Thermal Physics Tutorial 2 - The Equation of State for an Ideal Gas

 Contents
##### The Ideal Gas

We will be looking at gases and their behaviour under different conditions. The behaviour of ideal gases is quite easily predictable.  However real gases tend to behave slightly differently, especially those that consist of molecules.  So the ideal gas we will consider is:

• monatomic (the molecules are single atoms, for example Ar);

• at low temperature;

• at low pressure.

There is no interaction between the molecules of an ideal gas except for collisions, which are always perfectly elastic. Helium is a good example of an ideal gas.

 Question 1 Why is air not an ideal gas? There are three gas laws:

• Boyle's Law

• Charles' Law

• Pressure Law

These gas laws led to several important concepts in Physics.  Boyle's Law and Charles' Law are required practicals.

Boyles Law:

Pressure is inversely proportional to the volume: We can plot the data as a graph: We can see that the data fit into a pattern called a hyperbola.  If, however we plot pressure against 1/volume we get a linear (straight line) graph that shows direct proportionality. Since the line goes through the origin, we say that the two quantities are directly proportional.

• So we can say that

P µ 1/V

• Therefore:

P = k/V where k is a constant.

• Rearranging:

PV = constant.

The picture below shows a data-logging experiment for Boyle's Law. Charles' Law

The volume of an ideal gas is proportional to its Kelvin temperature.

The traditional way was to use a small drop of concentrated sulphuric acid in a capillary tube and heat it in water, watching it move up the capillary as the temperature got higher.  Getting good thermal contact is quite difficult in the experiment, so there is quite a lot of uncertainty.  We can use data logging equipment to show the experiment. This is the background picture on all the tutorial pages.

Whichever way we get the data, the ideal graph is like this: Whatever the gas we use, we find that the line always, without exception passes through the temperature axis at a very particular value, -273.15 oC.  This led to the concept of absolute zero, discovered by William Thomson, Lord Kelvin (1824 - 1907).  If we put the absolute zero point, 0 Kelvin, we get: This allows us to say:

V µ T

Therefore:

V = kT where k is a constant

Rearranging:

V/T = constant

To convert from Celsius to Kelvin, we use the following equation:

T = q + 273

where:

• T is the temperature in K;

• q is the temperature in oC.

It is not possible to get below 0 K.  You cannot get -274 oC.  At 0 K all molecular motion stops completely.

The Pressure Law:

This law tells us that pressure is proportional to the Kelvin temperature.  The traditional way to demonstrate this is with a large glass sphere immersed in water, connected to a manometer, a rather low tech (but remarkably accurate) way of detecting small differences in gas pressure. We can demonstrate the same using data-logging equipment like this: We can show this on a graph like this: So we can write:

P µ T

Therefore

P = kT

Rearranging:

P/T = constant

These three relationships can be combined to give:

PV = constant

T

From the gas laws we can write:

The value of the constant depends on how much gas is being considered.  If we are looking at one mole of gas (which we will define later), then the constant is the universal molar gas constant. This is given the code R and has the value 8.31 J mol-1 K-1.

For n moles of a gas, we can write: Where:

• p - pressure (Pa)

• V - volume (m3)

• n - number of moles

• R - molar gas constant ( 8.31 J mol-1 K-1)

• T - Temperature (K)

This is called the equation of state of an ideal gas. In this equation, SI units must be used, i.e. volume in m-3, pressure in Pa.  Temperature must be in Kelvin (K).  0 K = -273 oC. The temperature quoted in degrees Celsius is commonly set as a bear trap.  Make sure you don't fall into it.

 What is the volume of 2 moles of helium atoms at a temperature of 20 oC and a pressure of 100 kPa?

A more useful version of the ideal gas equation is this: The term p1 stands for pressure in container 1 while p2 stands for pressure in container 2.

 Question 3 A SCUBA* diver is deep underwater at a point where the pressure is 3.03 × 105 Pa, and the temperature is 10 oC.  The volume of her lungs is 3.0 litres.  She spots a dangerous animal and rises very rapidly to the surface where the temperature of the water is 20 oC and the air pressure is 1.01 × 105 Pa.  What is the volume of air in her lungs now? *SCUBA stands for Self-Contained Underwater Breathing Apparatus. ##### The Mole

The mole is defined as:

the same number of particles of a substance as there is in 12 g of 12C.

This number is called the Avogadro constant, and is given the codes L or NA.

L = 6.02 x 1023 mol-1

The molar mass Mm is the mass of 1 mole of the substance.  Chemists quote this in grams per mole; for physics we need to convert to kilograms per mole by dividing by 1000 (or multiplying by 10-3).

 A sealed container of volume 0.8 × 10-3 m3 contains gas at a temperature of 320 K and a pressure of 1.5 × 106 Pa.  Calculate: a)      The number of moles and molecules of the gas.                                      b)      The mass of the gas if its molar mass is 32.0 × 10-3 kg                           c)      The mass of a single molecule of gas. A rule that is important for chemists is that equal numbers of moles occupy the same volume of space.  1 mole of any gas occupies 0.0224 m3 at Standard Temperature and Pressure (STP), where temperature is 273 K and pressure is 1.01 × 105 Pa.

Investigating Boyle's and Charles' Law

This is a required practical (Number 8).  We have discussed the theory above.  Apparatus that is suitable for investigating Boyle's Law is like this: You need to follow a procedure like this:

• Clamp the clamp stand to the bench to stop it rocking.

• Use a micrometer to measure the diameter of the plunger.  Convert the reading into metres.

• Calculate the area of the plunger. (A = pD2/4).

• Replace the plunger and draw in about 4 to 5 cm3 of air.

• Place the rubber tubing over the nozzle.  Tighten up the thumbscrew clip (pinch-clip).

• Clamp the syringe.  This should not be too tight, as it will distort the syringe and could make it stick.  It should not be too loose either, because the weight of the slotted masses could pull it away from the clamp.

• The slotted masses should be attached to the plunger using a loop of copper wire which can be wrapped around the plunger. (I have found that string can easily come undone, which is a bit of a pain.)

• Move the plunger up and down to make sure it moves freely.

• Add 200 g at a time to a maximum of 1000 g and measure the new volume.

• Do at least one more set of repeat readings.

You will need to covert the mass to weight, using g = 9.81 N kg-1.  You will then need to get the pressure using: You then need to subtract this from the standard atmospheric pressure (= 100 kPa).  This is the pressure you will plot on a graph of 1/V against P.  This should be a straight line passing through the origin.

Having done this experiment with my students, I would say that you would be lucky to get a good straight line.  The plungers on cheap 10 ml syringes tend to be rather sticky.  If your school or college has a class set of glass gas syringes, you will get better results.  However, if you break one, it's expensive!  The risk of this is reduced by holding the syringe horizontally, and running the string over a bench pulley. Charles' Law

The capillary tube will be set up for you.  If it isn't you will be given instructions on how to do it by your tutor.  (I would be highly surprised if this were the case.  In the vast majority of schools and colleges, the technician or tutor would do this.)

The apparatus below has a small bead of concentrated sulphuric acid in the capillary tube.  Concentrated sulphuric acid is highly dangerous.  You will need to wear gloves, a lab coat, and goggles.  You must keep your face well away from the apparatus. You will need to do the following:

• Fill the beaker from a kettle.

• Stir the water with a glass rod (I was always taught not do do this with the thermometer - I did learn something from Mr Maslin).

• Read the value of length of the air sample, l, and the temperature on the thermometer, q.

• Allow the water to cool, taking readings every 5 oC fall.  You can speed up the process by removing a little of the water with a syringe, and placing the same amount of cold water.

• Plot a graph of l against q.

You may want to heat the water up with an immersion heater instead.  This would allow you more control of the temperature, so that you could do repeats.

You can then use your graph to calculate the value of absolute zero.  Your graph will look something like this: You need to work out the gradient, m.  The equation will be: At absolute zero, l = 0.  Therefore: Therefore: This, hopefully, should be -273 oC as shown on this graph: 