Thus, the molar mass of Carbon 12 is 12[g], the molar mass of water is close to 18[g]. One way to obtain a good approximation of the molar mass is through the sum of the mass numbers of the elements that make up the compound. For example, for water we have:
H_2 O = {}^{1}H + {}^{1}H + {}^{16}O
That is, two isotopes of Hydrogen with a single proton and one of Oxygen, which contains 8 protons and 8 neutrons. Thus, the molar mass will be 18[g].
Another more precise way to do the same is by using the periodic table, which considers the molar mass of atoms, taking into account the slight mass difference that exists between protons and neutrons.
If we determine the molar mass of water using the data from the periodic table, we will find that each mole of water weighs 2\cdot 1,00794 + 15,9994 [g]=18,01448[g].
The mass of a particle (molecule or atom) of a substance is, therefore, the molar mass divided by Avogadro’s number
\textnormal{Particle Mass} = \dfrac{\textnormal{Molar Mass}}{\text{Avogadro's Number}}