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

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.