GaleDoesThermo

Sorry guys! I know my queue ran out. I had 3 exams this week + a lab report and another 3 next week. I should be able to make at least one post this weekend between shifts.

Isothermal Case

By isothermal we mean that temperature is held constant. The following is true:

∆U =  ∆H = 0

∆S = R * ln( V2/V1 ) = R * ln( P1/P2 )    (substitutions with ideal gas law)

q = w = RT * ln( V2/V1 ) = RT * ln( P1/P2 )    (substitutions with ideal gas law)

w =  ∫ PdV

P1V1 = P2V2 (substitutions with ideal gas law)

Isobaric Case

By isobaric we mean that pressure is held constant. The following is true:

∆U = ∫ CvdT

∆H = ∫ CpdT

∆S = ∫ CpdT / T

q =  ∆H

w =  ∫ PdV

Isochoric Case

By isochoric we mean that volume is held constant. The following is true:

∆U = ∫ CvdT

∆H = ∫ CpdT

∆S = ∫ CvdT / T

q =  ∆U

w = 0

Adiabatic Case

By adiabatic we mean that heat flow is held constant. This is also referred to as reversible adiabatic. The following is true

∆U = ∫ CvdT

∆H = ∫ CpdT

∆S = 0

q = 0

w = - ∆U

P1(V1^ɣ) = P2(V2^ɣ)    ( ɣ is lowercase greek letter gamma)

ɣ = Cp/Cv = (R + Cv) / Cv

Enthalpy

Enthalpy(H) is the total amount of heat in a system. The equation is as follows:

H = U + PV (PV is the pressure and volume of the environment acting on the system)

Heat Capacity

Heat Capacity(C) is how much heat flow is required to change the temperature of a substance by 1 Kelvin ( dq / dT). This value changes depending on whether the system is being considered at constant volume or constant pressure, and can be used to find internal energy and enthalpy! This is usually much easier than trying to find them from the equations already given, especially for enthalpy.

Cv = ( dU / dT )v

Cp = ( dH / dT )p

For ideal gases, we can make the following assumptions:

Cv = 3R / 2

Cp = 5R / 2

[We use R = 8.314 J / mol * K for heat capacity(same SI units)]

Because C depends on temperature changing, we can say that  ∆U and  ∆H are zero if the temperature is held constant. We will go more into other things that we know to be true by definitions in the next post, as well as nomenclature for when different variables are held constant.

Reversible vs. Irreversible Changes

A reversible change is considered to happen at equilibrium, when the energy changes so incrementally that a system can exist as two different phases at the same amount of energy. This means that it doesn’t generate or consume heat ( q = 0). For that to happen, the system must be perfectly insulated. This is impossible. However, it helps with simplifying calculations to consider something to be reversible.

The Zeroth Law

If two systems are in equilibrium with a third system, then they must also be in equilibrium with each other. This is the merely the transitive property from math courses applied to physics.

Fun fact, this is called the zeroth law because it was discovered after all the other three but is more fundamental!

The First Law

Energy cannot be created or destroyed(like matter). The internal energy of a system(U) is equal to the heat flow(q) into the system minus the work(w) done by the system. Why? Heat flow adds energy, and work expends it! 

∆ U = q - w    *

w =  ∫ PdV

*This is a state function. That means that instead of relying on one variable, it relies on two or more. Both q and w can change.

The Second Law

The total entropy(S) of a system cannot decrease over time. Therefore, a system at equilibrium will be at a maximum entropy.

The heck is entropy??? It’s a fancy name for disorder, or the different number of ways that the molecules within a system can be oriented. This increases as heat flow increases; likewise, it will also increase with temperature (the following equation assumes constant temperate. The equation for changing temperature will be introduced later).

∆ S = ∆ q / T

The Third Law

The entropy of the system of a pure substance approaches zero as temperature approaches absolute zero. At absolute zero, the molecules within the system won’t have energy to move around. This means that there is only one state in which they can exist. For systems that are impure (more than one molecule or phase), the entropy will instead approach a constant value called the residual entropy. Entropy can be related to the number of possible microstates as follows:

S = k * ln(Ω)

k = Boltzmann’s constant = 1.380 E-23 J / K

Extensive and Intensive Properties

What is thermodynamics?

Thermodynamics is about how matter responds to changes in its environment or system! This could mean temperature, pressure, volume, or even composition. They could all change, or only one could change. We also use terms derived from these bases.

All in all, it’s the study of the relationships between the equilibrium of a system and whatever changes happen to it. This is super duper important to remember!

What is a state?

A state has a particular mass, velocity, position, mode(s) of motion, etc... Basically, it’s like looking at a picture of a boat with a little description of everything going on in that picture. Where does the boat exist? How hot is it? How big is it? Are there other boats? Is it moving? Specifically, this is a macroscopic state description. If you knew about all of the particles within that boat, that would be a microscopic state.

What do you mean by fixed state?

How can a state be broken? The heck?

What we mean by fixed is that there are no changes to the state. Remember those base terms I mentioned earlier? Fixing a few of those will fix everything else in the state because of how they are related to each other (this can get complicated depending on the system). Below is an example of the notation for pressure at a fixed volume:

PV = !Equation bits!

What’s that ideal gas law again?

I’m so glad you asked! It’s an equation used for gases in which we assume that:

A. There are a lot of particles spread far apart

B. The collisions between the gas and the walls of the container are elastic (no loss of kinetic energy)

C. The particles are in continuous, rapid, and random motion

D. There are no forces of attraction between particles

E. The temperature of the gas depends only on the kinetic energy of any moving object

These are a lot of assumptions, but personally I think assumptions B and D are the most important to remember (these are arbitrarily assigned, there’s no standard order of lettering/numbering). Below is the equation:

PV’ = nRT

This describes the state of the system. That is, V’ is the volume of the entire system! Without the apostrophe, it is the molar volume (Volume per mole). P is pressure, n is moles, R is the gas constant, and T is temperature in Kelvins. We have two different values for R depending on the units of your terms.

(Ideal Gas Law with V’)

PV = RT

R = 0.082057 L*atm / mol*K

R = 8.3144 J / mol*K

That’s it for now! We’ll continue with more important definitions in future posts!