5.1 Ideal gas law and general gas equation  (Page 4/4)

Molar volume of gases

It is possible to calculate the volume of a mole of gas at STP using what we now know about gases.

1. Write down the ideal gas equation pV = nRT, therefore V = $\frac{nRT}{p}$
2. Record the values that you know, making sure that they are in SI units You know that the gas is under STP conditions. These are as follows: p = 101.3 kPa = 101300 Pan = 1 mole R = 8.3 J.K ${}^{-1}$ .mol ${}^{-1}$ T = 273 K
3. Substitute these values into the original equation.
   $$V=\frac{nRT}{p}$$
   $$V=\frac{1mol\times 8.3J.K^{-1}.mo{l}^{-1}\times 273K}{101300Pa}$$
4. Calculate the volume of 1 mole of gas under these conditions The volume of 1 mole of gas at STP is 22.4 $\times$ 10 ${}^{-3}$ m ${}^3$ = 22.4 dm ${}^3$ .

Ideal gases and non-ideal gas behaviour

In looking at the behaviour of gases to arrive at the Ideal Gas Law, we have limited our examination to a small range of temperature and pressure. Almost all gases will obey these laws most of the time, and are called ideal gases . However, there are deviations at high pressures and low temperatures . So what is happening at these two extremes?

Earlier when we discussed the kinetic theory of gases, we made a number of assumptions about the behaviour of gases. We now need to look at two of these again because they affect how gases behave either when pressures are high or when temperatures are low.

1. Molecules do occupy volume This means that when pressures are very high and the molecules are compressed, their volume becomes significant. This means that the total volume available for the gas molecules to move is reduced and collisions become more frequent. This causes the pressure of the gas to be higher than what would normally have been predicted by Boyle's law ( [link] ).

   

2. Forces of attraction do exist between molecules At low temperatures, when the speed of the molecules decreases and they move closer together, the intermolecular forces become more apparent. As the attraction between molecules increases, their movement decreases and there are fewer collisions between them. The pressure of the gas at low temperatures is therefore lower than what would have been expected for an ideal gas ( [link] ). If the temperature is low enough or the pressure high enough, a real gas will liquify .

Summary

• The kinetic theory of matter helps to explain the behaviour of gases under different conditions.
• An ideal gas is one that obeys all the assumptions of the kinetic theory.
• A real gas behaves like an ideal gas, except at high pressures and low temperatures. Under these conditions, the forces between molecules become significant and the gas will liquify.
• Boyle's law states that the pressure of a fixed quantity of gas is inversely proportional to its volume, as long as the temperature stays the same. In other words, pV = k or p ${}_1$ V ${}_1$ = p ${}_2$ V ${}_2$ .
• Charles's law states that the volume of an enclosed sample of gas is directly proportional to its temperature, as long as the pressure stays the same. In other words,
  $$\frac{V_1}{T_1}=\frac{V_2}{T_2}$$
• The temperature of a fixed mass of gas is directly proportional to its pressure, if the volume is constant. In other words,
  $$\frac{p_1}{T_1}=\frac{p_2}{T_2}$$
• In the above equations, temperature must be written in Kelvin . Temperature in degrees Celsius (temperature = t) can be converted to temperature in Kelvin (temperature = T) using the following equation:
  $$T=t+273$$
• Combining Boyle's law and the relationship between the temperature and pressure of a gas, gives the general gas equation , which applies as long as the amount of gas remains constant. The general gas equation is pV = kT, or
  $$\frac{p_1{V}_1}{T_1}=\frac{p_2{V}_2}{T_2}$$
• Because the mass of gas is not always constant, another equation is needed for these situations. The ideal gas equation can be written as
  $$pV=nRT$$
  where n is the number of moles of gas and R is the universal gas constant, which is 8.3 J.K ${}^{-1}$ .mol ${}^{-1}$ . In this equation, SI units must be used. Volume (m ${}^3$ ), pressure (Pa) and temperature (K).
• The volume of one mole of gas under STP is 22.4 dm ${}^3$ . This is called the molar gas volume .

Summary exercise

1. For each of the following, say whether the statement is true or false . If the statement is false, rewrite the statement correctly.
   1. Real gases behave like ideal gases, except at low pressures and low temperatures.
   2. The volume of a given mass of gas is inversely proportional to the pressure it exerts.
   3. The temperature of a fixed mass of gas is directly proportional to its pressure, regardless of the volume of the gas.
2. For each of the following multiple choice questions, choose the one correct answer .
   1. Which one of the following properties of a fixed quantity of a gas must be kept constant during an investigation f Boyle's law?
      1. density
      2. pressure
      3. temperature
      4. volume
      ( IEB 2003 Paper 2 )
   2. Three containers of EQUAL VOLUME are filled with EQUAL MASSES of helium, nitrogen and carbon dioxide gas respectively. The gases in the three containers are all at the same TEMPERATURE. Which one of the following statements is correct regarding the pressure of the gases?
      1. All three gases will be at the same pressure
      2. The helium will be at the greatest pressure
      3. The nitrogen will be at the greatest pressure
      4. The carbon dioxide will be at the greatest pressure
      ( IEB 2004 Paper 2 )
   3. One mole of an ideal gas is stored at a temperature T (in Kelvin) in a rigid gas tank. If the average speed of the gas particles is doubled, what is the new Kelvin temperature of the gas?
      1. 4T
      2. 2T
      3. $\surd$ 2T
      4. 0.5 T
      ( IEB 2002 Paper 2 )
   4. The ideal gas equation is given by pV = nRT . Which one of the following conditions is true according to Avogadro's hypothesis?

      a	p $\propto$ 1/V	(T = constant)
      b	V $\propto$ T	(p = constant)
      c	V $\propto$ n	(p, T = constant)
      d	p $\propto$ T	(n = constant)

      ( DoE Exemplar paper 2, 2007 )
3. Use your knowledge of the gas laws to explain the following statements.
   1. It is dangerous to put an aerosol can near heat.
   2. A pressure vessel that is poorly designed and made can be a serious safety hazard (a pressure vessel is a closed, rigid container that is used to hold gases at a pressure that is higher than the normal air pressure).
   3. The volume of a car tyre increases after a trip on a hot road.
4. Copy the following set of labelled axes and answer the questions that follow:

   

   1. On the axes, using a solid line , draw the graph that would be obtained for a fixed mass of an ideal gas if the pressure is kept constant.
   2. If the gradient of the above graph is measured to be 0.008 m ${}^3$ .K ${}^{-1}$ , calculate the pressure that 0.3 mol of this gas would exert.
   ( IEB 2002 Paper 2 )
5. Two gas cylinders, A and B, have a volume of 0.15 m ${}^3$ and 0.20 m ${}^3$ respectively. Cylinder A contains 1.25 mol He gas at pressure p and cylinder B contains 2.45 mol He gas at standard pressure. The ratio of the Kelvin temperatures A:B is 1.80:1.00. Calculate the pressure of the gas (in kPa) in cylinder A. ( IEB 2002 Paper 2 )
6. A learner investigates the relationship between the Celsius temperature and the pressure of a fixed amount of helium gas in a 500 cm ${}^3$ closed container. From the results of the investigation, she draws the graph below:

   

   1. Under the conditions of this investigation, helium gas behaves like an ideal gas. Explain briefly why this is so.
   2. From the shape of the graph, the learner concludes that the pressure of the helium gas is directly proportional to the Celcius temperature. Is her conclusion correct? Briefly explain your answer.
   3. Calculate the pressure of the helium gas at 0 ${}^{\circ }$ C.
   4. Calculate the mass of helium gas in the container.
   ( IEB 2003 Paper 2 )
7. One of the cylinders of a motor car engine, before compression contains 450 cm ${}^3$ of a mixture of air and petrol in the gaseous phase, at a temperature of 30 ${}^{\circ }$ C and a pressure of 100 kPa. If the volume of the cylinder after compression decreases to one tenth of the original volume, and the temperature of the gas mixture rises to 140 ${}^{\circ }$ C, calculate the pressure now exerted by the gas mixture.
8. In an experiment to determine the relationship between pressure and temperature of a fixed mass of gas, a group of learners obtained the following results:

   Pressure (kPa)	101	120	130.5	138
   Temperature ( ${}^{\circ }$ C)	0	50	80	100
   Total gas volume (cm ${}^3$ )	250	250	250	250

   1. Draw a straight-line graph of pressure (on the dependent, y-axis) versus temperature (on the independent, x-axis) on a piece of graph paper. Plot the points. Give your graph a suitable heading. A straight-line graph passing through the origin is essential to obtain a mathematical relationship between pressure and temperature.
   2. Extrapolate (extend) your graph and determine the temperature (in ${}^{\circ }$ C) at which the graph will pass through the temperature axis.
   3. Write down, in words, the relationship between pressure and Kelvin temperature.
   4. From your graph, determine the pressure (in kPa) at 173 K. Indicate on your graph how you obtained this value.
   5. How would the gradient of the graph be affected (if at all) if a larger mass of the gas is used? Write down ONLY increases , decreases or stays the same .
   ( DoE Exemplar Paper 2, 2007 )