thermodynamics

Specific heat capacity

$$Q=mc\Delta T$$

atm to Pa

1atm=1.01×10^5 Pa

Ideal gas law

pV=nRT, pV=NkT

(On formula sheet)

What is an isochoric process

Volume is constant

What is an isobaric process

Constant pressure

What is an isothermal process

Constant temperature

Work done on a system

$$\int_{V_1}^{V_2}pdV$$

work is negative if the gas is compressed

work is positive is the gas expands

Change in internal energy

dU=dQ-dW

Where dQ is the change in heat

is path independent, constant in an isolated system

(On formula sheet)

work done in an isothermal process

dW=dQ so dU=0

$$W=nRT\ln\frac{V_2}{V_1}$$

Work done in an isochoric process

no work is done

$$dU=nC_{v}dT_{}$$

where C is molar heat capacity at a constant volume (C=Molar mass x c)

Work done in isobaric process

$$dW=p(V_2-V_1)$$

$$dU=ndT(C_{p}-R)$$

relationship between molar heat capacity at a constant pressure and at constant volume

$$C_{p}=C_{V}+R$$

Adiabatic constant

$$\gamma=\frac{C_{p}}{C_{V}}$$

Adiabatic process

no heat flow into or out of system (dQ=0)

dU=-W

$$TV^{\gamma-1}=const$$

$pV^{\gamma}=const$ (On formula sheet)

$$W=\frac{p_1V_1-p_2V_2}{\gamma-1}$$

Internal energy in a cycle

No net change in internal energy

Heat in a heat engine

$$Q=W=\left\vert Q_{H}\right\vert-\left\vert Q_{C}\right\vert$$

thermal efficiency of a heat engine

$$\eta=1-\left\vert\frac{Q_{C}}{Q_{H}}\right\vert$$

the Carnot cycle

Contains two isothermal and two Adiabatic processes

Efficiency for the carnot cycle

$$\eta=1-\frac{T_{C}}{T_{H}}$$

Energy in a refrigerator

$$\left\vert Q_{H}\right\vert=\left\vert Q_{C}\right\vert+\left\vert W\right\vert$$

Coefficient of performance in refrigerators

$$K_{r}=\frac{\left\vert Q_{C}\right\vert}{\left\vert Q_{H}-Q_{C}\right\vert}=\frac{T_{C}}{T_{H}-T_{C}}$$

coefficient of performance in a heat pump

$$K_{r}=\frac{\left\vert Q_{H}\right\vert}{\left\vert Q_{H}-Q_{C}\right\vert}=\frac{T_{H}}{T_{H}-T_{C}}$$

Zeroth law

When two bodies are each separately in thermodynamic equilibrium with a third body they will be in thermodynamic equilibrium which each other

First law

For a closed system, the change in internal energy is the difference between the heat accumulated by the system and the work done by it

Second law

It is impossible to construct a cyclic engine that only converts thermal energy from a reservoir at a given temperature into mechanical heat. A portion of the thermal energy must be transferred to a second reservoir at a lower temperature.

what is entropy

Increasing entropy is an increase in disorder

Adiabatic and reversible processes have no change in entropy

Increased by:

Add heat

Increase volume

Change in entropy

$\Delta S=S_{B}-S_{A}=\int_{A}^{B}\!\frac{1}{T}\,dQ$ $(JK^{-1})$

CONVERT dQ TO dT USING SHC FORMULA

Positive value=decrease in order

Entropy for microstates

$$S=k_{B}\ln w$$

w is number of microstates

change in s: kB ln(w2-w1)

Third law

The entropy of a perfect crystal at absolute zero is equal to zero

(Absolute zero is impossible to reach)

Kinetic theory of gases

• molecules move randomly

• total number of molecules is large

• volume occupied by molecules is small compared to that of the gas

• no forces between molecules unless they collide

• collisions are elastic and occur instantaneously

kinetic pressure

$$p=\frac13\rho v_{rms}^{_2}$$

(On formula sheet)

root mean squared speed

$$v_{rms}^2=\frac{3P}{\rho}$$

P is pressure

Average kinetic energy

$$KE=\frac32kT$$

each degree of freedom the molecule has contributes 1/2kT

Eg translational x,y,z =3/2kT

translational and two rotational=5/2kT

Average velocity

$$\overline{v}=\sqrt{\frac{8kT}{\pi m}}$$

rms velocity

$\overline{v_{rms}}=\sqrt{\frac{3kT}{m}}$ (On formula sheet)

most probable velocity

$$v_{mp}=\sqrt{\frac{2kT}{m}}$$

How are the velocities related

$$v_{mp}<\overline{v}<v_{rms}$$

mp<avg<rms

van der waals equation

ideal gas law corrected

$$(p+a(\frac{n}{V})^2)(\frac{V}{n}-b)=RT$$

a- corrects pressure/ intermolecular forces

b- corrects volume/ volume of molecules