Peter's Physics Pages

A Semester of First Year Physics with Peter Eyland

Lecture 4 (Temperature)

In this lecture the following are introduced:
• Temperature as the average kinetic energy per molecule.
• The zeroth law of thermodynamics
• Converting between Celsius and Fahrenheit
• Converting between Celsius and Kelvin
• The Constant-Volume Gas Temperature
• Comparing Temperature Scales

Temperature

Some definitions:
The heat that flows to or from a body is a total amount of kinetic energy.
A diathermal wall is one that heat can pass through [Greek "dia" = through]. It is made from a thermal conductor.
An adiabatic wall is one that heat cannot pass through [Greek = not cross through]. It is made from a thermal insulator.
The temperature of a body is the average kinetic energy per molecule.

Thermodynamics is the study of the patterns of energy change. The "thermo" part of the word is the energy bit, and "dynamics" is the change bit.

The Zeroth Law of Thermodynamics

When a system is thermally isolated (by placing it inside closed adiabatic walls), it will come to an equilibrium, or stable state. In this state the average kinetic energy per molecule remains constant in time. In the diagram, the particles have double ended arrows to indicate that they are moving randomly at high speed.

When two (otherwise isolated) systems are placed in contact with each other through a mutual diathermal wall, heat will flow to achieve a new stable state where the average kinetic energy per molecule is the same for each system.

The systems are then said to be in thermal equilibrium, or have equal temperatures.
The zeroth law can be stated this way: If two bodies are each in thermal equilibrium with a third body then they are in thermal equilibrium with each other, and their temperatures are the same.

Temperature Scales

When an object receives heat and its temperature increases, several of its physical properties can change. A property which changes with temperature is called a thermometric property. These can include volume, pressure, electrical resistance, emf and colour. With one of these properties, a thermometer can be made. Two fixed points will also be needed to create a temperature scale to display numbers.

Gabriel Fahrenheit devised a temperature scale in 1709 or 1714 in Holland. He chose two fixed points which were easily identifiable and reproducible, and a linear thermometric property (linear expansion).
He set 00 as the coldest temperature that he could get. To do this he used a mixture of ice and ordinary salt (brine). He set the normal blood temperature of a person as 1000. He chose the assumed linear expansion of Mercury along a capillary tube in glass as the thermometric property thus creating the "Mercury in glass thermometer". Unfortunately he didn't get blood temperature quite right (it is 98.60F) but the freezing point of water was officially set at 320F and its boiling point at 2120F.

Anders Celsius (1701-1744) in 1742 also devised a temperature scale that had 1000 separating the boiling and freezing points of water. His original scale had the boiling (or steam) point as 00 and the freezing (or ice) point as 1000 but this was reversed in 1948 when it was adopted internationally.

Conversion between Fahrenheit and Celsius Temperatures

You can convert between these scales by comparing the size of the degrees and the starting points.

 100 C0 corresponds to 180 F0 Then the zero points have to be aligned.

Neither of these temperature scales have a proper zero.
For example, if it was 100C yesterday and 200C today, it is not twice as hot.

Example:
It is a well established principle that when the temperature on the Celsius scale equals the temperature on the Fahrenheit scale you do not have to go to work. Find the temperature.

The Constant-Volume Gas Thermometer
 By international agreement, the standard thermometric property is the pressure of a gas at constant volume. The apparatus is shown on the right. A bulb is immersed in a liquid at the temperature to be measured. By raising or lowering the mercury reservoir the column of Mercury on the left can be set to the zero mark and the volume of gas thus kept constant. The pressure is then obtained from the difference between Mercury heights. The temperature is defined by being proportional to pressure.

The Kelvin Temperature Scale

Now since (absolute) pressure can go to 0 Pa but cannot go negative, there must be an (absolute) temperature zero at this point. The Kelvin temperature scale (devised by William Thomson, Lord Kelvin) uses this as its zero. The other fixed point of the Kelvin scale was chosen to be the triple point of water.
 The triple point is the easily identifiable and reproducible temperature where solid ice, liquid water and water vapour coexist in thermal equilibrium.

The temperature of the triple point of water was set (from Charles' gas law) at 273.16 Kelvin (= 0.01 0C)
(For information on Charles' law go here). The conversion between Celsius and Kelvin is: , where Celsius and Kelvin degrees are the same size.

Comparing the three scales:

The Constant Volume Gas Temperature

For the constant volume gas thermometer, with the triple point now set to 273.16K, the temperature of an object can now be given a numerical value from measured pressures:

Questions now arise about how much gas is needed and what kind of gas is appropriate.
By using different amounts of different gases and first immersing them in boiling water (for a fixed constant temperature) and then in a triple point reference cell, the following graphs are obtained.

The numbers obtained for boiling water are all different as they depend on the type and quantity of gas used. However, the graphs converge to the same temperature number as the pressure at the triple point goes to zero. In this limit there is no gas and it doesn't matter which gas you have none of.

A unique constant-volume gas temperature is now given by:

Example
A constant-volume gas thermometer is used to measure the temperature of a liquid. For a triple point reference pressure of 100 kPa the gas pressure of a hot body is measured as 137 kPa. For a triple point reference pressure of 60 kPa the gas pressure of the same hot body is now measured as 81 kPa. Find the constant-volume gas temperature.

By equating slopes:

Comparing Temperature Scales

An undiscovered material called Iradion is used as an electrical resistance thermometer. When placed in a mixture of ice and water at a stable temperature its resistance is 800 Ohms. When placed in boiling water its resistance is 4000 Ohms. The resistance of an insulated beaker of warm water is measured as 1450 Ohms. Assume a linear relationship between resistance and temperature and find the temperature of the warm water on this Iradion scale.

 Assuming a linear relationship between temperature and resistance gives the graph on the right. The Iradion Temperature is given by equating slopes: 100/(4000 - 800) = T/(R - 800) T = (R - 800)/32

A measured resistance of 1450 Ohms gives an Iradion temperature of (1450 - 800)/32 = 200.

The Temperature of the insulated beaker of warm water was also measured by a Mercury Thermometer and found to be 450C. Explain the difference.

When the Resistance was calibrated against the Mercury Thermometer, the following table was recorded.

 T0 (Mercury) Resistance (Iradion) T0 (Resistance) 0 800 0 10 830 1 20 930 4 30 1090 9 40 1310 16 50 1600 25 60 1950 36 70 2370 49 80 2850 64 90 3390 81 100 4000 100

Clearly the scales are not linear in relation to each other. The resistance is parabolic with Mercury temperature and vice-versa. 450C on the Mercury scale does correspond to an Iradion temperature of about 200R.

Summarising:

A diathermal wall is one that heat can pass through
An adiabatic wall is one that heat cannot pass through.
Temperature is the average kinetic energy per molecule.
The zeroth law of thermodynamics:
If two bodies are each in thermal equilibrium with a third body then they are in thermal equilibrium with each other.
The conversion between Celsius and Fahrenheit is:

The conversion between Celsius and Kelvin is: , where Celsius and Kelvin degrees are the same size.

The constant-volume gas temperature is given by:

Temperature scales using different thermometric properties may not be linear with each other.

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