Chapter 13
The Giancoli onLine Tutor

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Temperature and Kinetic Theory

This Chapter is more straight forward than most of the others we have covered. The Topics Covered are:

I. Temperature Scales

Temperature determines Thermal Equilibrium. Two objects are in thermal equilibrium if when placed in contact with each other no changes occur. If two objects have the same temperature they will be in thermal equilibrium if placed in contact. Thermometers measure temperature. Thermometers are based on two reference points and an 'arbitrarily' chosen number of divisions between these points.

Celsius Scale: based on water freezing at 0 ^oC and boiling at 100 ^oC.

Fahrenheit Scale: based on the freezing point of super-saturated salt water at 0 ^oF and the average human body temperature at 96 ^oF. The upperpoint was later adjusted to 98.6 so that there would be 180 divisions between freezing and boiling. On this scale the freezing point of water is 32 ^oF and the boiling point is 212 ^oF. Why 96 and 180?

Interactive Examples
To convert between the Fahrenheit Scale and the Celsius Scale use:
^oF= 9^oC/5+32
^oC= 5(^oF-32)/9
Try them out
(need a calculator-> get one here)
If the temperature is 68 ^oF what is the Celsius temperature to the nearest degree? Enter your answer below and click outside.

If the temperature is 80 ^oC what is the Fahrenheit temperature to the nearest degree?

At what temperature are the Fahrenheit and Celsius temperature equal?

Kelvin Scale: based on absolute zero as 0 K and freezing at 273 K. The Kelvin division is the same size as the Celsius division so boiling is at 373 K.
^oC= K -273
(The Kelvin Scale is an absolute measure of a physical quantity and as a result the "^o" sign is never used with it.)

What is the temperature to the nearest kelvin of the freezing point of saturted salt water?

It is always safe and wise to convert all temperatures to Kelvin before working a problem.
^oC or ^oF specify a temperature, C^o or F^o specify a temperature difference.
A temperature difference in Celsius and Kelvin are the same.

II. Ideal Gas Equation

The ideal gas equation

PV=nRT

P is the pressure on the gas, it must be measured in N/m².
V is the volume of the gas, it must be measured in m³.
n is the number of mols.
R is the Universal Gas Constant. We use 8.315 Nm/molK for R in this course.
T is the temperature of the gas , it must be measured in K.

is useful in describing gases under normal conditions.

Using the Ideal Gas Equation

If the pressure is given as gauge pressure you must add atmospheric pressure (for the purposes of this tutorial 1 atm =1.01x10⁵ N/m²). In most physics problems n is constant and a convenient way to work most problems is to use
P₁V₁/T₁= P₂V₂/T₂
Even if n is not constant, you can still use:
P₁V₁/(T₁n₁)= P₂V₂/(T₂n₂)
Normally for the ideal gas equation you need to change the units as described above, however, used in this manner any units for pressure and volume can be used as long as the same units on both sides of the equation. Temperature always must be converted to kelvin.

Interactive Examples
The pressure of a gas with a volume of 5 L , a pressure of 10 atm and a temperature of 250 K is doubled and at the same time the temperature is increased to 400 K. What is its new volume in liters? Enter your answer below and click outside.

What would be the change in volume of the gas in the above question in m³?

If the pressure of a gas at T= 47 ^oC is doubled without a change in volume, what is its new temperature to the nearest ^oC?

III. Kinetic Theory

Kinetic theory is based on the assumptions that a gas is composed of negligibly sized molecules in random motion. This model establishes that temperature is a measure of the average kinetic energy per molecule (1 kelvin= 2.1x10^-23 Joules) This is usually expressed in terms of Boltzmannās constant, k, as
Average kinetic energy per molecule= <mv²/2>= 3kT/2.
The value of k is 1.38x10^-23 J/K, k is related to the universal gas constant R by
k= R/N_A where N_A is Avogadro's number.
This can be used to rewrite the Ideal gas equation as
PV= NRT
where N is the total number of molecules in the gas. The relationship mv²/2>= 3kT/2 can be used to define a root-mean-square (rms) velocity, v_rms, (The square root of the average of the squared velocity or speed). To do this multiply both sides of
<mv²/2>= 3kT/2
by 2/m and you obtain
<v²>= 3kT/m
. Take the square root of both sides and you have
v_rms=√<v²>= √(3kT/m)
The rms velocity is always a good measure of the gas particles motion, however, it is always greater than the average velocity. Consider, for example, two particles, one with a speed of 2 and the other with the speed of 3. The average speed is
(2+3)/2= 2.5,
the average of the squared speed is
(2²+3²)/2= 6.5
and the square root of that gives
v_rms= 2.55

Interactive Examples
Three molecules have speeds of 300 m/s, 400 m/s and 500 m/s what is their rms velocity to the nearest m/s?

What is the rms velocity to the nearest m/s of a gas molecule (m= 9x10^-27 kg) at 300 K?

The kinetic Energy of a gas at temperature 47 ^oC is doubled, what is the new temperature to the nearest ^oC?

III. Thermal Expansion

All substances (with the exception of water between 0 and 4^oC) expand when heated. The change in length of any line on a solid (including the length, width, and depth) is given by
DL= a*L*DT
, where L is the original length, alpha, a, is the coefficient of linear expansion, and DT is the change in temperature in either C^o or kelvin. The coefficient of linear expansion depends on the nature of the material, and is normally a given. Liquids are described by a similar relationship to the above
DV= b*V*DT,
where V is the original length, beta, b, is the coefficient of volume expansion, and DT is the change in temperature in either C^o or kelvin. The coefficient of volume expansion depends on the nature of the material, and is normally a given. The volume expansion formula can be applied to solids by replacing b with 3*a.

Interactive Examples
A particular metal bar has a length of 3 meters and a coefficient of linear expansion of 10x10^-6 in metric units. If its temperature is increased by 100 C^o, what is its new length?

A 50 liter container made of this substance is subject to the same tempersture increase by what percentage does its volune increase?