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

Internal Energy – Cambridge International AS & A Level Physics (9702)

Objective

To relate a rise in temperature of an object to an increase in its internal energy, to understand the limits of this relationship, and to connect the concept with kinetic theory, heat capacities and the first law of thermodynamics.

1. Definitions & Core Concepts

• Internal energy, \(U\) – total kinetic + potential energy of all particles in a system. It is a state function; therefore \(\Delta U\) depends only on the initial and final states, not on the path taken.
• Temperature, \(T\) – a measure of the average kinetic energy of the particles.
• Specific heat capacity, \(c\) – energy required to raise the temperature of 1 kg of a substance by 1 K (units J kg⁻¹ K⁻¹).
• Heat capacity, \(C\) – energy required to raise the temperature of the whole body by 1 K (units J K⁻¹).
  \(C = mc\)
• First law of thermodynamics (closed system) – \[ \Delta U = q + w \] where \(q\) is heat added to the system (positive when heat enters) and \(w\) is work done on the system (positive when work is done on the system). Cambridge sign convention is used throughout.
• Work term for gases – at constant external pressure \(p\), the PV‑work is \[ w = -p\Delta V \] (negative because the system does the work).

2. When No Work Is Done (solids & liquids)

For most solid and liquid examples the volume change is negligible, so \(w = 0\). The first law reduces to

\[ \Delta U = q = mc\Delta T \]

Boxed remark: The simple relation \(\Delta U = mc\Delta T\) is valid **only when no PV‑work is performed**. If the object expands while heating, an additional work term must be included.

3. Internal Energy of an Ideal Gas

3.1 Derivation of \(\Delta U = nC_{V}\Delta T\)

1. For an ideal gas at constant volume, \(w = 0\) (no expansion work).
2. Apply the first law: \(\Delta U = q_{V}\).
3. By definition, the molar heat capacity at constant volume is \(C_{V} = \dfrac{q_{V}}{n\Delta T}\).
4. Hence \(\boxed{\Delta U = nC_{V}\Delta T}\).

3.2 Enthalpy Change at Constant Pressure

When the gas is heated at constant pressure, heat supplied is \(q_{P}=nC_{P}\Delta T\). The energy balance is

\[ \Delta U = q_{P} - p\Delta V \qquad\text{or}\qquad \boxed{\Delta H = nC_{P}\Delta T} \]

Because \(p\Delta V\) is positive (the system does work), \(C_{P} > C_{V}\).

3.3 Kinetic‑Theory Link

For a mono‑atomic ideal gas the internal energy is purely translational kinetic energy:

\[ U = \frac{3}{2}nRT \;\;\Longrightarrow\;\; \Delta U = \frac{3}{2}nR\Delta T \]

This expression is consistent with \(\Delta U = nC_{V}\Delta T\) because \(C_{V}= \tfrac{3}{2}R\) for a mono‑atomic gas.

3.4 Solids & Liquids

For condensed phases the volume change on heating is very small, so \(C_{V}\approx C_{P}\). Consequently the same simple relation \(\Delta U = mc\Delta T\) (or \(\Delta U = nC_{V}\Delta T\) per mole) is usually sufficient.

4. Practical Determination – Calorimetry (AO3)

4.1 Typical Set‑up

A known mass of a hot metal is placed in a calorimeter containing a known mass of water. The temperature rise of the water (and calorimeter) is recorded.

4.2 Energy Balance (neglecting heat loss)

\[ q_{\text{metal}} + q_{\text{calorimeter}} = q_{\text{water}} \] \[ m_{\text{metal}}c_{\text{metal}}\Delta T_{\text{metal}} + C_{\text{cal}}\Delta T_{\text{cal}} = m_{\text{water}}c_{\text{water}}\Delta T_{\text{water}} \]

4.3 Solving for the Metal’s Specific Heat

\[ c_{\text{metal}} = \frac{m_{\text{water}}c_{\text{water}}\Delta T_{\text{water}} - C_{\text{cal}}\Delta T_{\text{cal}}} {m_{\text{metal}}\Delta T_{\text{metal}}} \]

4.4 Error Analysis

• Random errors: repeat the experiment, calculate the mean and standard deviation of the obtained \(c\) values.
• Systematic errors: heat loss to surroundings, inaccurate mass/temperature readings, calorimeter heat capacity not accounted for – discuss the direction of the resulting bias.

5. Example Calculations

5.1 Solid – Aluminium Block

Given: \(m = 2.0\;\text{kg}\), \(c = 900\;\text{J kg}^{-1}\text{K}^{-1}\), \(T_{i}=20^{\circ}\text{C}\), \(T_{f}=80^{\circ}\text{C}\).

1. \(\Delta T = 60\;\text{K}\)
2. \(\Delta U = mc\Delta T = (2.0)(900)(60)=1.08\times10^{5}\;\text{J}\)

5.2 Gas – 1 mol of Mono‑atomic Ideal Gas (constant volume)

Given: \(C_{V}= \dfrac{3}{2}R = 12.5\;\text{J mol}^{-1}\text{K}^{-1}\), \(\Delta T = 40\;\text{K}\).

\[ \Delta U = nC_{V}\Delta T = (1)(12.5)(40)=5.0\times10^{2}\;\text{J} \]

5.3 Gas – 1 mol at Constant Pressure (Cp)

For the same gas, \(C_{P}=C_{V}+R = \dfrac{5}{2}R = 20.8\;\text{J mol}^{-1}\text{K}^{-1}\).

\[ \Delta H = nC_{P}\Delta T = (1)(20.8)(40)=8.3\times10^{2}\;\text{J} \]

6. Common Specific Heat Capacities

Material	Specific heat capacity \(c\) (J kg⁻¹ K⁻¹)
Water	4180
Aluminium	900
Iron	450
Copper	385
Glass	840
Air (constant volume)	718

7. Conceptual (AO2) Questions

• Why does the internal energy increase even though the macroscopic shape of a solid does not change?
• How would the relationship \(\Delta U = mc\Delta T\) be modified for a material that expands while being heated?
• Explain why gases have larger molar heat capacities than solids.
• Discuss the significance of \(U\) being a state function when a system undergoes a heating‑then‑cooling cycle.
• Using kinetic theory, derive \(\Delta U = \frac{3}{2}nR\Delta T\) for a mono‑atomic ideal gas.

8. Links to Other Syllabus Sections

• Kinetic theory of gases (Section 15.3) – internal energy of an ideal gas is kinetic, \(U = \frac{3}{2}nRT\) for mono‑atomic gases.
• Phase changes (Section 14.3) – during melting or boiling the temperature remains constant while internal energy rises by the latent heat, \(Q = mL\).
• Energy transfer (Section 13) – heat flow \(Q = \dot{Q}t\) links power to heating rates.
• Thermodynamics (Section 16.2) – the first law \(\Delta U = q + w\) underpins later topics such as enthalpy, calorimetry of gases and the concept of state functions.

9. Summary Box

• Internal energy is a state function: \(\Delta U\) depends only on the initial and final states.
• No work (solids & liquids): \(\boxed{\Delta U = mc\Delta T}\).
• Ideal gas, constant volume: \(\boxed{\Delta U = nC_{V}\Delta T}\).
• Ideal gas, constant pressure (enthalpy): \(\boxed{\Delta H = nC_{P}\Delta T}\).
• For a mono‑atomic gas, \(C_{V}= \tfrac{3}{2}R\) and \(U = \tfrac{3}{2}nRT\).
• For liquids/solids \(C_{V}\approx C_{P}\) because volume change on heating is negligible.
• Calorimetry provides experimental values of \(c\) and requires careful error analysis.