Sections#

Review & Summary#

Temperature; Thermometers#

Temperature is an SI base quantity related to our sense of hot and cold. It is measured with a thermometer, which contains a working substance with a measurable property (such as length or pressure) that changes in a regular way as the substance becomes hotter or colder.

Zeroth Law of Thermodynamics#

When a thermometer and some other object are placed in contact, they eventually reach thermal equilibrium. The zeroth law of thermodynamics: If bodies \(A\) and \(B\) are each in thermal equilibrium with a third body \(C\) (the thermometer), then \(A\) and \(B\) are in thermal equilibrium with each other.

The Kelvin Temperature Scale#

In the SI system, temperature is measured on the Kelvin scale, based on the triple point of water (273.16 K). With a constant-volume gas thermometer, the temperature \(T\) (in kelvins) is defined by

(142)#\[ T = (273.16\,\mathrm{K}) \frac{p}{p_3} \]

where \(p_3\) and \(p\) are the pressures of the gas at the triple point and at the measured temperature. The Celsius scale is \(T_C = T - 273.15\); the Fahrenheit scale is \(T_F = \frac{9}{5}T_C + 32°\).

Thermal Expansion#

For a temperature change \(\Delta T\), a change \(\Delta L\) in any linear dimension \(L\) is given by

(143)#\[ \Delta L = L \alpha \Delta T \]

in which \(\alpha\) is the coefficient of linear expansion. The change \(\Delta V\) in the volume \(V\) of a solid or liquid is \(\Delta V = V \beta \Delta T\), where \(\beta \approx 3\alpha\) is the coefficient of volume expansion.

Heat Capacity and Specific Heat#

If heat \(Q\) is absorbed by an object of mass \(m\), the temperature change \(\Delta T = T_f - T_i\) is related to \(Q\) by

(144)#\[ Q = C \Delta T = cm \Delta T \]

where \(C\) is the heat capacity and \(c\) is the specific heat of the material. The molar specific heat is the heat capacity per mole.

Heat of Transformation#

Heat absorbed by a material may change its physical state. The heat of transformation \(L\) is the energy per unit mass required to change the state (but not the temperature). Thus \(Q = Lm\). The heat of vaporization \(L_V\) and heat of fusion \(L_F\) apply to vaporization/condensation and melting/freezing, respectively.

Work Associated with Volume Change#

The work \(W\) done by a gas as it expands or contracts from volume \(V_i\) to \(V_f\) is

(145)#\[ W = \int_{V_i}^{V_f} p\,dV \]

The integration is necessary because the pressure \(p\) may vary during the volume change.

First Law of Thermodynamics#

The principle of conservation of energy for a thermodynamic process is expressed in the first law of thermodynamics:

(146)#\[ \Delta E_{\mathrm{int}} = Q - W \]

Here \(E_{\mathrm{int}}\) is the internal energy (a state function); \(Q\) is the energy transferred as heat (\(Q > 0\) if absorbed); \(W\) is the work done by the system (\(W > 0\) if the system expands). \(Q\) and \(W\) are path dependent; \(\Delta E_{\mathrm{int}}\) is path independent.

Heat Transfer Mechanisms#

Conduction: \(P = kA \Delta T / L\), where \(k\) is thermal conductivity, \(A\) is area, \(L\) is thickness. Convection: heat transfer by mass motion of fluid. Radiation: \(P = \sigma \varepsilon A T^4\) (Stefan–Boltzmann law), where \(\sigma\) is the Stefan–Boltzmann constant and \(\varepsilon\) is emissivity.