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Because diamond is thermodynamically unstable with respect to graphite, this means that delta G is negative for this process and diamond will spontaneously turn into graphite. However, because this process is also kinetically stable, this means that the process will occur very slowly. The book is just pointing out that while conversion to graphite is favorable, it does not necessarily mean that the reaction will occur quickly. In fact, this reaction will occur very slowly.
Although diamond is thermodynamically unstable because delta G is negative (the reaction is spontaneous) and diamond has a higher energy than graphite, diamond is considered kinetically stable because the pathway in which diamond becomes graphite has a large activation energy barrier that prevents its ready conversion to graphite. Thus, under ambient conditions, diamond won't change to graphite even though graphite is thermodynamically more stable than diamond.
According to thermodynamics, the reaction from diamond to graphite is spontaneous and favorable. However, because kinetics rather than thermodynamics is controlling this reaction, it occurs extremely slowly. So, diamond is kinetically stable, but thermodynamically unstable.
Diamond is thermodynamically unstable because delta G for diamond to graphite is negative and therefore spontaneous, but the transition from diamond to graphite has a large activation energy barrier, so diamond is not likely to transition on its own to graphite and is therefore kinetically stable.
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