Specific heat capacity and calorimetry
Heat capacity and specific heat
Put a pan with 1 litre of water and a pan with 2 litres on identical burners. The bigger amount takes about twice as long to warm up by the same amount.
Now heat 1 kilogram of water and 1 kilogram of iron on the same burner. The iron gets hot much faster. So the heat needed depends on the substance too.
If a body needs heat ΔQ to raise its temperature by ΔT, its heat capacity is S equals ΔQ over ΔT, in joules per kelvin.
Divide by the mass and you get a property of the substance alone: the specific heat capacity, s equals ΔQ over m ΔT. Its unit is joule per kilogram per kelvin.
So the heat needed is ΔQ equals m s ΔT. For water, s is 4186 joules per kilogram per kelvin. For aluminium it is about 900, for copper 386, and for lead only 128.
We can also count the amount of substance in moles. The heat needed per mole per kelvin is the molar specific heat capacity, C equals ΔQ over μ ΔT.
For a gas, C depends on how it is heated. Keep the pressure constant and it is C P; keep the volume constant and it is C V. C P is always larger than C V.
Water has the highest specific heat of common substances. That is why it is used as a coolant in car radiators and in hot-water bags: it takes in or gives out lots of heat for a small change in temperature.
It also explains sea breezes. In summer the land warms up much faster than the sea, so the wind from the sea has a cooling effect.
Calorimetry: the method of mixtures
When a hot body touches a cold one in an isolated system, heat flows from hot to cold until both reach the same temperature, and no heat escapes outside.
The principle of calorimetry says: heat lost by the hot body equals heat gained by the cold body.
A calorimeter is built for this. Watch a copper vessel with a stirrer, set inside a wooden jacket packed with an insulator like glass wool, so that very little heat leaks in or out.
Here is an NCERT example. A 0.047 kilogram aluminium sphere is left in boiling water until it is at 100 degrees Celsius.
It is quickly moved into a 0.14 kilogram copper calorimeter holding 0.25 kilogram of water at 20 degrees. Stir, and watch the temperatures meet. The water settles at 23 degrees.
The water and the calorimeter both warm by 3 degrees. The water gains 0.25 times 4186 times 3, and the copper 0.14 times 387 times 3. Together that is about 3300 joules.
The sphere cooled from 100 down to 23 degrees, a drop of 77 degrees. So it lost 0.047 times s times 77.
Set heat lost equal to heat gained: s equals 3300 divided by 0.047 times 77. The specific heat of aluminium comes out to about 911 joules per kilogram per kelvin.
The same idea works for any mixture, as long as we count every body that gains or loses heat, including the calorimeter itself.
Try it: the calorimeter
- This is the NCERT experiment. Press Drop it in (or Space).
- The two heat bars always end up equal: heat lost = heat gained.
- Change the metal and watch the final temperature change.
- Set the calorimeter mass to 0 to see how much it matters.
Aluminium or lead?
Equal masses of aluminium and lead, both at 100 °C, are dropped into two identical beakers of cold water. Which beaker of water ends up warmer?
- The one with lead, because lead is denser
- The one with aluminium, because aluminium has a higher specific heat
- Both the same, because both metals started at 100 °C
- The one with lead, because lead has a lower specific heat
Answer
B. Heat given out is m s ΔT. With the same m and nearly the same ΔT, aluminium (s ≈ 900 J kg⁻¹ K⁻¹) gives out about seven times as much heat as lead (s ≈ 128 J kg⁻¹ K⁻¹), so its water gets warmer. Starting at the same temperature does not mean carrying the same heat.
Mixing hot and cold water
200 g of water at 80 °C is mixed with 300 g of water at 20 °C in an insulated flask. Ignoring the flask, what is the final temperature?
- 44 °C
- 50 °C
- 56 °C
- 60 °C
Answer
A. Heat lost = heat gained: 0.2 × s × (80 − T) = 0.3 × s × (T − 20). The specific heat s cancels, so 16 − 0.2T = 0.3T − 6, giving 0.5T = 22 and T = 44 °C. The simple average, 50 °C, ignores the masses; 56 °C swaps them.
Summary
- Heat capacity:
S = ΔQ/ΔT - Specific heat capacity:
s = ΔQ/(m ΔT), so ΔQ = m s ΔT - Molar specific heat capacity:
C = ΔQ/(μ ΔT) - Principle of calorimetry:
heat lost = heat gained
- Specific heat is a property of the substance: water 4186, aluminium 900, copper 386, lead 128 J kg⁻¹ K⁻¹.
- For gases, C_P > C_V: the heat needed depends on whether the pressure or the volume is kept fixed.
- Water’s very high specific heat makes it a good coolant and keeps coastal climates mild.
- In a calorimeter, count every body that gains or loses heat, including the vessel.