Thermodynamics

Thermal expansion

$\Delta L=\alpha L_0\Delta T$ ; L=length, alpha=Coefficient of linear expansion, T=Temperature

$\Delta A=\gamma A_0\Delta T;2\alpha=\gamma$ ; gamma=Coefficient of areal expansion

$\Delta V=\beta V_0\Delta T;3\alpha=\beta$ ; beta=Coefficient of volumetric expansion

How do holes expand?

Holes in materials expand in the same way as they were filled with the surrounding material.

Internal energy

Total KE of all particles in a system

“u” measured in Joules

Heat

transfer of energy by random process (motion) (collisions, radiation)

“Q” measured in J

Work

Transfer of energy by bulk (i.e many particles moving the same way) motions.

Temperature

The average KE of a single particle

“T” measured in K

Process variables

Quantities that describe a process (way) taking a thermodynamic system from some initial state to some final state.

For example, Q, W

State variables

Quantities that describe the state of the thermodynamic system.

For example, T, P, V, n, N, S, u

Conduction

Transfer of KE of particles through random collisions (need material)

$T_{H},T_{C},\frac{Q}{\Delta t}$ ; these are constant if a thermodynamic system is in steady state

$P=\frac{Q}{\Delta t}=\frac{kA\left|\Delta T\right|}{d}$ ; Q=Heat, t=Time, k=Coefficient of thermal conduction, A=contact area, T=Temperature, d=Distance

Radiation

heat conduction through electromagnetic radiation (photons)

$P=eA\sigma T^4$ ; e=Emissivity coefficient, A=area, sigma=Stefan-Boltzmann’s constant, T=Temperature

$$0\le e\le1$$

If e=0, no power leaves; if e=1, all power leaves.

Heating curves

$Q=mc\Delta T$ ; equation for single phase; m=Mass, c=Specific heat, T=Temperature

$Q=Lm$ ; equation needed for 2 phases; L=Latent heat, m=Mass

Different ways to equate internal energy

$$u=\frac32PV=\frac32nRT=\frac32NK_{B}T=NK_{avg}=\frac12Nmv_{rms}^2$$

Thermal Equilibrium (between two thermodynamic systems) (Part of 0th Law)

No heat flows between them when they are in contact.

Thermodynamic Laws

0th Law: If two systems are in thermal equilibrium, then their temperatures are the same.

$$T_{A}=T_{B}$$

1st Law:$Q+W=\Delta U$ ; Q=Heat added, W=Work on gas

$W_{bygas}=-W_{ongas}$ ; Work on gas is the default in our class

2nd Law (v1): In the absence of external W, heat flows from hot to cold

2nd Law (v2): The most efficient engine is the Carnot engine, working between hot temperature and cold temperature reservoir

$$e=\frac{T_{H}=T_{C}}{T_{H}}$$

$$0\le e\le e_{c}<100\%$$

2nd Law (v3): A fridge needs to be plugged in. Q can flow from cold to hot, but only in the case of inputted work

2nd Law (v4): Systems undergo processes that take them from highly unlikely states to very likely states (probability wise).

2nd Law (v5-official): Entropy in a closed system can either increase or stay the same; it can never decrease.

Translating between Microscopic quantities and Macroscopic quantities

Microscopic quantities: $m,\overrightarrow{v},\overrightarrow{J},\overrightarrow{\Delta p},\Delta t,F,K_{avg}$

Macroscopic quantities: $P,V,T,N,n$ ; N=Number of particles, n=Amount

Maxwell’s Distribution

$N_{v}$ of particles moving at v

Area=N=const.

As Temperature decreases, the peak velocity decreases, the height increases, the width decreases.

Width and entropy have a proportional relationship.

Think of this as a histogram

Equation of State

$$PV=nRT$$

Isovolumetric/Isochoric process

V=const; n=const

$$\frac{P_1}{P_2}=\frac{T_1}{T_2}$$

W=0J

$$\Delta u=\frac32V\Delta P=Q$$

Lab design for Isovolumetric/Isochoric process

For Isochoric process with increasing pressure: Heat up gas in closed container

Isobaric process

P=const; n=const

$$\frac{V_1}{V_2}=\frac{T_1}{T_2}$$

$$W=-P\Delta V$$

$$\Delta u=\frac32P\Delta V$$

$$Q=\frac52P\Delta V$$

Lab design for Isobaric process

For Isobaric process with increasing volume: Heat up cylinder with piston attached. Add Mass to the top of piston (using sand). Apply heat.

Isothermal process

T=const; n=const

$$P_1V_1=P_2V_2$$

$$\Delta u=0J$$

$$Q=-W$$

Lab design for Isothermal process

For Isothermal process with decreasing volume and increasing pressure: Put container (w/piston) in boiling water/cold water. Add Mass to the top of piston (using sand).

Isoentropic/Adiabatic process

S=const; n=const

$$Q=0J$$

$$\Delta u=W$$

The slope of an Adiabatic graph is always steeper than an isothermal one

Lab design for adiabatic process

For adiabatic process with increasing volume and decreasing pressure: Surround container (w/piston) with a blanket (to prevent heat transfer). Remove Mass to the top of piston (remove sand).

Generic (applies to all previous PV graphs) process

W=area on PV graph; “+” when change in volume<0 (gas is compressed) and “-” when change in volume>0 (gas is expanding).

$$\Delta u=\frac32\left(P_{f}V_{f}-P_{i}V_{i}\right)$$

$$Q=\Delta u-W$$

Engine Efficiency

$$e=\frac{\left|W_{cyc}\right|}{Q_{+}}=\frac{\left|Q_{+}\right|-\left|Q_{-}\right|}{\left|Q_{+}\right|}=\frac{\left|W_{cyc}\right|}{\left|W_{cyc}\right|-\left|Q_{-}\right|}=\frac{P_{+}-P_{-}}{P_{+}}$$

Carnot Engine

Isothermal Expansion—>Adiabatic Expansion—>Isothermal Compression—>Adiabatic Compression (cycle repeats)

Entropy

a state variable that grows when the probability of finding a system in that state grows. S tells how many microstates compared to a given

Symbol: S

Most likely state=Thermal equilibrium

Irreversible process

$$\Delta S_{Tot}>0$$

Reversible process

$$\Delta S_{Tot}=0$$

Microstate

State of the gas as described at the microscopic level (displacement, velocity) for all particles.

Macrostate

State in terms of Macroscopic quantities, such as P, V, T, n