First, we examine how 10 molecules are distributed in space. We then count the ways energy can be distributed and consider why the original temperature difference does not return.
Adjust conditions:
The sliders adjust the values
平衡状態で、10個全部が左にいる確率は？
左側が箱全体の90％の体積なら、10個全部が左にいる確率は？
The equations explain the phenomenon
1Temperature is related to molecular motion.½ m ⟨v²⟩=(32) k T……
2Thermal contact brings temperatures toward the same value.Thigh→T←Tlow……
The following treatment assumes no phase change and a constant specific heat capacity. Work due to changes in the volume of the water is neglected.During a phase change, such as melting or boiling, this equation cannot be used to calculate a temperature rise.
3Heat transfer is related to temperature change.Q=m c ΔT……
Heat exchange with the surroundings and heat absorbed by the container are neglected. Both samples are water, and their specific heat capacity is assumed constant.
4The final temperature after mixing can be calculated.m₁ c (T₁ − T)=m₂ c (T − T₂)……
5Heating ice reveals changes of phase.Q=m L……
6Heat can be converted into work.W=Qin−Qout……
7The thermal efficiency can be calculated.e=1 − QoutQin＜1……
8The probability that all molecules are on the left at equilibrium can be calculated.P(all on the left)=(12)N……
9The probability can be calculated for larger numbers of molecules.(12)100=7.9×10⁻³¹……
10The ways of distributing energy can be counted.Ω=ΩA(qA)×ΩB(q − qA)……
11Explain the natural direction of heat flow.Thigh→Tlow……
What this chapter has shown
The probability that all 10 molecules are on the left is 1 in 1024. This probability falls rapidly as the number of molecules increases. Similarly, overwhelmingly fewer energy distributions correspond to a restored temperature difference than to equal temperatures. In objects containing many molecules, a return to the original temperature difference is not observed in practice.
0.098％All 10 molecules are on the left in a box divided into equal volumes
If the left side occupies 90％ of the total volume, the probability that all 10 molecules are on the left is about 35％.
In everyday lifeThe scent of perfume spreads through a room, but it does not gather back into the bottle.
Detailed summary
Temperature measures the intensity of molecular motion, while heat is energy transferred because of a temperature difference. Energy is conserved, and objects in thermal contact approach the same temperature. The process does not reverse because so few possible distributions of energy correspond to the original temperature difference. The same second law of thermodynamics also prevents a heat engine operating in a cycle from converting all the heat it receives into work.
Recall
左右の体積が等しい箱で分子を20個にすると、全部が左にいる確率は10個のときの何倍？
容器や外へ熱が逃げないとき、80 °C のお湯200 gに20 °C の水100 gを混ぜると何°C？