Question

The storage modulus \( G' \) of a collagen gel scales as \( G' \sim [C]^3 \), where [C] represents the collagen concentration in mg/ml. If the storage modulus of a 1 mg/ml collagen gel is 100 Pa, then the storage modulus of a 3 mg/ml collagen gel is _______ Pa. (rounded off to the nearest integer)

Hint:

When a property scales with concentration as a power, use the scaling relationship to find the property at different concentrations. In this case, the storage modulus scales as the cube of the concentration.

Answer 1

We are given that the storage modulus \( G' \) scales with collagen concentration \( [C] \) as follows: \[ G' \sim [C]^3 \] This means that: \[ \frac{G'(C_2)}{G'(C_1)} = \left( \frac{C_2}{C_1} \right)^3 \] Where:
\( G'(C_1) = 100 \, {Pa} \) is the storage modulus at \( C_1 = 1 \, {mg/ml} \),
\( C_2 = 3 \, {mg/ml} \) is the new collagen concentration. Substituting the known values into the equation: \[ \frac{G'(3)}{100} = \left( \frac{3}{1} \right)^3 \] \[ \frac{G'(3)}{100} = 27 \] \[ G'(3) = 100 \times 27 = 2700 \, {Pa} \] Thus, the storage modulus of the 3 mg/ml collagen gel is: \[ \boxed{2700 \, {Pa}} \]