One end of a copper rod of uniform cross section and length 1.5 m is kept in contact with ice and the other end with water at `100^@C`. At what point along its length should a temperature of `200^@C` be maintained so that in the steady state, the mass of ice melting be equal to that of the steam produced in same interval of time. Assume that the whole system is insulated from surroundings:
`[L_("ice")=80 cal//g,L_("steam")=540 cal//g]`

Options

A. 8.59 cm from ice and
B. 10.34 cm from water end
C. 10.34 cm from ice end
D. 8.76 cm from water end

Answer

B

Explanation

If the point is at a distance x from water at `100^@C`, heat conducted to ice in time t,
`Q_(ice)=KA((200-0))/((1.5-x))xxt`
So ice melted by this heat
`m_(ice)=(Q_(ice))/(L_F)=(KA)/(80)((200-0))/((1.5-x))xxt`
Similarly heat conducted by the rod to the water at `100^@C` in time t,
`Q_(water)=KA((200-100))/(x)t`
Steam formed by this heat
`m_(steam)=(Q_(water))/(L_V)=KA((200-100))/(540xxx)t`
According to given problem `m_(ice)=m_(steam)`,
i.e., `(200)/(80(1.5-x))=(100)/(540xx x)x=(6)/(58)m=10.34cm`
i.e., `200^@C` temeprature must be maintained at a distance 10.34 cm from water at `100^@C`.