For a perfectly cyrstalline solid `C_(p,m)= a T^(3) + bT`, where a and b are constants. If `C_(p,m)` is 0.40 J/K mol at 10K and 0.92 J/K mol at 20K, then molar entropy at 20K is `0.2x xx R` joules. Then the value of x is

Options

A. 0.92 J/K mol
B. 8.66 J/K mol
C. 0.813 J/K mol
D. none of these

Answer

C

Explanation

`S_(20)- S_(0) = underset(0)overset(20)int (C_(P))/(T) dT = underset(0)overset(20)int (aT^(3) + bT)/(T) dT`
`=[(aT^(3))/(3) + bT]_(0)^(20)`
`S_(20) = (a)/(3) (20)^(3) + b(20)`
`C_(10) = a(10)^(3) + b(10) =0.4`
`C_(20) = a(20)^(3) + b(20) = 0.92`
`rArr a = 2 xx 10^(-5) , b= 0.038`
`rArr S_(20) = (2)/(3) xx 10^(-5) xx (20)^(3) + 0.038 xx 20`
= 0.813 J/mol -K