# The growth of ice on a pond

An interesting application of thermal conductivity is the calculation of the rate
of growth of ice on top of a pond.

Let the air temperature be q
and the water temperature just below the ice be 0 ^{o}C. At a certain time let the
thickness of the ice be x, and let it increase by a further thickness dx in a time dt. The latent
heat released on melting has to be conducted away through the ice layer as the water freezes
and therefore we have:

quantity of heat lost due to increase dx = rLAdx

where r is the density of ice, L
the specific latent heat of fusion of water and A the area of the ice surface. Then, if x is the
thickness of the ice after a time t,

rate of loss of heat = LArdx/dt = kqA/x

Therefore dx/dt = kq/Lrx

Integrating gives:

New thickness of ice (x) after a time t is: x = [2kqt/Lr]^{1/2}
## Thermal conductivity and
kinetic theory

Consider three horizontal planes in the gas each of area A. The heat
conducted downwards through A per second is then -kAd

q/dx

However, each second a mass of
gas m at a temperature

q_{1} crosses A moving downwards and a mass of gas
m at a temperature

q_{2} crosses A moving upwards.

Now:
m =

rcA/6

q_{1}
=

q +

ld

q/dz and

q_{2} =

q +

ld

q/dz

Therefore, since heat = mc

q , the net transfer of heat
downwards is:

–[

rcA/6][2d

q/d

q]C

_{v}.

But this must equal –kAd

q/dz and therefore:

k =
1/3

rc

lC

_{v} and since

h = 1/3

rc

l

Thermal conductivity of a gas: k = hC_{v}