Intuition The one core ideaHeating a substance means pouring energy in to make its particles jiggle faster , and how fast the temperature climbs depends on the material, the amount, and how much energy you add. Calorimetry is nothing but honest energy accounting : in a sealed cup, every joule the hot thing loses is a joule the cold thing gains — none goes missing.
Before you can read Q = m c Δ T what every letter stands for and what picture lives behind it . This page builds each one from nothing, in an order where each idea leans on the previous. If you already met these in Heat and Internal Energy , this is your slow, visual refresher.
==Temperature T how vigorously the particles of a substance are moving (vibrating, if it's a solid). Hotter = faster jiggling. We measure it in degrees Celsius (°C) or in kelvin (K).
Picture a box of tiny balls (atoms). At a low temperature they twitch gently; at a high temperature they zip and bounce hard. Temperature is just a readout of that average liveliness — not how much stuff there is, just how energetic each piece is.
Intuition Why the topic needsT changing temperature by adding or removing energy. So T
Δ Δ "the change in" or "final minus initial" . So Δ T Δ T = T final − T initial
Look at a thermometer that started at 20 and ended at 75. The reading is a position; the change Δ T = 75 − 20 = 55
difference , not the raw temperature?
Heat cares about how much you moved the temperature, not where it started. Warming from 10→20 costs the same as 90→100 (both Δ T = 10 Δ T T
Common mistake "°C and K give differentΔ T Why it feels right: absolute-temperature formulas exist elsewhere.
Fix: the Celsius and kelvin scales are the same size step, just shifted by 273.15. A gap of 55 °C is a gap of 55 K. For a difference you may use either scale — the shift cancels out.
==Mass m how much substance you have, in kilograms (kg). More mass = more particles to heat.
Two cups of water side by side: one holds 0.5 kg, the other 1 kg. The big one has twice as many water molecules — twice as many little balls to get jiggling.
Intuition Why the topic needsm m
Q Q energy that flows from a hotter place to a cooler place simply because there is a temperature difference. Measured in joules (J). 1 J
Q > 0 into the object (it warms up).Q < 0 out (it cools down).
Picture heat as an invisible fluid trickling downhill from hot to cold. It always runs down the temperature slope, never up on its own.
Intuition Heat vs. temperature — don't confuse themTemperature is how hot (how fast the jiggling). Heat is how much energy moved . A swimming pool at 30 °C holds far more heat energy than a coffee at 80 °C, even though the coffee is hotter — because the pool has vastly more particles. This distinction is the heart of the chapter; see Heat and Internal Energy .
Q Q
Definition Specific heat capacity==Specific heat capacity c 1 kg of a substance by 1 K (= 1 °C) . Units: J kg − 1 K − 1
The "per kilogram per kelvin" is the recipe for one standard unit of heating: one kilo, one degree. Water's c ≈ 4186
c Δ T how much and how far ; they say nothing about which material . Different materials store energy differently, so we need a per-material constant. That constant, c
Definition Heat capacity (whole object)==Heat capacity C the entire object (whatever its mass) by 1 K. Units: J K − 1 no "per kg", because the mass is already baked in:
C = m c
Small c material ). Capital C specific object in front of you . A 3 kg copper block and a 1 kg copper block share the same c C
Recall Quick self-checkWhich one changes if you cut the block in half? ::: C c
Definition Water equivalentThe ==water equivalent w mass of water that would soak up the same heat as the object for the same temperature rise. Setting their heat capacities equal:
w c water = m c ⇒ w = c water m c
Trick: instead of tracking a copper cup with its own c
Definition Subscripts labelwhich object
A small number or word written low, like m 1 c water "this belongs to object 1 / the water" . It is a name tag, not a mathematical operation.
m 1 , c 1 , T 1 hot body.m 2 , c 2 , T 2 cold body.T f final temperature after they settle.
Intuition Why we need name tags at allCalorimetry mixes two or more things. Without labels we could not tell whose mass or whose temperature we mean. The subscript keeps each object's bookkeeping in its own column.
Intuition Nothing leaks, nothing appearsIn a perfectly insulated cup, energy is neither created nor destroyed (see Conservation of Energy and the First Law of Thermodynamics ). So the energy leaving the hot body has nowhere to go except into the cold body:
Heat lost by hot = Heat gained by cold Q = m c Δ T
Read each and try to answer before revealing. If any one stumps you, reread its section above.
What does the symbol Δ "Change in" — final minus initial.
What does Δ T T final − T initial
Is a gap of 40 °C the same as a gap of 40 K? Yes — the scales differ only by a constant offset, so differences match.
What does temperature T How vigorously the particles jiggle (their average liveliness).
What is the difference between heat Q T Q T
What is the sign of Q Negative — energy flows out.
What are the units of specific heat capacity c J kg − 1 K − 1
What is specific heat c Heat to raise 1 kg of a substance by 1 K.
How do C c C = m c c
What does a subscript like m 1 Which object the quantity belongs to — a name tag.
What is water equivalent w The mass of water that absorbs the same heat for the same Δ T w = m c / c water
Which law lets "heat lost = heat gained"? Conservation of energy in an insulated system.
