# What makes an increase in temperature change the reaction rate?

##### 1 Answer

*The higher the temperature, the faster the rate of (a non-biological) reaction.*

**CONCEPTUAL APPROACH**

**Temperature** is directly proportional to the average kinetic energy of the particles in a system.

An *increase* in temperature increases the **average kinetic energy** of the particles, which makes them move faster. That makes them collide more frequently and thus react more frequently.

Therefore, the **rate of reaction** increases at higher temperatures.

**MATHEMATICAL APPROACH**

A mathematical way of proving this is this. Suppose we have a reaction **rate law** for a basic first-order reaction is:

#r_2(t) = k_2[A]#

#r_1(t) = k_1[A]#

Let **rate constants** for temperature **Arrhenius equation**, we have:

#color(green)(k_2 = Ae^(-E_a"/"RT_2))#

#color(green)(k_1 = Ae^(-E_a"/"RT_1))# where:

#A# is thepre-exponential factor(it won't matter here), because it comes before#e# , the exponential function (kind of obvious where the name came from now, right?)#E_a# is theactivation energy, which does NOT depend on temperature.#R# is theuniversal gas constant#"8.314472 J/mol"cdot"K"# .#T_i# is thetemperatureof reaction#i# in units of#"K"# .

Now, if I divide these, I get a ratio between

#k_2/k_1 = (cancel(A)e^(-E_a"/"RT_2))/(cancel(A)e^(-E_a"/"RT_1))#

#ln\frac(k_2)(k_1) = ln\frac(e^(-E_a"/"RT_2))(e^(-E_a"/"RT_1))#

#= lne^(-E_a"/"RT_2) - lne^(-E_a"/"RT_1)#

#= -E_a/(RT_2) + (E_a)/(RT_1)#

#= (E_a)/(R)[-1/(T_2) + 1/(T_1)]#

#color(blue)(ln \frac(k_2)(k_1) = -(E_a)/(R)[1/(T_2) - 1/(T_1)])#

This is a useful equation; if you know the rate constants for a reaction (which you can get experimentally), and you know the starting and final temperature in a temperature ramp, you can determine the reaction's activation energy from the slope.

From this resultant equation, we see that if **negative** (since

Therefore, since **positive**,

From this, we see that since

*Thus, the higher the temperature, the faster the rate of (a non-biological) reaction.*