Q20PE - College Physics (Urone)

Question

(a) The density of water at 0ºC is very nearly 1000 kg/m³ (it is actually 999.84 kg/m³ ), whereas the density of ice at 0ºC is 917 kg/m³. Calculate the pressure necessary to keep ice from expanding when it freezes, neglecting the effect such a large pressure would have on the freezing temperature. (This problem gives you only an indication of how large the forces associated with freezing water might be.) (b) What are the implications of this result for biological cells that are frozen?

Step-by-Step Solution

Answer

(a) The pressure necessary to keep ice from expanding is 1.9×10⁸ Pa.

(b) Pressure has a keen role to play here so that cells and tissues do not expand and no icing in cells should be there

Step 1: Introduction

We calculate the volume change when water is converted to ice and hence by using the equation relating pressure to change in volume, we calculate pressure.

Step 2: Given parameters and formula linking pressure and change in volume

The bulk modulus of water  = 2.1 X 10⁹ Pa 

Let us take the mass of water to be 1000 kg.

The volume of 1000 kg of water will be 1 m³

The volume of 1000 kg of ice will be 1000/917 m³

$\begin{array}{rcl}\mathbf{Change}\mathbf{ }\mathbf{in}\mathbf{ }\mathbf{volume}\mathbf{ }\left(\frac{∆V}{V}\right)& \mathbf{=}& \left(\frac{1000 m^3}{9173 m^3}-1\right)\\ & \mathbf{=}& \frac{\mathbf{83}}{\mathbf{917}}\end{array}$

Change in the volume can be expressed as,

$\mathbf{∆}\mathbf{V}\mathbf{=}\frac{\mathbf{FV}}{\mathbf{BA}}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\left(1\right)$

Here, B is the bulk modulus of water, F/A is pressure, and $∆\mathrm{V}$ is volume change.

Step 3: Calculate pressure

$\begin{array}{rcl}\mathbf{P}& \mathbf{=}& \left(\frac{∆V}{V}\right)\mathbf{\times }\mathbf{B}\\ & \mathbf{=}& \left(\frac{83}{917}\right)\mathbf{\times }\left(2.1\times {10}^9 \mathrm{Pa}\right)\\ & \mathbf{=}& \mathbf{1}\mathbf{.}\mathbf{9}\mathbf{\times }{\mathbf{10}}^{\mathbf{8}}\mathbf{ }\mathbf{Pa}\end{array}$

Step 4: Implications of the result for frozen biological cells

Biological cells are frozen which are different parts of human organs & tissues are frozen so that the part which was performing some hazard to the body, its functioning can be stopped and hazard minimized. Pressure has a keen role to play here so that cells and tissues do not expand and no icing in cells should be there.

Therefore, the pressure required is 1.9 X 10⁸ Pa.