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. 10.34 cm from the end at `100^@C`
B. 10.34 mm from the end at `100^@C`
C. 1.034 cm from the end at `100^@C`
D. 1.034 m from the end at `100^@C`

Answer

A

Explanation

If the point is at a distance x from water at `100^@C`, heat transferred to ice in time t to melt it is
`m_1L_1=(KA(200-0)t)/((1.5-x))`
or `m_1=(KAxx200t)/(80(1.5-x))`
similarly, heat conducted by the rod to water at `100^@C` in time t is
`Q=(KA(200-100)t)/(x)=m_SL_S`
`m_S=(KA(200-100)t)/(xL_S)=(KAxx100t)/(x xx540)`
According to problem, `m_1=m_S`
i.e.,
i.e., `(KAxx200t)/(80(1.5-x))=(KAxx100t)/(x xx540)`
or `(2)/(8(1.5-x))=(1)/(54x)`
solving it, we get `x=0.1034m` or `10.34cm`