Andrew Mason said:
That is true, but it does not really explain why that occurs.
Heat flows spontaneously from a body at higher temperature to a body at lower temperature because the average translational kinetic energy of the molecules in the hotter body is greater than the average translational kinetic energy of the molecules in the cooler one. This is true not only for ideal gases but for all matter.
No, that isn't completely true. The statistical mechanics approach to thermodynamics explains thermal equilibrium this way:
Suppose that we have two systems A, and B, having total energy E. The two systems are in thermal contact (that is, they can exchange energy, but not particles). Because of thermal randomness, energy will move back and forth between the two systems, and the division of the total energy E between the two system will fluctuate. If we let E_1 be the energy of A and E_2 be the energy of B, what is the most likely value for E_1 and E_2?
Well, we can make a counting argument: Let W_1(E_1) be the number of possible ways to arrange A so that it has energy E_1. Let W_2(E_2) be the number of possible ways to arrange B so that it has energy E_2. Then the total number of ways to arrange the composite system so that it has total energy E = E_1 + E_2 is given by:
W(E) = W_1(E_1) \times W_2(E - E_1)
The most likely value for E_1 is assumed to be the value that makes W(E) largest. So to find E_1, we want to maximize the function:
f(E_1) = W_1(E_1) \times W_2(E - E_1)
where I used the fact that the total energy is E to rewrite E_2 = E - E_1.
To maximize a function, you take the derivative and set it to zero. So we have, for a maximum:
\dfrac{dW_1}{dE_1} W_2 - \dfrac{dW_2}{dE_2} W_1 = 0
We can rearrange this as:
\dfrac{\frac{dW_1}{dE_1}}{W_1} = \dfrac{\frac{dW_2}{dE_2}}{W_2}
which can be written, using properties of the logarithm, as:
\dfrac{d (ln(W_1))}{dE_1} = \dfrac{d (ln(W_2))}{dE_2}
The connection between these formulas and thermodynamics is to make the associations:
- The entropy of system A is a function of its energy E_1: S_1(E_1) = k ln(W_1(E_1)) where k is Boltzmann's constant.
- The temperature of system A is defined to be: T_1 = \dfrac{1}{\frac{dS_1}{dE_1}}
- The entropy of system B is a function of its energy E_2: S_2(E_2) = k ln(W_2(E_2)).
- The temperature of system B is defined to be: T_2 = \dfrac{1}{\frac{dS_2}{dE_2}}
- The systems are said to be "in thermal equilibrium" if E_1 andE_2 are at their most likely values, which is when T_1 = T_2.
This definition of temperature and entropy does not assume anything about the nature of the two systems. But if you do assume that the two systems are made up of particles, and that all the energy is kinetic energy of motion, then that allows you to calculate W_1 and W_2, and you find that the temperature is proportional to the average kinetic energy. So you don't have to
assum that.