relate a rise in temperature of an object to an increase in its internal energy

Internal Energy

Objective

Relate a rise in temperature of an object to an increase in its internal energy.

Key Concepts

• Internal energy ($U$) – the total kinetic and potential energy of the particles within a system.
• Temperature – a measure of the average kinetic energy of the particles.
• Specific heat capacity ($c$) – the amount of energy required to raise the temperature of 1 kg of a substance by 1 K.

Mathematical Relationship

The change in internal energy for a closed system with no work done is given by

$$\Delta U = Q$$

where $Q$ is the heat added. For a uniform substance:

$$\Delta U = mc\Delta T$$

Here $m$ is the mass, $c$ the specific heat capacity and $\Delta T$ the temperature change.

Derivation

1. Start with the definition of heat capacity: $C = \frac{Q}{\Delta T}$.
2. For a mass $m$, the specific heat capacity is $c = \frac{C}{m}$, so $C = mc$.
3. Substituting $C$ into the first equation gives $Q = mc\Delta T$.
4. Since $\Delta U = Q$ for a process with no work, we obtain $\Delta U = mc\Delta T$.

Example Calculation

How much does the internal energy of a 2.0 kg aluminium block increase when its temperature rises from $20^\circ\text{C}$ to $80^\circ\text{C}$? (Aluminium $c = 900\ \text{J kg}^{-1}\text{K}^{-1}$.)

1. Calculate the temperature change: $\Delta T = 80 - 20 = 60\ \text{K}$.
2. Apply $\Delta U = mc\Delta T$:
$$\Delta U = (2.0\ \text{kg})(900\ \text{J kg}^{-1}\text{K}^{-1})(60\ \text{K}) = 1.08\times10^{5}\ \text{J}.$$

The internal energy of the block increases by $1.08\times10^{5}\ \text{J}$.

Table of Common Specific Heat Capacities

Material	Specific heat capacity $c$ (J kg⁻¹ K⁻¹)
Water	4180
Aluminium	900
Iron	450
Copper	385
Glass	840

Conceptual Questions

• Why does the internal energy increase even though the macroscopic shape of the object does not change?
• How would the relationship change if the object does work on its surroundings while heating?
• Explain why gases generally have higher specific heat capacities per mole than solids.

Summary

For a closed system where no work is done, a rise in temperature directly reflects an increase in internal energy. The quantitative link is $\Delta U = mc\Delta T$, where $c$ encapsulates the microscopic ability of the material to store thermal energy.