Gases

Question 1

What is Boyle’s law?

Answer 1

Boyle observed that the product of pressure and volume is constant as long as temperature and number of moles are held constant

PV = C; C is constant at a given temperature but increases as temperature increases

Value of C for 1 mol at 0 celcius = 22.4 L x atm.

SATISFIED AT VERY LOW PRESSURES

Question 2

What is Torricelli’s Barometer?

Answer 2

Filled an open-ended glass tube with mercury and submerged it in an open dish of mercury

Mercury fell, but not all the way; occurred when the pressure of the air on the dish exactly balanced the pressure of the mercury on the surface

Stopped when top was about 76 cm above the surface of the mercury; 760 mmHg

Question 3

Name an equation relating pressure to density, g, and height

Answer 3

P = pgh

Derivation:

F = ma; P= F/A

So, F = ma/A

Volume of mercury in a barometer is V = Ah, so

P = mg/(V/h) = mgh/V = pgh

Plug in the density of mercury, g, and the height of the column to calculate the atmospheric pressure using Torricelli’s barometer

Question 4

Describe Boyle’s J-tube and what it means

Answer 4

Had a J-shaped tube, with gas trapped in one end and a column of mercury extending through the open end

When the level of mercury was the same in both sides, the pressure of the gas exactly equaled the atmospheric pressure

When the level of mercury was higher on the open end, that meant that the pressure in the closed end was greater than the atmospheric pressure by pgh

P = 1 atm + (h/760)

Question 5

What is Charles’ Law?

Answer 5

Charles observed that all gases expand by the same relative amount between two temperatures

Different ways of stating it:

V1/V2 = T1/T2, because volume is directly proportional to temperature

or

V = V0(1 + (t/273.15))

Question 6

What is the ideal gas law and under what conditions is it most valid?

Answer 6

PV = nRT

It is most valid under conditions of low pressure and high temperature, when intermolecular interactions between the gas molecules are weakest

Question 7

What is the definition of the partial pressure of a gas?

Answer 7

The pressure that the gas would exert if it alone were present in the container

Question 8

What is Dalton’s Law?

Answer 8

The total pressure is the sum of the partial pressures of the individual gases

Therefore,

P_A= n_ART/V, and

P_{A =}(n_a/n_tot)P_tot

Question 9

Name the four assumptions underlying the kinetic theory of gases

Answer 9

1. A pure gas consists of a large number of molecules that are separated by distances that are great compared with their size
2. The gas molecules are constantly moving in random directions with a distribution of speeds
3. The molecules exert no forces on each other between collisions, so between collisions they move in straight lines with constant velocities
4. The collisions of molecules with the walls of the container are elastic

Question 10

What is the average kinetic energy per mole of gas?

Answer 10

Mean E = (3/2)RT

If we divide by avogadro’s number to get the mean energy per molecule, we find that:

Mean E = (3/2)K_BT

Question 11

What is the force exerted by N gas molecules on a container of length L and cross-sectional area A?

Answer 11

Since ΔP = FΔT, the mean force exterted on the container = ΔP/ΔT

Change in momentum, ΔP to the box per collision = 2mV_x

Time between collisions, ΔT = 2L/V_x

Therefore, ΔP/ΔT = mV_x²/L

Therefore, the total force = NmV_x²/L

Then, because P = F/A, P = NmV_x²/LA, where LA = Volume

Question 12

How is the expression for the mean energy of a mole of gas derived?

Answer 12

We have that PV = NmV_x²; because there is no preference for any direction, V_x= 1/3 U²,

so PV = 1/3 NmU²

nRT = 1/3 NmU², which becomes RT = 1/3 N_amu²

KE = 1/2 mu², so KE = 3/2RT

Question 13

What is the equation for the mean-square speed of a gas?

Answer 13

Mean value of u² = 3RT/M, where M = molar mass

Derivation: 1/3 N_Amu² = RT, or

u² = 3RT/N_Am

Because N_Am = molar mass

Question 14

What parameters does the mean-square speed of a gas depend on?

Question 15

What is the van der Waals equation of state?

Answer 15

(P + a(n²/V²)(V-nb) = nRT

Question 16

Qualitatively, what are the two modifications that the van der Waals equation accounts for?

Answer 16

1. Molecules cannot occupy the same space, so the volume available to them is NOT V, but rather is V - nb, where n is the number of moles and b is the excluded volume per mole of molecules. This pushes the molecules away from each other, which has the effect of increasing the pressure above its ideal gas value
2. If molecules cluster together, it reduces the effective number of molecules which reduces collisions with the walls, which has the effect of reducing the pressure. Because this depends on pairs of molecules, it’s proportional to the square of molecules per unit volume

Question 17

What are the two characteristic parameters of a potential energy graph?

Answer 17

1. The location of the minimim
2. The depth of the minimum

Question 18

What is the compressibility factor?

Answer 18

z = PV/nRT

Since PV = nRT according to the ideal gas law, this theoretically should equal 1; if it doesn’t, must account for the parameters in the van der Waals equation

Question 19

When gases deviate from ideal gas behavior, what factor determines whether attractive or repulsive forces dominate?

Answer 19

At very high pressures, repulsive forces essentially always dominate and the compressibility factor >1

At lower pressures,TEMPERATURE dominates

When the temperature is low enough that K_bT (kinetic energy) < e, the depth of the potential energy well, then the pair of molecules will remain close together long enough for the pressure to be reduced; when the temperature is higher, the molecules experience only the repulsive part of the potential energy curve