# 5.7: Questions

- Page ID
- 20948

## Deeper questions

*The following questions require some thought and reaching the answer may require you to think beyond the contents of this TLP.*

1. Crystallization is usually fatal to cells because:

3. Crystallization of minerals in the cells can be limited by:

In some alpine plants, extracellular ice formation occurs at around –2°C. What is the critical radius for ice nucleation at this temperature?

In these plants, the extracellular ice forms on ice nucleating agents. If an ice nucleating agent is a circular disc and is a perfect template for ice, what size must it be for nucleation to occur at –2°C?

(For the ice-water interface, the interfacial energy *γ* = 0.028 J m^{-2}; the latent heat of freezing of ice is Δ*H*_{v} = -3.34 x 10^{8} J m^{-3}.)

hint: The critical radius for nucleation is *r**, where *r** = -2*γ* /Δ*G*_{v}, and *γ* is the solid-liquid interfacial energy and Δ*G*_{v} is the free energy of freezing per unit volume. For further details see the section on nucleation and crystallization.

**Answer**-
*r**, the critical nucleus for nucleation is given by,*r** = -2*γ*/Δ*G*_{v}where*γ*is the solid-liquid interfacial energy and Δ*G*_{v}is the free energy of freezing per unit volume.*γ*is given in the question and Δ*G*_{v}can be determined from Δ*H*_{v}. For freezing at a temperature*T*, the supercooling, Δ*T*is (*T*_{m}-*T*), where (*T*_{m}is the melting temperature. The entropy of fusion per unit volume is Δ*S*_{v}= Δ*H*_{v}/*T*_{m}. As shown in the section on Nucleation and Crystallization, Δ*G*_{v}= Δ*S*_{v}Δ*T*. Thus:\[r^{*}=-\frac{2 \gamma}{\Delta G_{v}}=-\frac{2 \gamma}{\Delta S_{v} \Delta T}=-\frac{2 \gamma T_{m}}{\Delta H_{v} \Delta T}\]

With

*γ*and Δ*H*_{v}as given,*T*_{m}taken to be 273 K, and*T*= 271 K, we find*r** = 22.9 nmFor ice to grow on the INA, the radius of the INA must be greater than or equal to

*r**, so, for ice to grow, the INA should have radius 22.9 nm.