Question

I am on summer break right now, but I am reading a chemistry book to get a little familiar with some chemical terms for next year: Thermodynamics. And i came upon a question that I have no idea how to answer. Would someone be kind enough to assist?

Calculate the rate of change in temperature per height for the rise of a 1g dry air in the lower atmosphere. The actual rate is 6.5EC/km Suggest reasons for the difference.

\[C _{p},_{m}=1.005kJ/kg/^{°}C\]

Answer 1

sorry but what is this unit EC/km?

Answer 2

You are probably looking for what's called the adiabatic lapse rate. This is the change (decrease) in temperature of the atmosphere, assumed to act as an ideal gas, with height. The reason the temperature declines can most simply be understood from a purely mechanical conservation of energy viewpoint: if a molecule rises higher in the Earth's gravitational field, then like a ball thrown up, it must lose speed as its potential energy increases. Since the average speed of molecules is what we mean by "temperature," more or less, then it follows that if a parcel of air rises it must cool.

From a strictly thermodynamic point of view, there are a number of ways to derive the adiabatic lapse rate. You will generally need to assume that entropy is constant, which is what "adiabatic" means -- no heat is transferred in or out, hence the entropy is fixed -- and that the atmosphere is ideal, which gives you a number of useful relationships among the pressure, volume, energy and temperature. Here's one derivation:

http://pds-atmospheres.nmsu.edu/education_and_outreach/encyclopedia/adiabatic_lapse_rate.htm

I can't actually find any much simpler. The bottom line, however, is extremely simple:

(dT/dz)_S = -g/C_p

Hence to find the adiabatic lapse rate all you need is the acceleration due to gravity g, and the constant pressure heat capacity C_p.

Answer 3

Oops, sorry, if you are working from thermodynamics, you will also need an equation from hydrostatic equilibrium. I should also mention this is why the actual lapse rate deviates from the ideal -- because most of the lower atmosphere is not really in hydrostatic or thermal equilibrium. There are substantial nonequilibrium movements ("fluctuations") of air and heat up and down, a result of winds, differential heating from above (by the Sun) and below (by the surface of the Earth), clouds, and the condensation and evaporation of water.