Leave a hot drink on the desk and it cools; drop an ice cube in it and the drink chills. In both cases thermal energy is quietly on the move — always in the same direction, from hotter to cooler — and it keeps flowing until everything settles at one shared temperature. That settled state is thermal equilibrium.
📚 What you need to know
Thermal energy always flows from the hotter region to the cooler region — never the other way on its own.
It keeps flowing until both reach the same temperature. At that point there is no more net transfer.
Thermal equilibrium: when two substances in thermal contact no longer exchange any net thermal energy and sit at an equal temperature.
The objects must be in thermal contact for energy to flow between them.
The hotter object cools and the cooler object warms until they meet somewhere in between.
At equilibrium energy is still swapped both ways — but in equal amounts, so the net transfer (and each temperature) stays put.
Energy always flows hot to cold
Put a hot object and a cold object in contact and thermal energy sets off in one direction only: from the hotter one to the cooler one. The hot object loses energy and cools; the cold object gains it and warms. This carries on until the temperature difference between them has vanished.
Net thermal energy flows from hot to cold until both objects reach the same temperature — thermal equilibrium.
It’s the temperature difference that drives the flow, not the amount of energy stored. A tiny spark at thousands of degrees will still lose energy to a huge, cooler room — because “hot to cold” is about temperature, not about which object holds more energy overall.
Reaching equilibrium
Track the two temperatures over time and you get a tell-tale shape: the hot object’s temperature falls, the cold object’s rises, and the two curves bend towards each other until they meet. Where they level off is the equilibrium temperature — and from then on, nothing changes.
The hotter object cools and the cooler object warms, both flattening out at a common equilibrium temperature. The transfer is fastest at the start, when the temperature gap is biggest.
Notice the curves are steepest at the start and flatten as they close in. That’s because the bigger the temperature gap, the faster the energy flows. As the gap shrinks the flow eases off, so the temperatures approach equilibrium quickly at first and then ever more gently — they glide in rather than snapping together.
What “equilibrium” really means
It’s tempting to picture equilibrium as everything going still — but that’s not quite it. The two objects are still in contact and still passing energy back and forth. The difference is that now they pass it in equal amounts both ways, so the net transfer is zero and neither temperature changes.
At equilibrium the exchange never stops — it just balances. Equal energy each way means zero net transfer, so both temperatures hold steady.
This balancing act is the whole reason a thermometer works. Slip it under your tongue and energy flows until the thermometer reaches equilibrium with you — so its reading becomes your temperature. It’s also the formal idea behind the “zeroth law”: if two things are each in equilibrium with a third, they’re in equilibrium with each other.
Worked examples
WE 1
A hot metal spoon is left standing in a cup of cooler tea. Describe the flow of thermal energy and the final state of the spoon and tea.
The spoon is hotter, so net thermal energy flows from the spoon to the tea.The spoon cools down; the tea warms up slightly.Both reach the same temperature — thermal equilibriumThe final temperature sits somewhere between the two starting temperatures. Flow stops once the temperature difference is gone.
WE 2
Two blocks that are already at the same temperature are placed in contact. What happens?
There is no temperature difference to drive a net flow.No net energy transfer — they’re already in thermal equilibriumEnergy is still swapped both ways, but equally, so neither block’s temperature changes. “In contact” doesn’t mean energy has to flow — a temperature difference does.
WE 3
Boiling water at 100 °C is mixed with an equal mass of cold water at 20 °C, with no heat lost to the surroundings. Without a full calculation, what is the final temperature?
Energy lost by the hot water = energy gained by the cold water.Equal masses of the same substance → each changes temperature by the same amount, so the final temperature is the midpoint.T = (100 + 20) ÷ 2T = 60 °CThe midpoint shortcut only works for equal masses of the same substance. Different masses or materials need the full mcΔT balance — that’s the specific heat capacity page.
🔧 Thinking through an equilibrium problem
Which is hotter? Net energy flows from hot to cold — every time.
In contact? No thermal contact means no transfer between them.
Track both: the hot object cools, the cold object warms.
Stop point: equal temperatures — net transfer becomes zero.
Final temperature lies between the two starting values (midpoint only for equal masses of the same substance).
hotter object cools
energy hot → cold
cooler object warms
until equal T
thermal equilibrium
Quick recap: Thermal energy flows from hot to cold whenever there’s a temperature difference and the objects are in thermal contact. The hot object cools and the cold one warms until they reach a shared equilibrium temperature, where the net transfer is zero. Energy is still exchanged both ways at equilibrium — just equally — so the temperatures hold. The final temperature lands between the two starting values.
💡 Top tips
Temperature drives the flow, not stored energy. Hot → cold, regardless of which object holds more energy.
Equilibrium = equal temperature, not equal energy and not “nothing happening”.
Say “net”. At equilibrium energy still flows both ways; it’s the net transfer that’s zero.
Contact matters. No thermal contact (a good insulator between them) means no transfer.
Final temperature is in between. It can only be the simple midpoint for equal masses of the same substance.
⚠ Common mistakes
Saying energy flows from the object with more energy to the one with less — it flows from hotter to cooler
Thinking all energy exchange stops at equilibrium — it continues, but the net transfer is zero
Assuming two touching objects must swap energy — not if they’re already at the same temperature
Expecting the final temperature to be the average when the masses or materials differ
Forgetting that thermal contact is needed for any transfer to happen
So temperatures always chase each other until they match. But how fast that happens, and how much energy it takes to shift a temperature at all, depends on the material — water is famously sluggish to heat, metals are quick. Next we put a number on it: specific heat capacity, and the mcΔT equation that finishes off mixing problems like the one above.
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