Question

In class we considered the situation of a parcel being warmed isobarically from below by a large thermal

reservoir. The parcel has temperature Tp initially and the surface has temperature Ts. We assumed that

Ts does not change appreciably if the surface is large compared to the parcel. Please show that this is true

for liquid water warming air: Assume that 1 cm3 of liquid is warming a 1 cm3 parcel of air that is directly

above it. If the parcel temperature rises by 1 K, how much does the water cool?

Answer 1

The process is considered isobaric, ie the pressure is constant. Therefore we will be considering the Heat Capacities at constant Pressure for the calculations.

The Heat capacity of air is,

The density of air is, 1.225 kg/m^3.

Therefore the mass of 1 cm^3 of air, is

Now, amount of heat required to change the temperature is given by,

Here the change in temperature is 1 K, Therefore the amount of heat energy absorbed by air is,

Now, this heat energy is absorbed from the 1cm^3 of water, The heat capacity of water at constant pressure is 4185.5 J/kg K and the density of water is 1000 kg m^3. Therefore the mass of water is,

The amount of heat absorbed by air is given so the change in temperature can be calculated by,

We can see that how small is the change in temperature of 1 cm^3 of water to heat a 1 cm^3 air parcel, So we can safely assume that the surface temperature of the water will not change when heating the air parcel from below.