3.2.2 (b) Gradient and Rate

Syllabus

(b) calculation of rate from the gradients of graphs measuring how a physical quantity changes with time

{Suitable physical quantities to monitor could include concentration, gas volume, mass, etc.}

What does this mean?

Oh dear.

This Maths.

And, therefore, even less interesting than Physics!

We should be used to graphs like the one on the right from GCSE.

Generally, all that was required was to say that the reaction was clearly fastest at the start when there are lots of reactant molecules and that this is reflected in a steep gradient when lots of gas was made in the first 1.5 seconds.

Between 1.5 and 3.0 seconds the gradient is lessening as the reactant were used up,producing a slowing reaction that is making less and less gas per second.

Until somewhere between 4.0 and 4.5 seconds the gradient is zero. The reaction is over because one of the reactants has run out. No more gas can be made.

Very occasionally GCSE examiners ask you to calculate a gradient yourself. A level examiners may well ask you to do this, although they frequently don't because they will have to accept a range of answers as correct since every pupil draws their tangent with a slightly different gradient.

Drawing and using Tangents

So long as your attempted tangent actually touches the line without cutting it then

Hopefully, you will remember the formula for the gradient of a straight line from GCSE maths:

Gradient = Difference of y's = y2 - y1 = dy

Difference of x's x2 - x1 dx

Once you've drawn your straight line you need to pick two points on it.

It really doesn't matter which two points you pick (so long as they are reasonably far apart).

So you might as well pick points that are exactly in the corner of a square on the graph paper.

If didn't have two such points you could just extend the tangent until you do.

In the case of these three points you may as well pick the two furthest apart.

In which case the gradient = (9--3) / (4 --2) = 12/6 = 2

Although you should get the same answer with any combination of points (or you've done it wrong).

Units

The x-axis will always be time and will generally be in seconds - although it may be in minutes.

Y-axis = Volume of gas

The y-axis will often be volume of gas in cm³ or dm³.

Gradient is dy/dx so the units will be cm³/s or dm³/s.

Y-axis = Concentration

The y-axis will often be concentration of a solution usually in mol/dm³ or g/dm³

Gradient is dy/dx so the units will be mol/dm³/s or g/dm³/s.

Y-axis = Mass

The y-axis may be mass of gas given off, or left in the reaction vessel in g.

Gradient is dy/dx so the units will be g/s.

Exam-style Questions

1. The curve below shows how the volume of oxygen evolved varies with time when 50 cm3 of a 2.0 mol dm–3 solution of Hydrogen Peroxide, H₂O₂, decomposes at 298 K.

(a) State how you could use the curve to find the rate of reaction at point A.

.........................................................................................................................................................................(1)

(b) Sketch curves, on the above axes, to illustrate how the volume of oxygen evolved would change with time if the experiment was repeated at 298 K using the following.

(i) 100 cm3 of a 1.0 mol dm–3 solution of H2O2. Label this curve X.

(ii) 25 cm3 of a 2.0 mol dm–3 solution of H2O2 in the presence of a catalyst.

Label this curve Y. (4)

(Total 5 marks)

Answers

1. (a) Gradient (or slope) (or draw a tangent) 1

(b) (i) Curve X is lower and starts at origin 1

And levels out at same volume as original curve 1

(ii) Curve Y is steeper than original and starts at origin 1

Then levels out at half the volume of the original 1

[5]

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