In Solids, Liquids and Gases you saw that particles never stop moving, and Temperature Scales and Absolute Zero gave you kelvins. Heat and temperature both describe that motion, but they are not the same thing.
Key idea
Temperature measures the average kinetic energy of the particles in a sample. Thermal energy is the total energy of their motion, so it also depends on how much material there is. Heat is energy that flows from a hotter object to a colder one because of their temperature difference, until they reach the same temperature (thermal equilibrium).
Why it works
Where a hot object touches a cold one, its fast particles bump into slower ones and, on average, pass energy to them. The reverse bumps also happen, but less energy goes that way, so the net flow is always from hot to cold. The flow stops when the averages match. Nothing called "cold" ever flows: a cold object only takes in less energy than it gives off, or gains energy from you.
Worked example: Mixing hot and cold water
$100\ \text{g}$ of water at $80^\circ\text{C}$ is mixed with $100\ \text{g}$ of water at $20^\circ\text{C}$ in an insulated cup. What is the final temperature?
- The hot water's energy loss equals the cold water's gain.
- Same substance, same mass: each changes by the same number of degrees, so they meet halfway.
- $\dfrac{80 + 20}{2} = 50^\circ\text{C}$.
Answer: $50^\circ\text{C}$.
Worked example: Bathtub or teacup?
Which has more thermal energy: a bathtub of water at $40^\circ\text{C}$ or a cup of tea at $80^\circ\text{C}$?
- The tea is at the higher temperature: its particles move faster on average.
- The bathtub holds hundreds of times more water, so its total energy of motion is far larger.
Answer: The bathtub, though the tea is hotter.
Common mistake
Treating heat and temperature as the same thing, or thinking cold flows from a cold object into a warm one. Ice in your hand feels cold because heat leaves your hand and goes into the ice. Check any heat question by asking which object is hotter: energy always flows away from it.
Watch two objects reach the same temperature in Temperature and the Zeroth Law.