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Hey, Guys, I have a 2010 TBR Gen Chem, i don't know if they have corrected this mistake in newer versions, but i point out here anyway. On page 122, figure 2-43 graphs the first order decay "rate" as a linear line with a negative slope. This is conceptually wrong. It is actually not linear line with a negative slope, actually dx/dt is still an exponential function scaled by a constant. below is the mathematical derivation.
For first order decay,
dx/dt=kx (k is a negative constant)
dx/x=kdt
ln(x)+c=kt
setting x=xi (xi=initial amount of x at t=0)
ln(xi)+c=0
c=-ln(xi)
so, ln(x)-ln(xi)=kt
x=xi*exp(kt)
We have derived the exponential function for decay, now lets derive dx/dt
dx/dt=xi*k*exp(kt)=kx
We see the first derivative is exponential when graphed with respect to time. It is linear only when graphed with respect to x. Figure 2-43 sets its horizontal axis as time, so dx/dt is supposed to be exponential rather than linear as it is drawn in 2-43.
Hope, this helps, if it has not yet made correction to it.
For first order decay,
dx/dt=kx (k is a negative constant)
dx/x=kdt
ln(x)+c=kt
setting x=xi (xi=initial amount of x at t=0)
ln(xi)+c=0
c=-ln(xi)
so, ln(x)-ln(xi)=kt
x=xi*exp(kt)
We have derived the exponential function for decay, now lets derive dx/dt
dx/dt=xi*k*exp(kt)=kx
We see the first derivative is exponential when graphed with respect to time. It is linear only when graphed with respect to x. Figure 2-43 sets its horizontal axis as time, so dx/dt is supposed to be exponential rather than linear as it is drawn in 2-43.
Hope, this helps, if it has not yet made correction to it.
