Everyone knows Temperature gradient is a vector quantity having direction from cold to hot.My confusion: why is temperature gradient vector if its direction is always fixed (as in the case of pressure) (Don't say that it is because it follows vector law of addition,I am searching for more concrete answer…)
[Physics] How temperature gradient is a vector
temperaturethermodynamics
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You asked for intuitive sense and I'll try to provide it. The formula is:
$$\Delta S = \frac{\Delta Q}{T}$$
So, you can have $\Delta S_1=\frac{\Delta Q}{T_{lower}}$ and $\Delta S_2=\frac{\Delta Q}{T_{higher}}$
Assume the $\Delta Q$ is the same in each case. The denominator controls the "largeness" of the $\Delta S$.
Therefore, $\Delta S_1 > \Delta S_2$
In each case, let's say you had X number of hydrogen atoms in each container. The only difference was the temperature of the atoms. The lower temperature group is at a less frenzied state than the higher temperature group. If you increase the "frenziedness" of each group by the same amount, the less frenzied group will notice the difference more easily than the more frenzied group.
Turning a calm crowd into a riot will be much more noticeable than turning a riot into a more frenzied riot. Try to think of the change in entropy as the noticeably of changes in riotous behavior.
The definition of an empirical temperature is basically what the Zeroth Law of Thermodynamics does.
Let us suppose we do not have any prior knowledge about temperature. What we do know is that if we put two bodies in contact with each other they may change some thermodynamic properties -volume, for instance - of one another. When such a thing happens we say the bodies are in thermal contact. After a while the thermodynamic properties stop changing and we say the bodies are in thermal equilibrium. The Zeroth Law consists on the empirical fact that if $A$ is in thermal equilibrium with $B$ and $B$ is in thermal equilibrium with $C$, then $A$ is in thermal equilibrium with $C$. This is an equivalence relation which classify a set of bodies into subsets called equivalence classes. Each class is labeled by a number $T>0$ which we shall call temperature. The Zeroth Law allows us establish thermal equilibrium just in terms of temperature.
To understand the importance of this empirical temperature, defined through the Zeroth Law, imagine a substance - a portion of Mercury would do well - characterized by its volume. You put it in thermal contact with a body $A$, wait for thermal equilibrium and measure its volume $V_1$. Then we freely assign a temperature $T_1$ to this volume and consequently to the body A. If the substance is then put in thermal contact with body $B$ and reaches thermal equilibrium, we measure the volume $V_2$ and arbitrarily assign a temperature $T_2$. The next step is to make use of an interpolation to obtain a function $T(V)$. This is most easily done with a linear interpolation. In this case $$T=a+bV.$$ In this case $$\frac{T}{T'}=\frac{a+bV}{a+bV'}\equiv \frac{f(V)}{f(V')}.$$ Therefore this works as a thermometer.
The drawback of this empirical temperature is that it is not absolute. One can define different scales based on different physical properties, reference points or interpolations. This difficulty is overcame by the thermodynamic temperature which can be defined through a reversible thermal engine operating between two sources. Any of such engine has efficiency $$\eta_R=1-\frac{T_2}{T_1},$$ where $T_1$ and $T_2$ are the temperatures of the sources. Given the universality of this result one can for instance arbitrarily define the temperature of the cold source $T_2$, measure - mechanically - the efficiency of the engine and then the temperature $T_1$ is determined by $$T_1=\frac{T_2}{1-\eta_R}.$$ Note that there is no longer arbitrariness about the concept of temperature, except for the choice the the temperature of the cold source. Therefore it is appropriate to use as reference point which is highly reproducible anywhere. A standard choice is the triple point of water which is defined to be at $273.16\, \mathrm K$.
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A maybe more mathematical awnser: You can define temperature as a scalar field (e.g. on earth). So given a certain position on the surface of the earth (or in three dimensions if you wish, it does not change anything) you have a scalar, the temperature on this position. Now you can take the gradient of this field, and now you have a vector.
More directly on your question: 1) A vector is still a vector, even if he has a constant value. 2) Why is its value fixed? You don't know where the temperature is highest, or you can even define a time dependend temperature field, (on earth temperature is not fix, it changes quite obviously), then its gradient is not fixed either