If an object has a multiplicity $\Omega$ that decreases as its thermal energy $U$ increases, its temperature would
A) be always zero.
B) be always positive (and nonzero).
C) be negative (and nonzero).
D) increase as $U$ increases.
E) decrease as $U$ increases.

Question

Quizplus · Accepted Answer

The temperature of an object is related to the change in its entropy (which is directly related to the logarithm of its multiplicity, $\Omega$) with respect to its internal energy $U$. Mathematically, this is given by $1/T = (\partial S/\partial U)_{N,V}$, where $S$ is the entropy, $T$ is the temperature, and $N,V$ are held constant. If the multiplicity $\Omega$ decreases as $U$ increases, it implies that the entropy decreases with increasing $U$, leading to a negative value for $\partial S/\partial U$, and thus a negative temperature.

If an Object Has a Multiplicity Ω\OmegaΩ That Decreases as Its Thermal Energy

If an Object Has a Multiplicity $\Omega$ That Decreases as Its Thermal Energy