### Video Transcript

The rate of a reaction doubles for
every 10 degrees Celsius increase in temperature. If the temperature increases from
25 degrees Celsius to 55 degrees Celsius, by what factor does the reaction rate
change?

The reaction in this question is
completely unknown. All we do know for this mystery
reaction is that if we increase the temperature by 10 degrees Celsius, we’ll double
the rate. Broadly speaking, the rate of a
reaction is the rate at which reactants are turned into products. What we’re being asked is by what
factor the rate changes when we increase the temperature from 25 degrees Celsius to
55 degrees Celsius.

When working out a factor, we need
to take the final rate, rate 𝑓, and divide it by the initial rate, rate 𝑖. So what we need to work out is how
many 10 degrees Celsius increases are there between 25 degrees Celsius and 55
degrees Celsius. 55 degrees Celsius minus 25 degrees
Celsius is 30 degrees Celsius. Dividing 30 degrees Celsius by 10
degrees Celsius gives us three. So our final rate is equal to our
initial rate times two once, twice, three times. So the final rate is equal to eight
times the initial rate.

If we substitute that into our
factor equation, we get eight times the initial rate divided by the initial
rate. Cancelling common terms gives us a
factor of eight. If you wanted to, you could use a
general equation for this relationship. The rate at temperature two is
equal to the rate at temperature one multiplied by two to the power of the
difference in temperature divided by 10 degrees Celsius. Whichever way you want to do it,
the rate at 55 degrees Celsius is eight times the rate at 25 degrees Celsius.

If the temperature decreases from
95 degrees Celsius to 25 degrees Celsius, by what factor does the reaction rate
change?

This question is about a decrease
rather than an increase in temperature. To figure this question out, we
have to flip the relationship on its head. The rate at a temperature 10
degrees below the starting temperature will be half. So we need to work out how many 10
degrees Celsius decreases there are between 95 degrees Celsius and 25 degrees
Celsius.

95 degrees Celcius minus 25 degrees
Celsius is 70 degrees Celsius. 70 degrees Celsius divided by 10
degrees Celsius is seven. So the rate of the reaction at 25
degrees Celsius is equal to the rate of the reaction at 95 degrees Celsius
multiplied by a half seven times. A half multiplied by itself seven
times is one over 128. So we can plug that into our
equation. The factor by which the rate
changes is equal to one over 128 multiplied by the initial rate over the initial
rate. The initial rates cancel, giving us
a final factor of one over 128.

If you know your laws of indices
well, you should be able to see that the general equation still applies here. △𝑡 in our general equation should
be the final temperature minus the initial temperature, so 25 minus 95. This gives us two to the power of
minus 70 degrees Celsius divided by 10 degrees Celsius, giving us the initial rate
multiplied by two to the power of minus seven, which can be rearranged as the
initial rate divided by two to the power of seven. And as we’ve already demonstrated,
two to the power of seven is 128.

Either way we do it, we’ve
demonstrated that for this reaction, when the temperature decreases from 95 degrees
Celsius to 25 degrees Celsius, the factor of the rate change will be one over
128. The reaction rate will decrease by
128 times.