Part 2 - Ideal Gases | rezarezvan.com

Ideal gases

Definition 1 (Ideal gas)

An ideal gas is a gas that is not subject to interparticle interactions. Also, commonly said to not react with other substances.

In many applications of thermodynamics, ideal gases often come up, so it’s natural to understand them within a thermodynamic perspective.

Gases and Pressure

Before we begin with the real thermodynamics of gases we need a bit of background with gases. Boyle’s law gives us a mathematical formula for our intuition of gases. Say we have set up an experiment where the mass of a gas and the temperature are constant. The only thing that could vary now is the pressure within the gas, and its volume. How we define pressure is the key here - pressure is, in non-science terms, the force applied per area unit. Usually we write pressure as $P$ , with the formula,

P = \frac{F}{A}.

So if we expand our container which the gas is held in, intuitively, the pressure should release/decrease. This is exactly what Boyle’s law tells us.

Theorem 1 (Boyle’s law)

PV = \text{constant}

This means we can, for example, write,

P_1 V_1 = P_2 V_2.

But in many cases we can’t keep the amount of particles (mass) and temperature constant - so how do we describe it then?

Ideal Gas Law

Suppose we have a cylinder with height $h$ and a surface area of $A$ , containing some gas. Simplified, the gas will have a gravitational force $F_{Mg}$ , since the gas has a mass $M$ .

We can therefore describe the pressure as,

\begin{align*} M & = \rho A h \newline F & = Mg \newline P & = \frac{F}{A} = \frac{\rho \cancel{A} h g}{\cancel{A}} = \rho h g. \end{align*}

From observations and experiments, scientists have found that $PV \propto T$ is a linear relationship and found that,

Theorem 2 (Ideal gas law)

PV = nRT,

where,

\begin{align*} P & \quad \text{Pressure } [\mathrm{N/m^2}, \mathrm{Pa}] \newline V & \quad \text{Volume } [\mathrm{m^3}] \newline n & \quad \text{Amount of moles } [\mathrm{mol}] \newline R & \quad \text{Gas constant, } 8.3145 \ [\mathrm{J/(mol \cdot K)}] \newline T & \quad \text{Temperature } [\mathrm{K}] \end{align*}

Moles may be something new that some haven’t encountered. It’s a way of describing how many particles there are in a given substance with some mass. The formula for moles is,

n = \frac{N}{N_A},

where,

\begin{align*} N & \quad \text{Amount of particles } [\text{count}] \newline N_A & \quad \text{Avogadro's constant, } 6.022 \cdot 10^{23} \ [\mathrm{1/mol}] \end{align*}

We can also define something called ‘particle intensity’ which is just $\frac{N}{V}$ , which can be useful sometimes.

If we use this definition of moles, we can rewrite the ideal gas law as,

PV = \frac{N}{N_A}RT \rightarrow P = \frac{N}{V} \frac{R}{N_A} T.

This constant $\frac{R}{N_A}$ has a name, Boltzmann’s constant, denoted by,

k_B = \frac{R}{N_A} = 1.38 \cdot 10^{-23} \ [\mathrm{J/K}],

therefore the equation becomes,

PV = N k_B T.

Kinetic Gas Theory

Our day to day definition of temperature is how ‘hot’ or ‘cold’ it is - but what is temperature really describing? One can think that temperature describes the average kinetic energy in an object. This of course varies with what substance we have and on many other factors - but in general we can say that a gas particle has an average kinetic energy of,

E_{\text{average}} = \frac{\alpha}{2} k_B T,

where $\alpha$ is the so-called ‘degrees of freedom’.

Therefore the total energy in a gas is,

E_{\text{total}} = N E_{\text{average}} = N \frac{\alpha}{2} k_B T.

In our course and its applications, where we will only encounter monoatomic and diatomic gases - $\alpha = 3$ for monoatomic and $\alpha = 5$ for diatomic.

There’s also a so-called ‘inner energy’ in the gas which we denote with $U$ - for a monoatomic gas, $U$ is generally,

U = n \frac{3}{2}RT.

Conductivity in Ideal Gases

Often when we encounter gases in a thermal system - we will encounter two different categories of them. One where we have constant volume, the gas can not expand more than its initial state. One where we have constant pressure, we may have a lid that can be expanded, therefore the volume will change.

For these kinds of problems the usual $Q = cm\Delta T$ becomes tricky. But luckily, the formulas for these are,

\begin{align*} Q & = n C_v \Delta T, \newline Q & = n C_p \Delta T, \end{align*}

where the subscript of the $C$ denotes which is constant and the other varying.

With this we can also get formulas for $C_v$ and $C_p$ for ideal gases,

\begin{align*} C_v & = \frac{\alpha}{2}R, \newline C_p & = C_v + R. \end{align*}

Work by Gases

Now that we’ve encountered the different categories of problems - we can understand which kind can perform work and which can not. If we have constant volume it means the gas cannot expand or ‘push’ its surroundings. Therefore it can’t do work!

But in the case where we have constant pressure, but not volume, the gas can perform work. As we remember from mechanics - work is defined as,

dW = F \, dx.

Since we have another equation for the force, $F = PA$ , we can write,

dW = P(A \, dx) = P \, dV.

We usually write this as,

W = \int_i^f P \, dV.

Now the ‘direction’ is really important here - and one may wonder what ‘direction’ we have when we’re dealing with thermodynamics? Direction within thermodynamics means if the gas expands $(W > 0)$ or if it compresses $(W < 0)$ .

Phase and Transition variables

In a given system, we may have a heat flux, $Q$ - we have an ‘inner energy’, $U$ , and the system may do work, $W$ . Notice how I used may for $Q$ and $W$ - this is because they are so-called ‘phase variables’, this is because they are only present during a ‘phase’ or ‘process’.

While the system always has an inner energy, $U$ .

Thermodynamics

Now we have finally arrived at what this section of the course is all about - ‘Thermodynamics’. So how would one describe thermodynamics? Our lecturer described it as, “A transaction system for energy”. Which I find a simple - yet good - explanation. It really is about how systems ‘handle’ energy in and out of it.

We’ve already defined a lot of variables that are essential in thermodynamics:

$Q$ - heat exchange with the system’s environment
$U$ - the system’s inner energy
$W$ - work which the gas performs - here we have to be careful, sometimes work is defined as the work done by the outside on the system.

0th Law of Thermodynamics

It’s quite a funny name since it’s a physics law - and not a Computer Science one - that it’s 0 indexed. But this law came after all the other laws, because people just assumed this was the case - but a proper formal law was needed.

The 0th law of thermodynamics states that,

Theorem 3 (0th law of thermodynamics)

Suppose we have 3 bodies, 1, 2, and 3. In that order, if 1 and 2 are in thermal equilibrium - 2 and 3 are also in thermal equilibrium. Then 1 and 3 are also in thermal equilibrium.

As stated - a quite ‘obvious’ law that is not so powerful, but needed.

1st Law of Thermodynamics

Now this is a really powerful law - the 1st law of thermodynamics states that,

Theorem 4 (1st law of thermodynamics)

Q = \Delta U + W

If we ‘talk’ through this formula we understand that - the heat flux/heat exchange in a system, will be the difference in inner energy added with the work done by the gas in the system.

Isoprocesses

As we already touched upon - thermodynamical gas problems often come in two categories. Here we will formally define them. The prefix ‘iso’ means that something is unchanged, so therefore an isobar process means that the pressure is constant. In all of these different processes, the ideal gas law and the difference in inner energy will be the same for all,

\begin{align*} PV & = nRT, \newline \Delta U & = n \frac{\alpha}{2} R \Delta T. \end{align*}

Definition 2 (Isobar process)

In an isobar process, the pressure will be held constant throughout the system. Which means,

\begin{align*} W_{if} & = P(V_f - V_i), \newline Q & = n C_p \Delta T, \newline \Delta U & = n C_v \Delta T. \end{align*}

Definition 3 (Isochoric process)

In an isochoric process, the volume will instead be held constant throughout the system. Which leads to,

\begin{align*} W_{if} & = 0, \newline Q & = n C_v \Delta T, \newline \Delta U & = n C_v \Delta T. \end{align*}

Definition 4 (Isothermal process)

In an isothermal process, the temperature will instead be held constant throughout the system. Which means,

\begin{align*} W_{if} & = \int P \, dV = nRT \ln\left(\frac{V_f}{V_i}\right), \newline \Delta U & = n C_v \Delta T = 0, \newline Q & = W. \end{align*}

Definition 5 (Adiabatic process)

In an adiabatic process, the heat exchange is 0 - this is quite a special case, but this means,

\begin{align*} Q & = 0, \newline W & = n C_v (T_i - T_f) = -n C_v \Delta T, \newline \Delta U & = n C_v \Delta T. \end{align*}

With this we can see that $\Delta U$ has to ‘pay the price’ since $Q = 0$ . This means the system takes its own inner energy to complete the transaction in energy.

We also have one more formula in the adiabatic case,

PV^\gamma = \text{constant}.

With the ideal gas law we can find that,

TV^{\gamma - 1} = \text{constant},

as well as,

TP^{\frac{1 - \gamma}{\gamma}} = \text{constant}.

Thermal Efficiency

Now that we have gone through almost all processes that we’re covering in this course - we will finally formally define thermal efficiency! As I promised in part 1!

Definition 6 (Thermal efficiency)

Thermal efficiency can be defined as,

e = \frac{\sum W_g}{\sum Q_{\text{pos}}} = \frac{\sum Q_{\text{total}}}{\sum Q_{\text{pos}}}.

Thermal efficiency can also sometimes be denoted as $\eta$ .

Conclusion

This concludes this part - in the next part (coming next week, probably) we’ll cover the 2nd law of thermodynamics, as well as summarize everything!