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PHYS 170

Lecture 16 - Temperature and Kinetic Theory

Temperature and Absolute Zero

The SI unit of temperature is K (Kelvin). We can convert a temperature from Celsius to Kelvin by adding 273 (not exact, see below):

TK=TC+273
It follows that 0C=273K and 273C=0K. The temperature of 0K is called absolute zero, and is believed to be the lowest temperature possible.

Absolute temperature could be identified by noticing how (at constant pressure) the volume of a gas decreases linearly with decreasing temperature. Extrapolating the linear relation shows that at 273.15C the volume would drop to zero. Clearly the volume of a gas cannot be negative, which implies temperature cannot fall below 273.15C.

Using different types of gases, different amount, and running the experiment at different pressure all lead to the same temperature of zero volume, so clearly there is something special about 273.15C, which we now call absolute zero. The Kelvin scale is defined so that absolute zero is exactly 0K.

One can see in the Kelvin scale, VT.

One can also plot the variation of gas pressure with temperature in a container at constant volume. Once again, the lines all crosses the T-axis at absolute zero. Therefore we find that PT in the Kelvin scale.

Ideal Gas Law

An ideal gas is a gas of point particles which has no other interactions besides collision among molecules. There is no intermolecular force. An ideal gas obeys the ideal gas law:

PV=nRT
where R=8.31JK1mol1 is the ideal gas constant, n is the number of moles.

SI Unit
P Pa
V m3
T K
n mol

Real gas molecules have non-zero volume (i.e. not point particles), and there is generally intermolecular force. However, for most gases at room temperature the size of the gas molecules are small compared to the whole volume of the gas, and the molecules are usually separated far enough for the intermolecular forces to be negligible (except when the gas is close to phase transition) so the ideal gas law usually works very well.

Review - Number of Moles n Vs Number of molecules N

n and N are related to each other via N=nNA, where NA=6×1023 is the Avogadro's number. NA is the number of molecules in 1 mole of gas.

A good analogy to Avogadro's number is the concept of dozen used in a bakery. You can think of NbakerA=12 for a baker. So if you buy 3 dozens of donuts, it means you are buying N=nNA=(3)(12)=36 donuts.

Chemistry
Number of moles, n Number of molecules, N=nNA
1 6×1023
2 12×1023
3 18×1023
4 24×1023
Bakery
Number of dozens, n Number of donuts, N=nNA
1 12
2 24
3 36
4 48

Review - Units of Volume and Pressure

The SI unit for volume is m3. Unit conversion is a very basic skill for a science major. Make sure you know how to convert units to and from cm3 and L (liter) correctly or you will be penalized heavily. Here is a quick reminder: 1m=100cm1m2=(100cm)2=104cm21m3=(100cm)3=106cm3 Using the above, we get: 1cm3=106m31L=1000cm3=1000×106m3=103m3

The SI unit of pressure is Pa (Pascals), which is equivaldent to N/m2.

A common non-SI unit for pressure is atm, or "atmospheric pressure". To convert to SI unit, use 1atm105Pa.

The term STP stands for standard temperature and pressure, meaning T=0C and P=1atm.



An alternative way of writing the ideal gas law is PV=NkT, where k=1.38×1023J/K is the Boltzmann constant. Compare with the version presented above PV=nRT, we could see: NkT=nRTNk=nRk=nRN=nRnNA=RNA=1.38×1023J/K

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A useful equation derived from the ideal gas law is:

P1V1T1=P2V2T2
The implicit assumption here is n1=n2. In other words, the container of the box is not leaking so the amount of gas stays constant. For the proof, start with PV=nRTnR=PVT. The assumption n1=n2 then gives the equation above.

Try It Yourself (click to show)

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Notations

Name Symbol Unit Meaning
Pressure P Pa force per unit area
Volume V m3 size of a gas
Temperature T K how hot an object is
Avogadro's number NA=6×1023 none number of molecules in one mole
Number of moles n mol amount of molecules as multiples of NA
Number of molecules N none N=nNA