Describe an increase in temperature of an object in terms of an increase in the average kinetic energies of all of the particles in the object

2.2.2 Specific Heat Capacity

Learning Objective

Describe an increase in temperature of an object in terms of an increase in the average kinetic energies of all of the particles in the object.

Key Concepts

• Temperature – a measure of the average kinetic energy of the particles in a substance.
• Average kinetic energy (\$\overline{KE}\$) – for a particle of mass \$m\$ moving with speed \$v\$, \$KE = \frac12 mv^2\$. In a solid or liquid the particles vibrate; the average of these motions determines the temperature.
• Specific heat capacity (\$c\$) – the amount of energy required to raise the temperature of 1 kg of a substance by 1 °C (or 1 K). Units: \$\text{J kg}^{-1}\text{K}^{-1}\$.
• Heat energy (\$Q\$) – the total energy transferred as a result of a temperature difference.

Relationship Between Heat, Temperature and Kinetic Energy

The internal energy \$U\$ of a macroscopic object is the sum of the kinetic energies of all its particles (translational, rotational, vibrational) and any potential energy associated with intermolecular forces. When heat \$Q\$ is added:

\$\Delta U = Q = m c \Delta T\$

Because \$U\$ is essentially the total kinetic energy of the particles, an increase in \$Q\$ leads to an increase in the average kinetic energy \$\overline{KE}\$ of each particle, which we perceive as a rise in temperature \$T\$.

Derivation of the Specific Heat Formula

1. Consider a mass \$m\$ of a substance with specific heat capacity \$c\$.
2. If an amount of energy \$Q\$ is supplied uniformly, the temperature rises by \$\Delta T\$.
3. By definition, \$c = \dfrac{Q}{m\Delta T}\$, rearranged to \$Q = mc\Delta T\$.
4. Since \$Q\$ changes the internal kinetic energy, \$\Delta U = m \Delta \overline{KE} = mc\Delta T\$.

Typical \cdot alues of Specific Heat Capacity

Connecting Temperature Change to Particle Kinetic Energy

For an ideal monatomic gas, the average translational kinetic energy per molecule is given by:

\$\overline{KE} = \frac{3}{2}k_{\mathrm{B}}T\$

where \$k_{\mathrm{B}}\$ is Boltzmann’s constant (\$1.38\times10^{-23}\,\text{J K}^{-1}\$). Although solids and liquids have more complex motions, the proportionality between temperature and average kinetic energy still holds.

Worked Example

Problem: How much heat is required to raise the temperature of a 250 g aluminium block from \$20^\circ\text{C}\$ to \$80^\circ\text{C}\$?

1. Identify the data:

   • Mass \$m = 250\ \text{g} = 0.250\ \text{kg}\$
   • Specific heat capacity of aluminium \$c = 900\ \text{J kg}^{-1}\text{K}^{-1}\$
   • Temperature change \$\Delta T = 80 - 20 = 60\ \text{K}\$

2. Apply \$Q = mc\Delta T\$:

   \$Q = (0.250\ \text{kg})(900\ \text{J kg}^{-1}\text{K}^{-1})(60\ \text{K}) = 13\,500\ \text{J}\$

3. Interpretation: The supplied \$13.5\ \text{kJ}\$ of energy increases the average kinetic energy of each aluminium atom, producing the observed temperature rise.

Common Misconceptions

• “Heat is a substance.” – Heat is energy in transit; it is not stored in the object.
• “All materials heat up at the same rate.” – The rate depends on specific heat capacity; water heats more slowly than metals for the same energy input.
• “Temperature and kinetic energy are unrelated for liquids.” – Even in liquids, temperature reflects the average kinetic energy of molecular motion.

Summary

Increasing the temperature of an object means that the average kinetic energy of its constituent particles has increased. The quantitative link is provided by the specific heat capacity \$c\$, which relates the amount of heat energy \$Q\$ added to the mass \$m\$ and the resulting temperature change \$\Delta T\$ through \$Q = mc\Delta T\$. Understanding this relationship allows us to predict how different substances respond to heat and to calculate energy requirements in practical situations.