Boyle's Law (Mariotte's Law) Chemistry Tutorial

Key Concepts

• Boyle's Law is also known as Mariotte's Law.
• Boyle's Law (Mariotte's Law) states that

  At constant temperature, the volume of a given quantity of gas is inversely proportional to its pressure :

  V

  ∝

  1   P

  
  where V = volume of the gas
  and P = pressure of the gas

  So at constant temperature, if the volume of a gas is doubled, its pressure is halved.

• This relationship can be written as an equation if we use a constant of proportionality, that is:

  At constant temperature for a given quantity of gas, the product of its volume and its pressure is a constant :

  PV = constant

  
  OR

  PV = k

  
  where k is a constant

• At constant temperature for a given quantity of gas :

  P_iV_i = P_fV_f

  
  where
      P_i is the initial (original) pressure
      V_i is its initial (original) volume
      P_f is its final pressure
      V_f is its final volume

  (a) P_i and P_f must be in the same units of measurement (eg, both in atmospheres or both in kPa)
  (b) V_i and V_f must be in the same units of measurement (eg, both in litres or both in millilitres)

• All gases approximate Boyle's Law at high temperatures and low pressures.

  (a) A hypothetical gas which obeys Boyle's Law at all temperatures and pressures is called an Ideal Gas.

  (b) A Real Gas is one which approaches Boyle's Law behaviour as the temperature is raised or the pressure lowered.

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Graphical Representations of Boyle's Law

Consider an experiment in which a known amount of hydrogen gas in a syringe has a volume of 23 mL at atmospheric pressure (760 mm Hg or 1 atm or 101.3 kPa).

You then apply an external pressure of 912 mm Hg (1.2 atmospheres or 121.6 kPa) by pressing down on the plunger in the syringe.

The volume of hydrogen gas is then recorded as 19.2 mL.

You continue to apply external pressure by pushing the plunger down further, recording the volume of hydrogen gas as shown in the table below:

If we plot these points on a graph, the graph looks like the one below:

Note that this is not a linear relationship, the line in the graph is curved, it is not a straight line.

But look what happens if we multiply volume and pressure (P × V):

For a given amount of gas at constant temperature we now we can write the equation:

P × V = constant

If we divide both sides of the equation by P, we get:

Recall that the equation for a straight line that runs through the point (0,0) is

y = mx

where m is the slope (or gradient) of the line

Then a graph of V against ¹/_P, should be a straight line with a slope (or gradient) equal to the value of the constant.

The table below shows what happens if we calculate ¹/_P for each volume, V, in the experiment above and then graph the results:

By plotting these points on a graph, we can see that the relationship is linear:

We now have a simple method for determining the value of the constant:

Recall that we can calculate the slope (gradient, m) of a straight line using two points on the line

m =	(y2 - y1) (x2 - x1)

Choosing the points (0.00094,16.4) and (0.0013,23)

m	=	(23 - 16.4)   (0.0013 - 0.00094)
	=	(6.6)   (0.00036)
	=	1.8 × 104

and the equation for this straight line is

This equation then allows us to calculate the volume of the gas at any pressure, as long as we use the same amount of gas and keep the temperature the same.

Let us say we have a specific amount of gas and keep the temperature constant, then initially at pressure P_i the gas has a volume of V_i and we know that:

P_iV_i = constant

If we maintain the same temperature and the same amount of gas, but change the pressure to P_f, then the new gas volume will be V_f, and

P_fV_f = the same constant

So, as we long as we use the same amount of gas at the same temperature:

P_iV_i = constant = P_fV_f

that is:

P_iV_i = P_fV_f

This means that if we know the initial conditions (P_i and V_i), and, we know the final pressure (P_f), we can calculate the final volume (V_f):

Vf =	Pi × Vi Pf

or we can calculate the final pressure (P_f) if we know the final volume (V_f):

Pf =	Pi × Vi Vf

Similarly, if we know the final conditions (P_f and V_f), and, we know the initial pressure (P_i), we can calculate the initial volume (V_i):

Vi =	Pf × Vf Pi

or we can calculate the initial pressure (P_i) if we know the initial volume (V_i):

Pi =	Pf × Vf Vi

Worked Example: Calculating the Final Pressure of a Gas Sample

Question : A certain mass of gas occupies a volume of 2.5 L at 90 kPa pressure.
What pressure would the gas exert if it were placed in a 10.0 L container at the same temperature?

Solution:

(Based on the StoPGoPS approach to problem solving.)

1. What is the question asking you to do?

   Calculate final gas pressure
   P_f = ? kPa

2. What data (information) have you been given in the question?

   Extract the data from the question:

   Conditions of the experiment: constant amount of gas at constant tempertaure.

   V_i = initial gas volume = 2.5 L
   P_i = initial gas pressure = 90 kPa
   V_f = final gas volume = 10.0 L

3. What is the relationship between what you know and what you need to find out?

   Because the amount of gas and temperature are constant, we can use Boyle's Law:

   P_iV_i = P_fV_f

   Rearrange this equation by dividing both sides by V_f:

   PfVf Vf	=	PiVi Vf
   Pf	=	PiVi Vf

4. Substitute in the values and solve for P_f

   Pf	=	PiVi Vf
   	=	90 × 2.5 10.0
   	=	22.5 kPa

5. Is your answer plausible?

   Consider that the volume has increased from 2.5 to 10, an increase of 10/2.5 = 4.
   If the volume increases 4 times, then the pressure must decrease and the new pressure will be ¹/₄ of the initial pressure, that is, ¹/₄ × 90 = 22.5 kPa
   Since this is the same value as we calculated above, we are reasonably confident that our answer is correct.

6. State your solution to the problem "final gas pressure":

   P_f = 22.5 kPa

Worked Example: Calculating the Final Volume of a Gas Sample

Question : 4.5 L of gas at 125 kPa is expanded at constant temperature until the pressure is 75 kPa.
What is the final volume of the gas?

Solution:

(Based on the StoPGoPS approach to problem solving.)

1. What is the question asking you to do?

   Calculate final volume
   V_f = ? L

2. What data (information) have you been given in the question?

   Extract the data from the question:

   Conditions: constant amount of gas and temperature.

   P_i = 125 kPa
   V_i = 4.5 L
   P_f = 75 kPa

3. What is the relationship between what you know and what you need to find out?

   Because the amount of gas and temperature are constant, we can use Boyle's Law:

   P_iV_i = P_fV_f

   Rearrange this equation by dividing both sides by P_f:

   PfVf Pf	=	PiVi Pf
   Vf	=	PiVi Pf

4. Substitute in the values and solve for V_f

   Vf	=	PiVi Pf
   	=	125 × 4.5 75
   	=	7.5 L

5. Is your answer plausible?

   Pressure has decreased by a factor of 75/125 = 0.6
   Therefore volume must have increased by a factor of 1/0.6 = 1.667
   Final volume must be 1.667 × 4.5 = 7.5 L
   Since this value is the same as the one we calculated above, we are reasonably confident that our answer is plausible.

6. State your solution to the problem "final gas volume ":

   V_f = 7.5 L