Specific heat, calorimetry and change of state

Heat given to a body does one of two things: it raises its temperature, or it changes its state (melting, boiling). Never both at once.

For a change of temperature, Q = msΔT:

• Specific heat capacity s = Q/(mΔT), in J kg⁻¹ K⁻¹: the heat to warm 1 kg by 1 K.
• Heat capacity of a whole body = ms (J K⁻¹). Molar specific heat C = Q/(μΔT), per mole.
• Water's s is very high, 4186 J kg⁻¹ K⁻¹, so water warms and cools slowly. That is why seas keep coastal places mild, and why water is used to cool car engines.

Calorimetry: when hot and cold bodies are mixed in an insulated vessel, heat lost by the hot = heat gained by the cold. (A change of 1 °C is the same as 1 K, so temperature differences can be used in either.)

Example: 0.2 kg of iron (s = 450) at 200 °C is dropped into 0.5 kg of water at 20 °C. Let the final temperature be T: 0.2 × 450 × (200 – T) = 0.5 × 4186 × (T – 20). That gives 90(200 – T) = 2093(T – 20), so T ≈ 27.4 °C. The water barely warms, because its s is so large.

During a change of state the temperature stays constant until the change is complete. The heat needed is Q = mL, where L is the latent heat:

• Fusion (melting ice at 0 °C): L_f = 3.33 × 10⁵ J kg⁻¹.
• Vaporisation (boiling water at 100 °C): L_v = 22.6 × 10⁵ J kg⁻¹, almost seven times larger.
• A heating curve is a staircase: sloping parts where the temperature rises (msΔT), flat parts where the state changes (mL).

This is why a steam burn is worse than a burn from boiling water: each gram of steam at 100 °C first gives up 2260 J as it condenses on the skin, and only then the heat of the hot water.

Melting and boiling points depend on pressure. More pressure raises the boiling point (a pressure cooker cooks hotter than 100 °C) and lowers the melting point of ice: a loaded wire sinks through a block of ice, which refreezes above it (regelation).

Gases made of two-atom molecules (oxygen, nitrogen) have larger molar specific heats than one-atom gases (helium, argon), because their molecules can also rotate, and sometimes vibrate, so they soak up more energy for each degree.