Video: Calculating the Change in Reaction Rate from the Change in Reaction Temperature

The rate of a reaction doubles for every 10°C increase in temperature. (a) If the temperature increases from 25°C to 55°C, by what factor does the reaction rate change? (b) If the temperature decreases from 95°C to 25°C, by what factor does the reaction rate change?

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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.

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