- #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##
##\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##