Physics Bootcamp

Subsection 19.3.1 Thermal Equilibrium

Temperature is actually a property associated with the state of thermal equilibrium. Suppose you place a wooden block, initially at room temperature, into an oven whose temperature is set at \(300^{\circ}\text{F}\text{.}\) After some time has passed, you will notice that the wooden block's temperature is also \(300^{\circ}\text{F}\text{.}\) We say that the system (the block) has reached in thermal equilibrium with the environment (the oven) due to the fact that wooden block was not insulated from the oven, but could exchange heat with it.

Generally, all systems that are in thermal contact with each other will reach thermal equilibrium with every system having the same temperature.

One important consequence of the definition of temperature in terms of the thermal equilibrium is that if a system is in thermal equilibrium internally, the temperature at all points inside the system will be the same. When this happens we say that the system is at a particular temperature. If a system is not in thermal equilibrium internally, then the system would have different temperatures at different points inside the system. For instance, when you heat water on a hot plate, the water near the plate is usually at a higher temperature than the water at points further away.

Subsection 19.3.2 Zeroth Law of Thermodynamics

The zeroth law of thermodynamics states that if a system A is in thermal equilibrium with a system B, which, in turn, is in thermal equilibrium with another system C, then systems A and C are also in thermal equilibrium with each other regardless of whether or not A and C are in direct contact.

Thus all systems at the same temperature can be thought to be in thermal equilibrium with each other, or equivalently, all systems in thermal equilibrium must have the same temperature. If two systems, which are not in thermal contact but have the same temperature, are brought in thermal contact, then there will be no heat transfer between them.

Subsection 19.3.3 Temperature Scales

To make temperature a more quantitative property we need an operational definition of temperature which we can implement with real materials. In particular, we need to find reproducible thermal equilibrium situations, to which numerical values of temperature may be assigned, and track a property that varies with temperature as a conduit of the value of temperature.

Historically, freezing and boiling points of water at atmospheric pressure were selected as highly reproducible equilibrium states of matter. The two common scales of temperature, the Celsius or centigrade (C) and Fahrenheit (F), use these reference states.

In the Celsius scale, we assign a value of zero degree to the freezing point of water and a value of 100 degrees to the boiling point of water. In the Fahrenheit scale, the freezing point is assigned 32 degrees and the boiling point 212 degrees. There are 180 degrees in Fahrenheit scale between the freezing and boiling points of water while there are only 100 degrees in the Celsius scale.

Therefore, each degree change in the Celsius scale is equal to \(\dfrac{180}{100}\) or \(\dfrac{9}{5}\) degrees change in the Fahrenheit scale. If a particular temperature reads C degrees in Celsius and F degrees in Fahrenheit, then their relation will be:

In scientific work, another temperature scale, called the Absolute or kelvin scale is often used, especially in low-temperature physics or cryogenics. This scale has a strong theoretical basis with claims that zero temperature in this scale may never be breached. A reading in kelvin scale to be denoted in this book by the capital letter \(T\) is related to the Celsius scale reading \(\text{C}\) for the same condition as follows.

Since 1954, the temperature scale has been fixed by first assigning a temperature of 273.16 degrees to the triple point of water in the Absolute or kelvin (K) temperature scale. The triple point is defined as the state when all three phases of water, namely ice, liquid water and vapor are in equilibrium at atmospheric pressure, i.e., a mixture of iced water with vapor above it. We assign it a value of \(273.16K\text{.}\)

We assign temperature for other conditions by the process described in Subsection 19.3.4 below.

In nature we find a wide range of temperature as shown in the figure. the temperature of the universe as seen in the background radiation is approximately \(2.7\text{K}\) and \(\gt 10\text{ MK}\) inside stars. In laboratories physicists have been able to push the temperatures to as low as \(100\text{ pK}\) in a piece of rhodium metal and as high as \(1\text{ GK} \) in quark-gluon plasma.

Subsection 19.3.4 Calibration Using The Triple Point of Water

Since 1954 the triple point of water, a state in which a mixture of ice, liquid water and vapor coexist in equilibrium, has been chosen as the standard reference which is arbitrarily assigned a value of 273.16 K.

Let \(X\) be a property that we use for the thermometer, e.g., the volume of a mercury thermometer. Let the value of \(X \) at the triple point be \(X = X_\text{TP}\text{.}\) Then, a thermometer that is in equilibrium with triple point water will have

for some value of \(a\text{.}\) Now, we want to know the temperature of something else. Suppose when we place the thermometer in contact with the unknown, we find that value of \(X=X_\text{T}\text{,}\) then

Dividing this equation by (19.3.4) we remove the unknown constant \(a\) and obtain the value of the unknown temperature from measured value of X.

Subsection 19.3.5 Change in Temperature in Different Scales

Many thermodynamic properties are stated as per unit change in temperature. For instance, we would like to know how much length changes if we change the temperature by \(1^{\circ}\text{C}\) versus by \(1^{\circ}\text{F}\text{.}\) They are not the same change.

Let \(\Delta C \) be a change in temperature stated in the Celsius scale, and similarly \(\Delta F \) for Fahrenheit and \(\Delta T \) in kelvin scale. Their relation will be