Show it, then

Okay, here goes. You will first need read the explanation of how the equilibrium constant K depends on k_f and k_r.

We are interested in the rate of change of the concentration of the reactants, which we will write like so:

  .      
[ R ]

  . 
[ Y ]

for the rate of change with time of the concentration of Y.

Now what determines these rates of change? Having read the explanation of how K depends on k_f and k_r you will know that a reasonable guess is:

  .       .
[ R ] = [ Y ] = k_r [ G ] [ B ] - k_f [ R ] [ Y ]

Now in the special case we're considering here, the reverse reaction rate constant k_r = 0, hence:

  .       .
[ R ] = [ Y ] =  - k_f [ R ] [ Y ]

These are two ``first order differential equations.'' Let's write the derivatives out:

d [ R ]   d [ Y ]
------- = ------- =  - k_f [ R ] [ Y ]
  d t       d t

Looking at these equations you can see that what we need is a function (appearing on the right) which has the exact same form as its first derivative (appearing on the left). There is only one such function, namely the exponential function, for which we have:

   a t
d e           a t
------  =  a e
  d t

            
                  a_r t
[ R ] (t) =  b_r e

                  a_y t
[ Y ] (t) =  b_y e

in the differential equations above, with a_r, b_r, a_y, and b_y all being constants, then we will find that we can satisfy the differential equations only if a_r = a_y = -k_f. Thus we have:

            
                  - k_f t
[ R ] (t) =  b_r e

                  - k_f t
[ Y ] (t) =  b_y e

Now we need to also satisfy the initial conditions, that is, these equations must be true at t = 0. If we substitute t = 0 above we have:

            
                      
[ R ] (t = 0) =  b_r
                      
[ Y ] (t = 0) =  b_y

Hence our final results are:

            
                            - k_f t
[ R ] (t) =  [ R ] (t = 0) e

                            - k_f t
[ Y ] (t) =  [ Y ] (t = 0) e

If you plot the function e^-x on your graphing calculator, you will see that the curve looks exactly like the curves you see on the stripchart that plot the concentration of R and Y versus time.