 #1
EEristavi
 108
 5
 Homework Statement:

A liquid has a density ##\rho##.
Show that the fractional change in density for a change in temperature ##\Delta##T is
##\frac {\Delta \rho} {\rho} = \beta \Delta T##
 Relevant Equations:
 ##\Delta V = \beta V \Delta T##
##\Delta V = \beta V \Delta T##
##\rho _2 = \frac m {V + \Delta V} = \frac { \rho V} {V(1+\beta \Delta T)} = \frac \rho {1+\beta \Delta T}##
##\Delta \rho = \rho _2  \rho = \frac \rho {1+\beta \Delta T}  \rho = \rho (\frac 1 {1+\beta \Delta T}  1) = \rho \frac {\beta \Delta T} {1+\beta \Delta T}##
##\frac {\Delta \rho} {\rho} = \frac {\beta \Delta T} {1+\beta \Delta T}##
I can't figure out were I make the mistake...
It should be:
##\frac {\Delta \rho} {\rho} = \beta \Delta T##
##\rho _2 = \frac m {V + \Delta V} = \frac { \rho V} {V(1+\beta \Delta T)} = \frac \rho {1+\beta \Delta T}##
##\Delta \rho = \rho _2  \rho = \frac \rho {1+\beta \Delta T}  \rho = \rho (\frac 1 {1+\beta \Delta T}  1) = \rho \frac {\beta \Delta T} {1+\beta \Delta T}##
##\frac {\Delta \rho} {\rho} = \frac {\beta \Delta T} {1+\beta \Delta T}##
I can't figure out were I make the mistake...
It should be:
##\frac {\Delta \rho} {\rho} = \beta \Delta T##