Temperature gradient

Temperature

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Temperature

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One of the central concepts in thermal physics is \textbf{temperature}. We all have an intutive understanding of what temperature means | hotter objects have high temperatures and cooler objects have low temperatures. But if our understanding of what is hot and what is cold is based merely on our sense of touch, we are likely to be mislead. Our sense of touch can only sense a change in the degree of hotness, more precisely it allows us to sense whether or not something is hot relative to say what temperature our hands were at initially.

We therefore use thermometers to measure temperature. We could even define temperature as the quantity measured by a thermometer. Much like, without getting into philiosophical musings, we define time as what is measured by a clock. But in order to fully grasp this (what we shall call) \textbf{operational defintion}, we must understand on what principle does a thermometer work.

If observed closely, what happens when a mercury thermometer is dipped into a hot cup of water (say) is that the mercury inside the thermometer heats up and expands (as most materials do), rising up in the capilary column of the thermometer. More specifically, we say that the temperature of the mercury increases until it equals that of the water. The expanded volume occupied by this hotter mercury indicates the temperature of water in the cup. We might hence define temperature as a quantity that becomes the same for two objects when they are kept in contact long enough. This we shall call a \textbf{theoretical defintion} of temperature.

Introducing more jargon, we say that when two objects are kept in contact long enough, heat flows between them and they attain \textbf{thermal equilibrium}. And the time it takes for thermal equilibirum to establish is called the \textbf{relaxation time}.

When a hot drop of ink is dropped in a cold cup of water, there are two distinct things happening. There is exchange of heat between the ink drop and the water. The ink drop itself is diffusing throughout the water. Initially, the molecules of the ink have a tendency to spread outwards. But once the ink has been spread throughout, the molecules of the ink no longer have a spcial direction in which to travel and their motion is identically random as that of the water molecules. We then speak of a diffuse equilibrium established between ink and water. Here molecules themselves are exchanged, unlike in thermal equilibrum where ebnergy alone is exchaged (and not molecules).

Further, when different parts of a system stop moving, without diffusion, we speak of \textbf{mechanical equilibrium} being established. The quantity exchanged in this case is volume.

Also, when we speak of two objects brought in contact, we mean an arrangement where energy in the form of heat is allowed to exchange between them. This could be by actually touching the surfaces of the two objects or even with empty space separating the two, when heat is transferred through radiation.

The fact that heat flows from one object to another when brought in contact with one another means that one of the objects spontaneously emits the energy and the other absorbs it. This allows us to define temperature yet again, but more precisely as a measure of the tendency of objects to emit heat to their surroundings. The object that emits heat (and hence loses energy) is at a higher temperature than the one that absorbs it (and hence gains energy).

With the theoretical defintion of temperature seeming more satisfactory, we turn towards the operational defintion. It stilll isn’t clear how must we assign values for temperature when a thermometer is dipped in a material. Say we have a mercury thermometer, we dip it in water just when it freezes into ice and mark the level of mercury (at sea level). We do the same when water just begins boiling. We assign the numbers 0 and 100 to each of these marks. We then divide the entire length of mercury between these two points into 100 equal parts. Then each tiny division will mark one unit of temperature. This is the Celsius or the centigrade scale.

We could as well use a gas thermometer where a gas is filled in place of liquid mercury. With such thermometers, it is observed that while $0^o$C indicates freezing point of water, further reduction in temperature means a shrinking of the volume of the gas. And extrapolation will lead to the gas shrinking to zero volume at a temperature of $-273^o$C (though in reality the gas liquifies first). Also, if the volume of the gas is fixed, lowering the temperature leads to a reduction in the pressure of the gas. And at $-273^o$C, we have zero pressure. This special value of temperature is fixed as the zero of the absolute scale. It is the lowest possible temperature attainable. The unit temperature is the same in both the centigrade and the absolute scale. But whater freezes at $273$K and boils at $373$K in the absolute scale. Almost all equations in physics yield correct values when the absolute scale is used.

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