Let's break down the equation for the shear modulus $\mu_0$ and analyze each statement step by step.

The given formula for $\mu_0$ is:

$\mu_0 = \frac{\sqrt{3} k_B T}{4 p l_m x_0} \left( \frac{x_0^2}{(1 - x_0)^3} - \frac{1}{4(1 - x_0)^2} + \frac{1}{4} \right) + \frac{\sqrt{3} k_p (n+1)}{4 l_0^{m+1}}$

We are told to assume:

• $x_0 = \frac{1}{2}$,
• $k_p = 0$.

Step 1: Substitution of $x_0 = 1/2$

When $x_0 = \frac{1}{2}$, let's simplify the term inside the parentheses:

$\frac{x_0^2}{(1 - x_0)^3} = \frac{\left(\frac{1}{2}\right)^2}{\left(1 - \frac{1}{2}\right)^3} = \frac{\frac{1}{4}}{\frac{1}{8}} = 2$ $\frac{1}{4(1 - x_0)^2} = \frac{1}{4\left(1 - \frac{1}{2}\right)^2} = \frac{1}{4 \times \frac{1}{4}} = 1$ $\frac{1}{4} = 0.25$

Substituting these into the equation for $\mu_0$, we get:

$\mu_0 = \frac{\sqrt{3} k_B T}{4 p l_m \times \frac{1}{2}} \left( 2 - 1 + 0.25 \right)$ $\mu_0 = \frac{\sqrt{3} k_B T}{2 p l_m} \times 1.25$ $\mu_0 = \frac{5 \sqrt{3} k_B T}{8 p l_m}$

Step 2: Analyze the statements

(i) The graph of $\mu_0$ as a function of $T$ is linear.

From the simplified formula, $\mu_0 = \frac{5 \sqrt{3} k_B T}{8 p l_m}$, it is clear that $\mu_0$ is directly proportional to $T$. Since this is a linear equation in $T$, statement (i) is true.

(ii) Viewing $\mu_0$ as a function of $p$, we can say that $\mu_0$ is proportional to $T$.

In the equation, $\mu_0 = \frac{5 \sqrt{3} k_B T}{8 p l_m}$, $\mu_0$ is proportional to $T$ and inversely proportional to $p$. Therefore, statement (ii) is true.

(iii) The graph of $\mu_0$ as a function of $T$ has a slope of $\frac{5 \sqrt{3} k_B}{8 p l_m}$.

The equation for $\mu_0$ is $\mu_0 = \frac{5 \sqrt{3} k_B T}{8 p l_m}$, which is in the form of $\mu_0 = mT$, where the slope $m = \frac{5 \sqrt{3} k_B}{8 p l_m}$. This matches the slope given in the statement, so statement (iii) is true.

Conclusion:

All three statements (i), (ii), and (iii) are true.

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Would you like further details on any part of this explanation?

Here are some related questions to deepen your understanding:

1. What is the physical significance of the shear modulus $\mu_0$ in this context?
2. How would the expression for $\mu_0$ change if $x_0$ were different from $\frac{1}{2}$?
3. What impact does the parameter (