The densities of wood and benzene at `0^@C` are `880 kg m^(-3) and 900 kg m^(-3)` respectively. The coefficients of volume expansion are `1.2 xx10^(-3)C^(-1)` for wood and `1.5 xx 10^(-3)C^(-1)` for benzene. At what temperature will a piece of wood just sink in benzene?

Explanation

Given `f_w = 880 kg/m^3,`
` f_b= 900kg/m^3`
`T_1=0^@C,`
`gamma_w = 1.2 xx 10^-3/^@C,`
` gamma_b = (1.5 xx 10^-3/^@C)`
The sphere begins to sink when
` rArr (mg)_sphere = displaced water `
` (Vf'_w)g = (Vf'_b)g`
` rArr f_w /(1+(gamma_w) Delta theta)= f_b/(1+(gamma_b) Delta theta)`
` rArr (880/(1+1.2 xx (10^-3) Delta theta)) = (900/ ( 1+1.5 xx (10^-3) Delta theta))`
` rArr (880+880 xx 1.5 xx (10^-3) Delta theta )`
` (900+900 xx 1.2 xx (10^-3) Delta theta)`
`rArr (880 xx 1.5 xx (10^-3)- 900 xx 1.2 xx (10^-3))Delta theta`
` =20`
` rArr (1320-1080) xx (10^-3)Delta theta = 20`
` rArr Delta theta = 83.3^@C`
` T_2- T_1 = 83 rArr T_2- 0^@C= 83`
` rArr T_2 = 83^@C`