Ch 13
instead of using moles, can use
#of atoms/molecules in a gas N= n * N_A N= # molecules n= moles N_A= avagardos (number of moles)
delta V=
(Beta)(V_0)(delta T) -Beta is coefficient of volume (-valid when expansion is small?)
alpha has units of
(degrees C)^-1
objects at different temperatures emit different
(infared) lights
1 u=
1.6605 x 10^-27 kg
KE=
1/2m(avgv^2)=3/2kT
alpha (l_0)(delta T)=
1/E*(F/A)(l_0)
thermal stress: delta l=
1/E*(F/A)(l_0)
volume of 1 mol of any ideal gas is
22.4 L conversion factor
what would be the water's freezing point in Kelvin?
273.15 K
KE= 1/2(m)(avg v^2)=
3/2(k)(T)
What would be the human body temperature in Kelvin?
310.15 K
normal body temperature is 98.6 F. what is this celsius?
37 C
Two gas containers A and B have the same volume. A is filled with 1 mol of Oxygen at 27 degrees celsius and B is filled with 1 mol of CO2 at 127 Celsius. The pressure ratio (PA:PB) is
3:4
water turns out to be special: it has max density at
4 centigrades -increases towards 4 (inc in density and dec in volume then opposite after 4)
1 mol of substance=
6.02x10^23 objects (Avagadro's number)
Universal Gas Constant R=
8.314J/(K*mol)
two identical cylinders at the same temperature contain the same gas. If A contains three times as much gas as B, which cylinder has the higher pressure?
A -more n
Two gas containers A and B have the same volume and temperature. A is filled with 100 g of oxygen and B is filled with 100 g of CO2. compare their pressure
A has a higher pressure
Charles's Law
A principle that describes the relationship between the temperature and volume of a gas at constant pressure
two identical cylinders at the same pressure contain the same gas. If A contains 3 times as much as gas as B, which cylinder has the higher temperature?
B -less moles
gas tank in the sun: the 70 liter (L) steel gas tank of a car is filled to the top with gasoline at 20 degrees C. The car sits in the sun and the tank reaches a temperature of 40 C (104 F). How much gasoline do you expect to overflow from the tank?
Both the gasoline and the tank expand as the temperature increases, and we assume they do so linearly. The volume of overflowing gasoline equals the volume increase of the gasoline minus the increase in volume of the tank. delta V= Beta(V_0)(delta t)= 1.3 L the tank expands also: Beta= 35x10^-6 delta V= 0.049 L the tank has little effect. more than a liter of gas could spill out.
a grandfather clock uses a brass pendulum to keep perfect time at room temperature. if the air conditioning breaks down on a very hot summer day, how will the grandfather clock be affected?
Clock will run slower than usual. The pendulum will expand, so its length will increase. The period of a pendulum depends on the length as shown below, so the period will also increase. Thus, the clock will run slow.
Metals such as brass expand when heated. The thin brass plate in the movie has a circular hole in its center. When the plate is heated, what will happen to the hole?
Gets larger. Imagine drawing a circle on the plate. This circle will expand outward along with the rest of the plate. Now replace the circle with the hole, and you can see that the hole will expand outward as well. Note that the material does NOT "expand inward" to fill the hole!!
P1V1/T1=
P2V2/T2
Two identical cylinders at the same temperature contain the same gas. If B has twice the volume and half the number of moles as A, how does the pressure in B compare with the pressure in A?
PB= 1/4(PA)
ideal gas law with n replaced
PV=(N/N_A)(RT)
ideal gas law with Boltzmann constant that results
PV=NkT -N (number of molecules) k= R/N_a N_a (avagadro's)
Ideal Gas Law Equation
PV=nRT (because PV is proportional to mT)
Bi-metallic strip
Two metals fastened together, but they expand differently -shape change (curve)
mass of air in a room. estimate the mass of air in a room whose dimensions are 5 m x 3 m x 2.5 m high, at STP. air is about 20% oxygen (O2) and 80% nitrogen (N2).
V= 22.4L-> 22.4 x 10^-3 m^3 (1 mol at STP) n= 5*3*2.5/ 22.4x10^-3 = 1700 mol air is about 20% oxygen (O2) and 80% nitrogen (N2). The molecular masses are 2 * 16 u= 32 u and 2 x 14 u= 28 u. An avg of about 29 u. Thus, 1 mol of air has a mass of about 29 g = 0.029 kg. m= (1700)(0.029)= 50 kg
helium balloon: a helium party balloon, assumed to be a perfect sphere, has r= 18 cm. at room temperature (20 C), its internal pressure is 1.05 atm. Find the number of moles of helium in the balloon and the mass of helium needed to inflate the balloon to these values. molecular mass of helium = 4 g/mol
V= 4/3pir^3= 0.0244 m^3 P= 1.064 x 10^5 N/m^2 T= 293 K R= 8.314 J/mol*K n=PV/RT= 1.066 mol mass= n * molecular mass mass= 4.26 g (4.26x10^-3 kg)
volume of one mole at STP. Determine the volume of 1 mol of any gas assuming it behaves like an ideal gas at STP
V=nRT/P= 22.4 x 10^-3 m^3 Since 1L is 1000 cm^3= 1 x 10^-3 m^3, 1 mol of any ideal gas has V= 22.4 at STP
distance between atoms: the density of copper is 8.9 x 10^3 kg/m^3 and each copper atom has a mass of 63 u. Estimate the average distance between the centers of neighboring copper atoms.
We consider a cube of copper 1 m on a side. From the given density (rho) we can calculate the mass of a cube of V= 1 m^3. We divide this mass by the mass of one atom (63 u) to obtain the number of atoms in 1 m^3. the mass of 1 copper atom is 63 u x 1.66 * 10^-27 kg= 1.05 * 10^-25. this means that in a cube of copper 1 m on a side (volume= 1 m^3), there are 8.9x10^3 kg/(1.05x10^-25) = 8.5 x 10^28 atoms the volume of a cube of side l is V=l^3, so on one edge of the 1 m long cube there are (8.5 * 10^28)^[1/3] atoms= 4.4 * 10^9. Hence the distance between neighboring atoms is 1m/(4.4 x 10^9) = 2.3x10^-10 m
how many molecules in one breath? estimate how many molecules you breathe in with a 1 L breath of air. At stp
We determine what fraction of a mole 1 L is using the example that 1 mole has volume of 22.4 L at STP, and multiply that by Na to get the number of molecules in this number of moles. 1.0 L/(22.4 L/mol)- 0.045 mol then 1 L of air contains (0.045 mol)(6.02x10^23 molecules/1 mole)= 3x10^22 molecules
T (degrees F)=
[(9/5)T(degrees C)]+32
T(degrees C)=
[5/9][T(degrees F)-32]
V is proportional to
[T(C) + 273(C)] [-273(C)... absolute 0.... V=0]
all systems at the same temperature are
all in equilibrium with each other
Beta is about 3x
alpha (for solids)
the stress is: F/A=
alpha*E*deltaT
linear thermal expansion delta l=
alpha*l_0(deltaT) -Alpha is coefficient of expansion; table
bridge expansion: the steel bed of a suspension bridge is 200 m long at 20 C. If the extremes of temperature to which it may be exposed are -30 C to 40 C, how much will it contract and expand?
alpha= 12x 10^-6 for steel at 40: delta l= 4.8 x 10^-2 -> increase of 4.8 cm at -30: -12x10^-2 m -> decrease of 12 cm total range of expansion 12+4.8= 17cm
an increase of temperature by 1 centigrade is equivalent to
an increase of temperature by 1.8 in Farenheit
different phases: different ways of
arranging and organizing atoms and molecules
matter is made of a large number of tiny particles
atoms, molecules
molecular kinetic energy: what is the average translational kinetic energy of molecules in an ideal gas at 37 C?
avg KE= 3/2kT= 3/2(1.38 x 10^-23 J/K)(310K)= 6.42 x 10^-21 J (note: a mole of molecules would have a total translational kinetic energy equal to (6.42 x10^-21) (6.02 x 10^23)= 3860 J)
Gases are very different from solids and liquids in one aspect: they are highly
compressible and their volume depends strongly on pressure in addition to temperature
derivation of gas pressure
consider on molecules bounded back from wall: delta (mv)= mv_x - (-mv_x)= 2mv_x average time between successive collisions on the same wall of the same single molecule: delta t= 2l/v_x average force between wall and molecule: F= delta(mv)/ delta t = 2mv_x/(2l/V_x) =m*v_x^2/l [due to one molecule] total force on the wall from all molecules F=m/l*(v_x1^2 + v_x2^2 .... etc) introducing an average for N molecules: avg v_x^2= avg v^2= 3v^3_x we obtain the for force on a wall to be: F= m/l*N*(avgv^2) P= F/A= 1Nmavgv^2/A*l= 1/3Nm*avgv^2/V *pressure in ideal gas^* *PV= 1/3(Nm*avgv^2/V)* *PV= 2/3N(1/2m*avgv^2)* PV= NkT
standard kind of thermometer must be chosen so that all temperatures can be precisely defined. the chosen standard for this purpose is the
constant-volume gas thermometer
the strategy to fix one variable and check how the other two are
correlated with each other when we change them
exploding coke can to freezer: room temp down to freezing volume
decreases then INCREASES to cause explosion
the average translational kinetic energy of molecules in random motion in an ideal gas is
directly proportional to the absolute temperature of the gas
the relationship between volume, pressure, temperature, and mass of a gas is called
equation of state
by changing temperature you can change
forms and properties of matter
metals expand a lot more than
glass
most materials expand when they are
heated
the volume is linearly proportional to the temperature when the pressure is
held constant
the zeroth law of thermodynamics
if two systems are in thermal equilibrium with a third system, then they are in thermal equilibrium with each other
do holes expand or contract? if you heat a thin, circular ring in the oven, does the ring's hole get larger or smaller?
if you guessed the metal expands into the hole (hole gets smaller), it is not so. Imagine the ring is solid. When the metal expands, the material inside the circle will expand along with the rest of the metal; so the dashed circle expands. cutting the metal where the circle is shows that the hole increases in diameter
Pressure is proportional to T when T is
in absolute temperature scale (K) and volume is held constant
V is proportional to T when T
is in Kelvins
Boltzmann constant
k=1.3806505x10(-23) J K(-1) k=R/N_A
l=
l_0(1+alpha(deltaT)) -lower temp= shrinks
assumptions of kinetic theory:
large number of molecules, moving in random directions with a variety of speeds molecules are far apart, on average molecules obey laws of classical mechanics and interact only when colliding collisions are perfectly elastic
speeds of air molecules: what is the rms speed of air molecules (O2 and N2) at room temp of 20 C?
m(O2)= 32(1.66 x 10^-27 kg)= 5.3 x 10^-26 kg m(N2)= 28(1.66 x 10^-27 kg)= 4.6 x 10^-26 kg Thus, for oxygen vrms= 480 m/s and for nitrogen= 510 m/s
kinetic theory provides a microscopic understanding of the
macroscopic ideal gas law
n mole=
mass (g)/ molecular mass (g/mol)
with a large number of molecules, what is the speed distributions?
maxwell distribution Cv^2exp(-1/2mv^2/kT)
water doesn't follow the usual uniform pattern of expansion: if water at 0 C is heated, it actually decreases in volume until it reaches 4 C. Above 4 C, water behaves
normally and expands in volume as the temperature is increased *greatest density. at 4 C* *important for aquatic life* *can expand as it freezes to ice* -low volume= higher density
which weighs more a mole of N2 or a mole of Oxygen (O2)?
oxygen
heating or changing temp can induce the change in the
phases of matter
which has more molecules- a mole of nitrogen (NN2) or a mole of oxygen gas?
same
equal temperature in gases means
same KE -not necessarily same speed
thermal equilibrium-> two systems reach the
same temperature
which one (1 mol O2 or 1 mol N2) takes up more space under the same pressure and temperature?
same volume
Coefficient of volume is very different between
solids, liquids, and gases
v_rms=
square root of avg v or sqrt[3kT/m]
kinetic energy near absolute zero
the equation leads us to believe KE = 0 when T=0 but it is not the case... KE becomes very small nonzero minimum value
ring on a rod. an iron ring is fit snugly on a cylindrical iron rod. At 20 C, the diameter of the rod is 6.445 cm and the inside diameter of the ring is 6.420 cm. to slip over the rod, the ring must be slightly larger than the rod by about .008 cm. to what temperature must the ring be brought if its hole is to be large enough so it will slip over the rod?
the hole must be increased from 6.420 cm to 6.453 cm. the ring must be heated since the hole diameter will increase linearly with temperature delta t= delta l/alpha*l_0 delta t= 430 C so temperature= 450 C
why you should not put a closed glass jar into a campfire. what could happen if you did this?
the inside of the jar has air in it. as the fire heats the air inside, its temperature rises. the volume of the glass jar changes only slightly due to the heating. According to Gay-Lussac's law the pressure P of the air inside the jar can increase enough to cause the jar to explode, throwing glass pieces outward.
when the lid of a glass jar is tight, holding the lid under hot water for a short time will often make it easier to open. why?
the lid may be struck by hot water more directly than the glass and expand sooner, and if not that, metals generally expand more than glass for the same temp change.
hydrogen atom mass: use Avagadro's number to determine the mass of a hydrogen atom
the mass of one atom equals the mass of 1 mol/ number of atoms in 1 mol (Na) One mole of hydrogen atoms has a mass= 1.008 u and contains 6.02 x 10^23 atoms m of one atom= 1.67 x10^-24 kg (?)
object A in an Antarctic science station is measured to be at 0 C. Object B in a Hawaiian observatory is measured to be at 32 degrees Fahrenheit. Which of the following is true?
the objects are in thermal equilibrium
the volume of a given amount of gas is inversely proportional to
the pressure as long as the temperature is constant (pressure inversely proportional to V)
Avagadro's number (Na) represents the fact that the numbers of molecules in 1 mol is
the same for all gases
constant-volume gas thermometer. this thermometer consists of a bulb filled with a low-pressure gas connected by a thin tube to a mercury manometer. the volume of the gas is kept constant by raising or lowering the right-hand tube of the manometer so that the mercury in the left-hand tube coincides with the reference mark. an increase in temp causes a proportional increase in pressure in the bulb. the tube must be lifted higher to keep the gas volume constant. the height of the mercury in the right-hand column is then a measure of the temperature. this thermometer gives the same results for all gases in the limit of reducing the gas pressure in the bulb toward zero. the resulting scale serves as a basis for
the standard temperature scale
zeroth law allows to define temperature as a unique meaningful property of a
thermal system
thermal expansion can be used to make
thermometers -when temp changes, mixture expands and column of liquid goes up
linear expansions has no meaning for liquids and gases because
they don't have fixed shapes
atomic and molecular masses are often measured in ----. This unit is defined so that the carbon- 12 atom has a mass of exactly 12 u. what does this show?
unified atomic mass units (u); huge differences in the micro and macro world 1 large rain drop-> 1 gram-> 6*10^23 u-> 3 * 10^22 molecules
root-mean-square (rms) speed of a gas molecule
v_rms= sqrt[avg v^2]= sqrt[3kT/m]
check tires cold. An automobile tire is filled to a gauge pressure of 210 kPA (30 psi) at 10 C. After a drive of 100 km, the temperature within the tire rises to 40 C. what is the pressure within the tire now?
we do not know the number of moles of gas or the volume of the tire, but we assume they are constant. we use the ratio form of the ideal gas law. Since V1= V2, then P1/T1= P2/T2 P1= 210 kPa + 101 kPA= 311 kPa (absolute pressure 1) P2= P1(T2/T1)= 344 kPa Resulting in a gauge pressure of 243 kPA... 16% increase
Maxwell distribution and temperature
with higher temp, distribution is shifted toward higher speed
absolute temperature scale: T(K)=
T(C)+273.15
A helium filled balloon escapes a child's hand at sea level at 20 C. When it reaches an altitude of 3600 m, the pressure is 0.68 atm and the temperature is 5 C. what is the change in volume?
T1= 293 K P1= 1 atm -pressure is in balance with pressure outside V1=? (looking at change) T2= 278 K P2= 0.68 atm V2= ? V2/V1= P1*T2/P2*T1 V2/V1= (P1/P2)*(T2/T1) V2/V1= 1.4 1.4:1 -increases by 40%
before T1= 302 K P1= 205 atm after T2=? P2= 192 atm
T2= T1 x P2/P1 -volume cancels since it remains the same -can keep P in atm since atm will be canceled out T2= 283 K (= 10*C)
STP: T= P=
T= 273 K P= 1 atm or 1.013 x 10^5 N/m^2
useful facts: standard temp and pressure (STP)
T= 273 K (0*C) P= 1 atm= 1.013 x10^5 N/m^2= 101.3 kPa
Maxwell Distribution
The distribution of kinetic energy in a gas
