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.
\[
\Delta U = q + w
\]
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.
\[
w = -p\Delta V
\]
(negative because the system does the work).
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.
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), \(C{P} > C{V}\).
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.
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.
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.
\[
q{\text{metal}} + q{\text{calorimeter}} = q_{\text{water}}
\]
\[
m{\text{metal}}c{\text{metal}}\Delta T{\text{metal}} + C{\text{cal}}\Delta T{\text{cal}} = m{\text{water}}c{\text{water}}\Delta T{\text{water}}
\]
\[
c_{\text{metal}} =
\frac{m{\text{water}}c{\text{water}}\Delta T{\text{water}} - C{\text{cal}}\Delta T_{\text{cal}}}
{m{\text{metal}}\Delta T{\text{metal}}}
\]
Given: \(m = 2.0\;\text{kg}\), \(c = 900\;\text{J kg}^{-1}\text{K}^{-1}\), \(T{i}=20^{\circ}\text{C}\), \(T{f}=80^{\circ}\text{C}\).
Given: \(C_{V}= \dfrac{3}{2}R = 12.5\;\text{J mol}^{-1}\text{K}^{-1}\), \(\Delta T = 40\;\text{K}\).
\[
\Delta U = nC_{V}\Delta T = (1)(12.5)(40)=5.0\times10^{2}\;\text{J}
\]
For the same gas, \(C{P}=C{V}+R = \dfrac{5}{2}R = 20.8\;\text{J mol}^{-1}\text{K}^{-1}\).
\[
\Delta H = nC_{P}\Delta T = (1)(20.8)(40)=8.3\times10^{2}\;\text{J}
\]
| Material | Specific heat capacity \(c\) (J kg⁻¹ K⁻¹) |
|---|---|
| Water | 4180 |
| Aluminium | 900 |
| Iron | 450 |
| Copper | 385 |
| Glass | 840 |
| Air (constant volume) | 718 |
• 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.
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