A container of negligible heat capacity contains 1 kg of water. It is connected by a steel rod of length 10 m and area of cross-section 10 `cm^2` to a large steam chamber which is maintained at 100°C. If initial temperature of water is 0°C, find the time after which it becomes 50°C. (Neglect heat capacity of steel rod and assume no loss of heat to surroundings) (use table 3.1, take specific heat of water = 4180 J/kg°C)

Explanation

Let temperature of water at time t be T, then thermal current at time t,
`i=((100-T)/(R ))`
This increases the temperature of water from T to `T+dT`
`rArr i=(dH)/(dt)=ms(dT)/(dt)`
`rArr (100-T)/(R )=ms(dT)/(dt)`
`rArr underset(0)overset(50)(int)(dT)/(100-T)=underset(0) overset(t ) (int) (dT)/(Rms)`
`rArr l n (1/2)=(t)/(R ms)`
or `t=Rms ln 2 sec =(L)/(KA)ms ln 2 sec`
`=((10m )(1 kg)(4180 J//kg-^@ C ))/(46(w//m^@ C)xx(10xx10^(-4) m^2))`
`=(418)/(46)(0.69)xx10^(5)=6.27xx10^(5) sec`
`=174.16` hours