# Thermodynamics: Enthalpy and energy change of a column of air

1. May 28, 2014

### Nyfinscyf

1. The problem statement, all variables and given/known data
A column of dry air, of cross-sectional area 1m2, has a surface pressure of 1atm. Over the course of several hours during the day it gains 2MJ of energy from radiation and surface heating. There are no horizontal exchanges of energy of mass, and it remains in hydrostatic equilibrium.

Does the surface pressure change?
What is the change in enthalpy, internal energy, and potential energy?

Area: 1m2
Energy gain: 2 MJ
Pressure: 1 atm

2. Relevant equations
$H = U + pV$
$ΔH = ΔU + Δ(pV)$
$H=U+P = \int^{∞}_{0} ρ(c_v + R)T dz = \int^{∞}_{0} ρh dz$
${\frac{∂U}{∂T}}=\frac{3}{2} Nk=C_v$

Ideal gas - internal energy does not depend on volume.

3. The attempt at a solution
Surface pressure: Does not change as there is no change in mass of the column and so there is the same volume of air above the location.

I understand enthalpy increases because the internal energy increases from the gain in heating and radiation. But I don't understand how the potential energy increases nor do I understand how the math works. These formulas seem very vague to me.

How does the gain of energy increase the temperature mathematically?