- #1

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- 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##