I'm having a problem getting the "right" answer to the following entropy
change calculation. I want the net entropy change when water freezes in a
surrounding gas:
1) Water at 0 deg C freezes slowly and isothermally. I compute the entropy
change of this process (dS_ice) as -mL_f/T_0 where m is the mass of water
(or ice), L_f is the latent heat of fusion of water, and T_0 is the freezing
temperature (assumed constant). The thermal energy given off by the water
as it freezes is mL_f = delta_Q.
2) I assume a gas in a fixed volume, perfectly insulated container encloses
the ice. I assume that the thermal energy given off by the water as it
freezes is absorbed with no loss by the surrounding gas molecules.
Therefore, the temperature of the gas increases above its assumed initial
temperature of T_0 by the amount delta_T with mL_f = nC_v delta_T = delta_Q
,where n is the number of moles of gas present and C_v is the molar specific
heat of the gas at constant volume. The entropy gain of the gas (dS_gas) is
given by the integral of dS = dQ/T as the temperature increases from T_0 to
T_0 + delta_T. This gives dS_gas = nC_v ln[(T_0 + delta_T)/T_0]
3) The net entropy gain of ice plus surrounding gas (after a little
re-arranging) is dS_net = dS_ice + dS_gas = A [ln(1+x) - x] where A = m
L_f/delta_T and x = delta_T/T_0
Unfortunately, ln(1+x) - x is < zero for all x > 0, so I have calculated
that dS_net is < 0 rather than the answer I wanted (> 0)
Would appreciate your help in finding my error or errors.