2 deleted 2 characters in body edited Aug 4 '12 at 9:59 Christoph 9,77211 gold badge2323 silver badges4747 bronze badges This answer is somewhat hand-wavy, but I do believe it should help to grasp the concepts on an intuitive level. First of all, entropy is not a measure of randomness. For an isolated system in equilibrium under the fundamental assumption of statistical mechanics, the entropy is just $$S=k\ln\Omega$$ where $$\Omega$$ is the number of micro-statesmicrostates - microscopic system configurations - compatible with the given macro-statemacrostate - macroscopic equilibrium state characteristed by thermodynamical variables. It follows from the second law $$\delta Q = T\mathrm{d}S=T\mathrm{d}(k\ln\Omega)=kT\frac1\Omega\mathrm{d}\Omega$$ or equivalently $$\mathrm{d}\Omega = \Omega\frac{\delta Q}{kT}$$ The energy $$kT$$ is related to the average energy per degree of freedom, so this formula tells us that the transfer of heat into a system at equilibrium opens up a new number of microstates proportional to the number of existing ones and the number of degrees of freedom the transferred energy may excite. This answer is somewhat hand-wavy, but I do believe it should help to grasp the concepts on an intuitive level. First of all, entropy is not a measure of randomness. For an isolated system in equilibrium under the fundamental assumption of statistical mechanics, the entropy is just $$S=k\ln\Omega$$ where $$\Omega$$ is the number of micro-states - microscopic system configurations - compatible with the given macro-state - macroscopic equilibrium state characteristed by thermodynamical variables. It follows from the second law $$\delta Q = T\mathrm{d}S=T\mathrm{d}(k\ln\Omega)=kT\frac1\Omega\mathrm{d}\Omega$$ or equivalently $$\mathrm{d}\Omega = \Omega\frac{\delta Q}{kT}$$ The energy $$kT$$ is related to the average energy per degree of freedom, so this formula tells us that the transfer of heat into a system at equilibrium opens up a new number of microstates proportional to the number of existing ones and the number of degrees of freedom the transferred energy may excite. This answer is somewhat hand-wavy, but I do believe it should help to grasp the concepts on an intuitive level. First of all, entropy is not a measure of randomness. For an isolated system in equilibrium under the fundamental assumption of statistical mechanics, the entropy is just $$S=k\ln\Omega$$ where $$\Omega$$ is the number of microstates - microscopic system configurations - compatible with the given macrostate - macroscopic equilibrium state characteristed by thermodynamical variables. It follows from the second law $$\delta Q = T\mathrm{d}S=T\mathrm{d}(k\ln\Omega)=kT\frac1\Omega\mathrm{d}\Omega$$ or equivalently $$\mathrm{d}\Omega = \Omega\frac{\delta Q}{kT}$$ The energy $$kT$$ is related to the average energy per degree of freedom, so this formula tells us that the transfer of heat into a system at equilibrium opens up a new number of microstates proportional to the number of existing ones and the number of degrees of freedom the transferred energy may excite. 1 answered Aug 4 '12 at 9:54 Christoph 9,77211 gold badge2323 silver badges4747 bronze badges This answer is somewhat hand-wavy, but I do believe it should help to grasp the concepts on an intuitive level. First of all, entropy is not a measure of randomness. For an isolated system in equilibrium under the fundamental assumption of statistical mechanics, the entropy is just $$S=k\ln\Omega$$ where $$\Omega$$ is the number of micro-states - microscopic system configurations - compatible with the given macro-state - macroscopic equilibrium state characteristed by thermodynamical variables. It follows from the second law $$\delta Q = T\mathrm{d}S=T\mathrm{d}(k\ln\Omega)=kT\frac1\Omega\mathrm{d}\Omega$$ or equivalently $$\mathrm{d}\Omega = \Omega\frac{\delta Q}{kT}$$ The energy $$kT$$ is related to the average energy per degree of freedom, so this formula tells us that the transfer of heat into a system at equilibrium opens up a new number of microstates proportional to the number of existing ones and the number of degrees of freedom the transferred energy may excite.