Know that a rise in the temperature of an object increases its internal energy

2.2.2 Specific Heat Capacity

Learning Objective

Understand that a rise in the temperature of an object increases its internal energy and be able to use the quantitative relationship between heat energy, mass, specific heat capacity and temperature change.

Notation Box

Key Concepts

• Internal energy (U): the total kinetic energy of all the particles in a substance. When the temperature rises, the average speed of the particles increases, so the total kinetic energy – and therefore U – increases.
• Temperature (T): a measure of the average kinetic energy of the particles. Higher T means higher average particle speed.
• Specific heat capacity (c): an intrinsic property of a material that tells how much heat is needed to raise the temperature of 1 kg of that material by 1 K (or 1 °C).
• Heat energy (Q): energy transferred because of a temperature difference. Units: J.

Fundamental Relationship

The quantitative link between heat, mass, specific heat capacity and temperature change is

\[

Q = mc\Delta T

\]

Re‑arranged to find any missing quantity:

• \(m = \dfrac{Q}{c\Delta T}\)
• \(c = \dfrac{Q}{m\Delta T}\)
• \(\Delta T = \dfrac{Q}{mc}\)

Why a Rise in Temperature Increases Internal Energy

When heat \(Q\) is supplied, the particles gain kinetic energy and move faster. Because internal energy is the sum of the kinetic energies of all particles, any increase in temperature (i.e. an increase in average kinetic energy) directly raises \(U\). For the IGCSE level we can state:

Higher temperature → higher particle speeds → higher internal energy.

If no work is done, the change in internal energy equals the heat transferred:

\[

\Delta U = Q \quad (\text{no work})

\]

Typical Specific Heat Capacities

Measuring Specific Heat Capacity

1. Solid (Calorimetry with water)

1. Apparatus

   • Calorimeter (insulated container with lid)
   • Thermometer (±0.1 °C)
   • Electronic balance
   • Solid sample of unknown \(c\)
   • Known mass of water (reference material, \(c_{\text{w}} = 4180\) J kg⁻¹ K⁻¹)
   • Stirring rod, safety goggles, heat‑proof gloves

2. Procedure

   1. Weigh the empty calorimeter, \(m_{\text{cal}}\).
   2. Add a known mass of water, \(m{\text{w}}\), and record its initial temperature, \(T{\text{i}}\).
   3. Weigh the dry solid, \(m_{\text{s}}\).
   4. Heat the solid in a beaker of hot water until its temperature is a few degrees above \(T{\text{i}}\); record this temperature as \(T{\text{s}}\).
   5. Quickly transfer the hot solid into the calorimeter, close the lid and stir gently.
   6. When the temperature stabilises, record the final equilibrium temperature, \(T_{\text{f}}\).

3. Data handling (no heat loss assumed)

   \[

   m{\text{s}}c{\text{s}}(T{\text{s}}-T{\text{f}})

   = m{\text{w}}c{\text{w}}(T{\text{f}}-T{\text{i}})

   + m{\text{cal}}c{\text{cal}}(T{\text{f}}-T{\text{i}})

   \]

   \[

   c_{\text{s}}=

   \frac{m{\text{w}}c{\text{w}}(T{\text{f}}-T{\text{i}})

   + m{\text{cal}}c{\text{cal}}(T{\text{f}}-T{\text{i}})}

   {m{\text{s}}(T{\text{s}}-T_{\text{f}})}

   \]

4. Safety notes

   • Use heat‑proof gloves when handling hot water or the hot solid.
   • Wear safety goggles to protect against splashes.
   • Do not over‑fill the calorimeter; leave space for the solid.

2. Liquid (Direct heating method)

1. Apparatus

   • Insulated container (or calorimeter)
   • Thermometer
   • Electronic balance
   • Electric heater or spirit burner
   • Known mass of the liquid, \(m\)

2. Procedure

   1. Weigh the empty container, then add the liquid and re‑weigh to obtain \(m\).
   2. Record the initial temperature, \(T_{\text{i}}\).
   3. Supply a known amount of heat, \(Q\) (e.g., by measuring the electrical energy: \(Q = VIt\)).
   4. Stir continuously and record the final temperature, \(T_{\text{f}}\).

3. Calculation

   \[

   c = \frac{Q}{m\Delta T}\qquad\text{where }\Delta T = T{\text{f}}-T{\text{i}}

   \]

Worked Example (Heat Required for Water)

1. Calculate the heat energy required to raise 250 g of water from \(20 °C\) to \(80 °C\).
2. Given: \(c_{\text{water}} = 4180\; \text{J kg}^{-1}\text{K}^{-1}\).

Solution

• Convert mass: \(m = 250\;\text{g}=0.250\;\text{kg}\).
• Temperature change: \(\Delta T = 80-20 = 60\;\text{K}\).
• Apply the formula:

\[

Q = (0.250\;\text{kg})(4180\;\text{J kg}^{-1}\text{K}^{-1})(60\;\text{K})

= 6.27\times10^{4}\;\text{J}

\]

Therefore, \(62.7\;\text{kJ}\) of heat must be supplied.

Implications for Everyday Situations

• Heating water on a stove needs more energy than heating the same mass of oil because water’s \(c\) is larger.
• Metal cookware (low \(c\)) heats up and cools down quickly; ceramic or glass (higher \(c\)) changes temperature more slowly and retains heat longer.
• Thermal‑energy‑storage systems (e.g., solar‑heated water tanks, molten‑salt tanks) use materials with high specific heat capacities to store large amounts of heat.

Common Misconceptions

1. “All substances heat up at the same rate.” – Wrong. The rate of temperature change for a given amount of heat depends on the material’s specific heat capacity.
2. “Specific heat capacity is the same as heat capacity.” – Heat capacity \(C\) refers to a particular object (\(C = mc\)), whereas specific heat capacity \(c\) is an intrinsic property of the material.
3. “Temperature and internal energy are the same thing.” – Temperature measures the average kinetic energy per particle; internal energy is the total kinetic energy of all particles.

Practice Questions

1. A 1.5 kg aluminium block is heated from \(25 °C\) to \(75 °C\). Calculate the heat energy absorbed.

   (\(c_{\text{Al}} = 900\;\text{J kg}^{-1}\text{K}^{-1}\))

2. Why does a metal spoon become hot faster than a wooden spoon when placed in a cup of hot tea?
3. If \(2000\;\text{J}\) of heat is removed from 0.5 kg of copper, what is the resulting temperature drop?

   (\(c_{\text{Cu}} = 385\;\text{J kg}^{-1}\text{K}^{-1}\))

Summary

• Increasing the temperature of an object raises its internal energy because particle kinetic energy increases.
• The quantitative link between heat, mass, specific heat capacity and temperature change is \(Q = mc\Delta T\).
• Specific heat capacity is the amount of heat required to raise 1 kg of a material by 1 K (or 1 °C); the numerical value is the same for both units.
• Students should be able to describe a simple calorimetry experiment for a solid or a liquid to determine \(c\).
• Knowing typical values of \(c\) helps explain everyday phenomena and solve exam‑style problems.