Internal Energy

Internal energy is one of the most important concepts in thermodynamics. Energy changes in a body sliding with friction. Warming a body increases its internal energy and cooling the body decreases its internal energy. However, what is internal energy? We can look at it in various ways: let's start with one based on the ideas of mechanics. Matter consists of atoms and molecules, and these are made up of particles having kinetic energy and potential energy. We tentatively define the internal energy of a system as the sum of the kinetic energy of all its constituent particles, plus the sum of all the potential energy of interaction among these particles.

Note that internal energy does not include potential energy arising out of interaction between the system and it surroundings. If the system is a glass of water, placing it on a high shelf, increases the gravitational potential energy arising from the interaction between the glass and the Earth. However, this has no effect on the interaction between the molecules of water, and so the internal energy of water does not change.

We use the symbol 'U' for internal energy. During a change of state of the system, the internal energy may change from an initial value U₁ to a final value U₂. We denote the change in internal energy as .

Sub Topics

Thermometry
Constant volume gas thermometer
Thermal expansion
Linear expansion

Thermometry

The branch of physics dealing with measurement of temperature is called thermometry. A number of physical properties of a substance change almost linearly with temperature. This forms the basis or principle for construction of thermometers.

Different scales of temperatures have been adopted. For example, in the Celsius scale, the two arbitrary fixed point ice point and steam point were chosen to define the temperature scale. If K₀, K₁₀₀ and K represent any of thermometric properties (like length of a liquid, pressure of gas at constant volume, volume of a gas, resistance of a wire, etc.) at temperatures 0⁰C, 100⁰ C and T⁰ C, then

The modern thermometry, only one reference point is chosen i.e., triple point of water which the point or temperature at which ice, water and water vapors coexist. The temperature being 273.16 K.

Therefore,

where K and K_Tr are the thermometric property at T = 0 K and triple point.

Constant volume gas thermometer

At any temperature, the pressure of a gas depends upon its volume. If the volume is kept constant, the pressure depends upon the temperature and increases steadily with rising temperature. The constant volume gas thermometer uses the pressure at constant volume as the thermometric property.

The constant volume gas thermometer is shown diagrammatically in the figure. It consists essentially of a bulb C of glass, glazed porcelain, fused quartz, platinum or platinum-iridium (depending upon the temperature range over which it is to be used). The bulb is connected by a capillary tube to a mercury pressure gauge such as an open manometer. The bulb is immersed in the system whose temperature is to be measured.

The bulb contains some gas such as helium or hydrogen or nitrogen or even air. The mercury reservoir R is so adjusted that the mercury in the branch B of the U-tube is at a fixed reference mark E to keep the confined gas at a constant volume. Then, we read the height of the mercury in the A branch. The pressure of the confined gas is the difference of the heights of the mercury columns (times rg) plus the atmospheric pressure (as indicated by the barometer reading). In actual practice we have to apply corrections for the small volume change owing to slight contraction or expansion of the bulb. We have also to consider the fact that not all the confined gas has been immersed in the bath. Assume that these and other possible corrections have made. If P the corrected pressure at the temperature of the bath, then the temperature of the bath is given by

Thermal expansion

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Most materials expand when their temperatures increase. The railway tracks, need special joints and supports to allow for expansion. A completely filled and tightly capped bottle of water cracks when it is heated, but you can loosen the lid of a metal jar by running hot water over it. These are all examples of thermal expansion.

Linear expansion

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Assume a rod of material having a length L_o at initial temperature T_o. When the temperature changes by DT, the length changes by DL. Experiment shows that if DT is not too large (say, less than 100^oC or so), DL is directly proportional to DT. If two rods made of the same material have the same temperature change, but one is twice as long as the other, then the change in its length is also twice as great. Therefore, DL must also be proportional to L_o. Introducing a proportionality constant a (which is different for different materials), we may express these relations in an equation:

If a body has length L_o at temperature T_o_, then its length L at a temperature T = T_o + DT

The constant a, which describes the thermal expansion properties of a particular material, is the coefficient of linear expansion. The units of a are K^-¹ or (C^O)^-¹ (remember that a temperature interval is same in both the Kelvin and Celsius scales). For many materials, linear dimension changes according to Equation (i) or (ii). Thus, L could be the thickness of a rod, the side length of a square sheet, or the diameter of a hole. Some materials, such as wood or single crystals, expand differently in different directions.

We can understand thermal expansion qualitatively on a molecular basis. Picture the interatomic forces in a solid as in the above figure. Each atom vibrates about its equilibrium position. When the temperature increases, the energy and amplitude of the vibration also increase. The interatomic spring forces are not symmetrical about the equilibrium position. Their behaviour is like that of a spring, which is easier to stretch than to compress. As a result, when the amplitude of vibration increases, the average distance between molecules also increases. As the atoms go farther apart, every dimension increases.