How temperature changes with elevation?

When we hike up a mountain, it gets cooler as we climb higher. What is the quantitative relation between temperature and elevation?

Imagine a gas molecule. If it rises higher, its potential energy increases. Its kinetic energy should decrease. Temperature is the average kinetic energy of molecules. Temperature therefore should decrease.

Suppose a kilogram of gas rises for one thousand meters. Its gain in potential energy is 9.8*1000= 9800 J. Here 9.8 is the acceleration rate of the gravitational force. We might expect the corresponding kinetic energy drops 9800 J as well. For most ideal gases at constant pressure and volume, the specific heat capacity is approximately 1000 J/(kg·K) The drop of temperature should be 9800/1000= 9.8 degrees. This is a decrease of 9.8 degrees of temperature. 

In the above calculation, we assume no heat transfer. Since heat flows from hot place to cold place, the actual decrease of temperature should be less than 9.8 degrees. Typically, it is about 6.5 degrees.

A simple calculation estimates the relation between temperature and elevation to the correct magnitude.