Question

The density of water at 0°C is 0.998 g cm–3 and at 4°C is 1.000 g cm–1. Calculate the average coefficient of volume expansion of water in the temperature range of 0 to 4°C.

Answer 1

$\Large\underline{\underline{\sf \green{Given}:}}$

• The density of water at 0°C is 0.998 g cm–3

• And Density of water at 4°C is 1.000 g cm–1

$\Large\underline{\underline{\sf \green{To\:Find}:}}$

• Calculate the average coefficient of volume expansion of water in the temperature range of 0 to 4°C.

$\Large\underline{\underline{\sf \green{Formula\:Used}:}}$

$\large{\boxed{\sf \red{d=d_0(1+\gamma \triangle T}}}$

Here ,

d = Density of water at 4°C

$\sf{d_0}$=Density of water at 0°C

$\sf{\gamma}$=Coefficient of Volume expansion

$\Large\underline{\underline{\sf \green{Solution}:}}$

$\implies{\sf 1.000=0.998(1+\gamma×4) }$

$\implies{\sf 1+4\gamma=\dfrac{1}{0.998} }$

$\implies{\sf 1+4\gamma=\dfrac{1000}{998} }$

$\implies{\sf 1+4\gamma=1.002004}$

$\implies{\sf 4\gamma=0.002004}$

$\implies{\sf \gamma=\dfrac{0.002004}{4} }$

$\implies{\sf \gamma=5×10^{-4}°C^{-1}}$

As density Decrease

•°• $\implies{\sf \gamma=-5×10^{-4}°C^{-1}}$

$\Large\underline{\underline{\sf \green{Answer}:}}$

•°• Average coefficient of volume expansion of water in the temperature range of 0 to 4°C is $\bf{-5×10^{-4}°C^{-1}}$

Answer 2

The average coefficient of volume expansion of water in the temperature range of 0 to 4°C  is $\bold{v=-5 \times 10^{-4}}$

Explanation:

Given:

Water density  at 0°C, ()= 0.998 g cm-3

Water density  at 4°C, () = 1.000 g cm-3

Changing in temperature, (Δt) = 4°C

Solution:

Let the average volume expansion coefficient of water in the temperature range from 0 to 4°C be .

We know that  

$f_{0}=\frac{f_{4}}{1+\gamma \Delta t}$

$0.998=\frac{1}{1+\gamma \times 4}$

$0.998=\frac{1}{1+4 \gamma}$

$1+4 \gamma=\frac{1}{0.998}$

$4 \gamma=\frac{1}{0.998}-1$

$4 \gamma=\frac{2 \times 10^{-3}}{0.998}$

$4 \gamma=2.004 \times 10^{-3}$

$\gamma=\frac{2.004 \times 10^{-3}}{4}$

$\gamma=5.010 \times 10^{-4}$

As density shrinks $v=-5 \times 10^{-4}$

The average volume expansion coefficient of water in the temperature range of 0 to 4 ° C will therefore be $V_{m}=-5 \times 10^{-4}$