Gas Laws and the Kinetic Theory of Gases
Gas Laws
When an inflated balloon is left out in the sun, it increases in volume without extra air being pumped in. If the balloon is taken out of the sun and placed in a cold room, the volume reduces without any air escaping. Out in the sun, the temperature of air in the balloon rises and the gas expands leading to an increase in volume. In the cold room, the temperature of the air in the balloon reduces and consequently the air contracts hence volume reduces. This implies that when temperature of air (gas) increases, the volume of the gas increases and vice versa. For this to happen, the pressure and mass of the gas must be kept constant. Hence;
(i)
(ii)
(iii)
Equation (i-iii) represents Charles law which states that pressure constant, the volume of a given mass (moles) of a gas is directly proportional to the temperature of the gas.
Now, suppose that the balloon is loosely inflated. If the volume of the balloon is gradually decreased without letting out any air, the balloon becomes firmer and eventually bursts. The reason for these observations is that initially, the pressure of air inside the balloon is equal to the atmospheric pressure hence the balloon remains inflated. However, as the volume of the balloon is reduced, the trapped air particles are forced closer to each other (air is compressible). This leads to increased collisions of particles within the balloon leading to an increase in pressure. At some point, the pressure inside the balloon increases much higher than the atmospheric pressure causing the balloon to burst. Thus, when the volume of a fixed mass of gas at a constant temperature reduces, pressure increases and vice-versa, i.e.,
(iv)
(v)
(vi)
Equations (iv-vi) represent Boyles law and only hold if the temperature and mass (number of moles) of the gas are kept constant.
Suppose now that a rigid and empty metal can is sealed in such a way that the air inside remains trapped. When the can is gradually heated, it eventually explodes. The reason for this is that as the temperature of the air in the can increases, the kinetic energy of the air molecules increases leading to increased collisions and therefore an increase in pressure. When the air pressure exceeds the atmospheric pressure, the can eventually explodes. Thus, an increase in temperature of a fixed mass of gas, volume constant, leads to an increase in the gas pressure and vice versa. Hence;
(vii)
(viii)
(ix)
Equations (vii-ix) represents pressure law and only holds when volume and mass (number of moles) of the gas are constant
The three laws, Charles’s, Boyle’s and pressure laws, are collectively referred to as gas laws. Real gases obey the gas laws only at very low pressure. A hypothetical gas that obeys the gas laws perfectly at all time is referred to as an ideal gas.
For a gas system containing n moles of an ideal gas, equations (i) (iv) and (vii) can be combined into a single equation as;
(x)
(xi)
The constant is equal to the product of the number of moles of the gas (n) and a constant of proportionality R referred to as the universal gas constant. Equation (xi) may therefore be written as;
(xii)
Equation (xii) is referred to as the ideal gas equation. If for example the state of a gas system of n moles changes from (p1V1,T1) to (p2V2T2) then by equation (xii), it follows that;
(xiii)
(xiv)
The right-hand side of equation (xiii) equals the right-hand side of equation (xiv) hence;
(xv)
Gas laws verification
Charles' Law
According to Charles law, volume V of a fixed mass of gas is directly proportional to the temperature T, pressure constant.
(i)
For a column of air of length l and cross-sectional area A;
(ii)
If the cross-sectional area is kept constant, Charles law can be expressed as;
(iii)
To prove Charles’ law, air is trapped in a test-tube using a moveable index as shown in Figure 1. The test-tube is dipped in cold water and after the index has stabilized the temperature of the trapped air Ti (which is equal to the temperature water) and the corresponding position of the index Li obtained.
Figure 1
The water is gradually heated while being continuously stirred and the temperature of air in the tube, T and the corresponding position of the index l obtained periodically. A graph of l against T is then plotted. A straight line inclined to the horizontal proves Charles’ law.
Important points to note;
- To keep the pressure constant, the test tube is kept open and the index fixed in such a way that it is free to move up and down without letting trapped air out.
- The length of the air column trapped in the test-tube is used in place of volume since the cross-sectional area is constant.
- Water is continuously stirred to distribute heat evenly to ensure that the temperature of air is accurately measured.
Boyle’s law:
An air column is trapped in a test tube of cross-sectional area A by a mercury thread of length Lm that is free to move about without letting air in or out of the tube. The tube is placed horizontally and the length of the air column Lh measured.
If V1 be the volume of the trapped air, then
(i)
The mercury thread is not resting on the trapped air column hence does not exert pressure (pm) on the air. The atmosphere on the other hand exerts pressure pa in all directions. Considering that the system is at equilibrium, it follows that the pressure p1 the trapped air exerts is equal to the atmospheric pressure.
That is;
(ii)
Multiplying equations (i) and (ii) leads to;
(iii)
The tube is now placed in an upright position and the length Lu of the air column measured (the length of the air column should be observed to decrease, that is, Lu < Lh).
If V2 be the volume of the tapped air, then;
(iv)
The mercury thread this time rests on the trapped air column hence exerts pressure (pm) on it. The atmosphere too exerts pressure pa on the air column.
The resultant pressure pam on the trapped air is therefore equal to;
(v)
It is this increased pressure that leads to a reduction in length of the trapped air column. Since the system is in equilibrium, the pressure the mercury column and the atmosphere exert on the trapped air should be equal to the pressure p2 the gas exerts in the opposite direction.
That is;
(vi)
Multiplying ((i) by (vi) leads to;
(vii)
If the trapped air obeys Boyle’s law, then equations (iii) and (vii) should be equivalent, that is;
(viii)
Suppose now the tube is turned upside down.
The length of the air column, say Ld should be observed to increase, that is, Ld > Lh). If V2 be the volume of the tapped air, then;
(ix)
The mercury thread does not exert pressure on the trapped air. However, the thread exerts pressure pm on the atmosphere. This counters the pressure pa the atmosphere exerts on the trapped gas (pm and pa are in opposite direction).
The resultant pressure pam on the trapped air is therefore equal to;
(x)
The reduced resultant pressure on the trapped air is responsible for the longer column. If p2 be the pressure the trapped air exerts, then at equilibrium this pressure should be equal to the resultant pressure due to the atmosphere and the mercury column.
(x)
Multiplying ((ix) by (x) leads to;
(xi)
If the air obeys Boyle’s law, then equations (iii) and (xi) should be equivalent, that is;
(xi)
Kinetic theory of gases
According to the kinetic theory of gases, gas molecules with temperature above the absolute zero (0 K) are in constant random motion (called Brownian motion), colliding with each other and with the walls of the container they are placed in. These collisions account for gas pressure. If for example the temperature of a fixed mass of gas is increased, the kinetic energy of the particles increases and to keep pressure constant, the particles drift further apart leading to an increase in volume. This accounts for Charles law. If on the other hand the gas is restricted to a rigid container (constant volume), an increase in kinetic energy causes the gas molecules to collide faster leading to an increase in gas pressure in accordance with the pressure law. If the temperature is kept constant and the volume of the gas reduced (gases are compressible), the gas particles move closer to each other hence the collisions and consequently the gas pressure increases which is in line with Boyle’s law. The random movement of molecules of a gas above 0 K is the basis of the kinetic theory of gases. A number of assumptions are however made in relation to the kinetic theory of gases;
- Molecules of a given gas are identical
- Collisions between particles and the container are perfectly elastic and therefore energy and momentum are conserved.
- Molecules do not exert any force on other molecules except during collisions. The influence of gravity on the particles is also ignored.
- The number of particles is high enough for statistics to be meaningfully applied.
- The size of molecules is negligible compared to their separation.
- The laws of Newtonian (classical) mechanics apply (as opposed to quantum mechanics).
EXAMPLES
Example 1 KCSE 2021
(1) Figure 3(a) shows a horizontal tube containing air trapped by a mercury thread of length 5cm. The length of the enclosed air column is 7.5cm. The atmospheric pressure is 76cmHg. The tube is then turned vertically with its mouth facing down as shown in Figure 3(b).
(a) Determine the length l of the air column. (3 marks)
Solution
(b) State the reason why the mercury thread did not fall out in Figure 3(b).
The upward atmospheric pressure is greater than the downward pressure exerted by the trapped air and the mercury thread.
KCSE 2020
(1) State the kinetic theory of gases. (1 mark)
Gas molecules are in constant random motion, colliding with each other and with the walls of the container they are placed in.
(2) Figure 11 shows a setup that can be used to verify Charles’ Law.
(a) Explain how the:
(i) temperature of air in the tube is measured;(2 marks)
By measuring temperature of water since the temperature of water equals the temperature of the air trapped in the test-tube.
(ii) volume of air in the tube is measured. (2 marks)
According to Charles law, volume of a fixed mass of gas is directly proportional to the temperature, pressure constant.
For a column of air,
Where A is the cross-sectional area and l the length of the air column. If the cross-sectional area is kept constant, then Charles law can be expressed as;
Answer: By measuring the length of the air column trapped in the test-tube since the cross-sectional area is constant.
(b) State how the pressure is kept constant during the experiment.
The test tube is kept open and the index is free to move up and down.
(c) State how the measurements in (a) can be used to verify Charles’ law. (3 marks)
The test-tube is dipped in cold water and after the index has stabilized the temperature of air Ti (which is equal to the temperature water) and the corresponding position of the index Li obtained. The water is gradually heated while being continuously stirred and the temperature of air in the tube, Tf and the corresponding position of the index Lf obtained. A graph of (L) against (T) is plotted. A straight line inclined to the horizontal proves Charles’ law
(d) State one precaution that must be taken to ensure that the temperature of air is accurately measured. (1 mark)
Water should be continuously stirred
(2) A fixed mass of gas initially at 20°C is heated at constant pressure until its volume doubles. Determine its final temperature. (4 marks)
Temperature should be in Kelvin hence;
when
and
when
Hence
KCSE 2019
(1) Figure 9 shows a graph of pressure against temperature for a fixed mass of gas at constant volume
From the graph, determine the values of n and c given that P =nT + c where n and c are constants. (4 marks)
A linear graph is represented by the equation; y=bx+c
where m is the gradient of the graph and c is the value of y when x = 0.
From the equation P =nT + c
c is the value of P when T=0 hence
(2) Explain why it is not possible to obtain zero pressure of a gas in real life situation. (2 marks)
According to pressure law,
This means that as pressure reduces, so does the temperature. At very low temperature, gases turn to liquids.
NOTE: A gas can be made to liquify if exposed to extremely high pressure. For example, LPG gas used for cooking is liquid gas under pressure. LPG gas cylinders must therefore be strong enough to withstand the high pressure
(3) A fixed mass of a gas occupies 1.5 x 10-3m3 at a pressure of 760 mmHg and a temperature of 273 K. Determine the volume the gas will occupy at a temperature of 290 K and a pressure of 720 mmHg. (3 marks)
(4) State any three assumptions made in kinetic theory of gases. (3 marks)
- Molecules of a given gas are identical
- Collisions between particles and the container are perfectly elastic and therefore energy and momentum are conserved.
- Molecules do not exert any force on other molecules except during collisions. The influence of gravity on the particles is also ignored.
- The number of particles is high enough for statistics to be meaningfully applied.
- The size of molecules is negligible compared to their separation.
- The laws of Newtonian (classical) mechanics apply (as opposed to quantum mechanics).
KCSE 2018
(1) Figure 6 shows the relationship between volume and pressure for a certain gas.
Name the law that the gas obeys.
Boyles law
(2) State two quantities that must be kept constant in order to verify Boyle’s law. (2 marks)
- Temperature
- Number of moles (mass)
(3) An air bubble at the bottom of a beaker full of water becomes larger as it rises to the surface. State the reason why;
(a) the bubble rises to the surface, (1 marks)
The air in the bubble is less dense than water.
(b) it becomes larger as it rises. (1 marks)
Pressure exerted by a column of water reduces with the reduction in depth (distance below the surface). A bubble at the bottom of the beaker is under the pressure of the water above it. As it rises on account of being less dense, the pressure exerted by water on the bubble reduces. Since the temperature is constant, then by Boyle’s law, the volume subsequently increases. If the pressure on the bubble falls below the pressure of the air inside the bubble, the bubble extends to the limit and eventually bursts.
(4) Figure 11 shows an incomplete experimental set up that was prepared by a student to verily one of the gas laws.
(a) State with a reason which one of the laws may be verified using the set up. (2 marks)
Pressure gauge measures pressure
Thermometer measures temp
Container does not expand.
Hence
Pressure law
(b) State what the student left out in the diagram of the set up. (1 mark)
Source of heat to vary temperature
(5) The volume of a fixed mass of a gas reduced from 500 cm3 to 300 cm3 at constant pressure. The initial temperature was 90K. Determine the final temperature. (3 marks)
Pressure constant - use Charles’s law:
(6) State two assumptions made in explaining the gas laws using the kinetic theory of gases. (2 marks)
Gas molecules with temperature above the absolute zero (0 K) are in constant random motion, colliding with each other and with the walls of the container they are placed in. These collisions account for gas pressure. If for example the temperature of a fixed mass of gas is increased, the kinetic energy of the particles increases and to keep pressure constant, the particles drift further apart leading to an increase in volume. This accounts for Charles law. If on the other hand the gas is restricted to a rigid container (constant volume), an increase in kinetic energy causes the gas molecules to collide faster leading to an increase in gas pressure in accordance with the pressure law. If the temperature is kept constant and the volume of the gas reduced (gases are compressible), the gas particles move closer to each other hence the collisions and consequently the gas pressure increases which is in line with Boyle’s law. The random movement of molecules of a gas above 0 K is the basis of the kinetic theory of gases. A number of assumptions are however made in relation to the kinetic theory of gases;
- Molecules of a given gas are identical
- Collisions between particles and the container are perfectly elastic and therefore energy and momentum are conserved.
- Molecules do not exert any force on other molecules except during collisions. The influence of gravity on the particles is also ignored.
- The number of particles is high enough for statistics to be meaningfully applied.
- The size of molecules is negligible compared to their separation.
- The laws of Newtonian (classical) mechanics apply (as opposed to quantum mechanics).
KCSE 2017
(1) Figure 2 shows a round bottomed flask fitted with a long capillary tube containing a drop of coloured water.
The flask is immersed in ice water for some time.
(a) State the observation made. (2 marks)
Air contracts (volume reduces) as temperature reduces - Charles's law
The ink drops further down the capillary tube -
(b) State one assumption for the experiments carried out to verify the gas laws. (1 mark)
- Air in the flask is an ideal gas (ideal gas is a gas that perfectly obeys the gas laws)
- The flask, the capillary tube and the ink drop do not loose/gain heat and therefore they do not contract/expand.
Practice Questions
KCSE 2016
(1) Explain why it is advisable to use a pressure cooker for cooking at high altitudes. (2 marks)
(2) State the meaning of the term ideal gas. (1 mark)
(3) The pressure acting on a gas in a cylinder was changed steadily while the temperature of the gas was maintained constant. The value of volume V of the gas was measured for various values of pressure. The graph in Figure 11 shows the relation between the pressure P, and the reciprocal of volume, 1/V.
(i) Suggest how the temperature of the gas could be kept constant. (2 marks)
(ii) Given that the relation between the pressure P and the volume, V of the gas is given by PV = K, where K is a constant, use the graph to determine the value of K. (4 marks)
(iii) What physical quantity does K represent? (1 mark)
(iv) State one precaution you would take when performing such an experiment. (1 mark)
(4) A gas occupies a volume of 4000 litres at a temperature of 37°C and normal atmospheric pressure. Determine the new volume of the gas if it is heated at constant pressure to a temperature of 67°C (Normal atmospheric pressure, P = 1.01 X 105 Pa). (4 marks)
KCSE 2015
(1) Figure 11 shows a graph of pressure (p) against volume (v) for a fixed mass of a gas at constant temperature.
Plot a corresponding sketch of p against 1/V
(2) Explain the pressure law using the kinetic theory of gases. (3 marks)
(3) 20 cm3 of a gas exerts a pressure of 760 mmHg at 25°C. Determine the temperature of the gas when the pressure increases to 900 mmHg and the volume reduces to 15 cm3. (4 marks)
KCSE 2014
(1) A long horizontal capillary tube of uniform bore sealed at one end contains dry air trapped by a drop of mercury. The length of the air column is 142 mm at l7°C. Determine the length of the air column at 25°C. (3 marks)
(2) The pressure of the air inside a car tyre increases if the car stands out in the sun for some time on a hot day. Explain the pressure increase in terms of the kinetic theory of gases. (3 marks)
KCSE 2013
(1) Figure 11 shows an insulated cylinder fitted with a pressure gauge, a heating coil and a frictionless piston of cross-sectional area 100 cm2.
(a) While the piston is at position O, the pressure of the enclosed gas is 10 Ncm-2 at a temperature of 27°C. When a 10 kg mass is placed on the piston, it comes to rest at position A without change in the temperature of the gas.
(i) Determine the new reading on the pressure gauge. (4 marks)
(ii) State with a reason how the value obtained in (i) compares with the initial pressure. (2 marks)
(b) The gas is now heated by the heating coil so that the piston moves back to the original position O.
(i) State the reading on the pressure gauge. (1 mark)
(ii) Determine the temperature of the gas in °C. (4 marks)
(Take g = 10 Nkg-1).
KCSE 2012
(1) A balloon is filled with a gas which is lighter than air. It is observed to rise in air upto a certain height. State a reason why the balloon stops rising. (1 mark)
(2) In verifying the pressure law of gases, the temperature and pressure of a gas are varied at constant volume. State the conditions necessary for the law to hold. (1 mark)
(3) Figure 6 shows a graph of volume against temperature for a given gas.
Use the graph to determine the absolute zero temperature in 0C. (2 marks)
(4) Figure 7 shows a horizontal tube containing air trapped by a mercury thread of length 24 cm. The length of the closed air column is 15 cm. The atmospheric pressure is 76 cm Hg.
(a) State the pressure of the enclosed air. (1 mark)
(b) The tube is now held in a vertical position with the open end facing upwards as shown in figure 8.
Determine:
(i) The pressure of the enclosed air. (1 mark)
(ii) The length l of the enclosed air column. (3 marks)
KCSE 2011
(1) When the temperature of a gas in a closed container is raised, the pressure of the gas increases. Explain how the molecules of the gas cause the increase in pressure. (2 marks)
(2) When the temperature of water reaches the boiling point, bubbles rise to the surface.
(a) State what is contained in the bubbles. (1 mark)
(b) State the reason why bubbles rise to the surface only at the boiling point. (1 mark)
(3) Figure 14 shows a graph of vapour pressure against the temperature of water vapour, in a laboratory where a mercury barometer indicates a height of 61.8 cm.
(a) Determine the atmospheric pressure in the laboratory in Nm2. (Take g = 10m/s2 and density of mercury = 13600 kg/m3). (3 marks:
(b) Use the graph to determine the boiling point of water in the laboratory. (1 mark)
KCSE 2010
(1) When a bicycle pump was sealed at the nozzle and the handle slowly pushed towards the nozzle, the pressure of the air inside increased. Explain this observation. (1 mark)
(2) Explain why it is advisable to use a pressure cooker for cooking at high altitudes. (2 marks)
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