An aluminum rod, thermal conductivity 0,5 cal/s.cm.°C, is in contact at one end with melting ice and the other with boiling water vapor under normal pressure. Its length is 25 cm and the cross section area 5 cm². The bar being isolated laterally and given the latent heat of ice melting 80 cal/g and latent heat of water vaporization 540 cal/g, determine:
a) The mass of ice melting in half an hour;
b) The mass of vapor that condenses at the same time.


Problem data

• length of bar:    \( e=25\;\text{cm} \);
• bar cross section area:    \( A=5\;\text{cm}^{2} \);
• thermal conductivity of aluminum:    \( K=0.5\;\text{cal/s.cm.°C} \);
• temperature of boiling water:    \( t_{\text{e}}=100\;text{°C} \);
• temperature ofmelting ice:    \( t_{\text{f}}=0\;\text{°C} \);
• latent heat of melting ice:    \( L_{\text{F}}=80\;\text{cal/g} \);
• latent heat of water vaporization:    \( L_{\text{V}}=540\;\text{cal/g} \).

Problem diagram Solution

First we calculate the rate of heat flow through the bar, assuming that it is in a steady state (heat flow is constant across the bar) The flow gives us the amount of heat (Q) that crosses a cross section of the bar per unit of time (1 s), as we want the heat going through the bar in half an hour (30 min = 30x60 = 1800 s), we use a cross-multiplication In half an hour 18000 cal of heat flows from the vessel containing boiling water vapor, which condenses in liquid water, to the container with ice, which melts in liquid water. Therefore there is phase change in the two containers.

a) The melting ice mass can be calculated from the received heat calculated above and from the latent heat of fusion
b) The condensed vapor mass can be calculated from the heat calculated above and from the latent heat of vaporization