Question

Derive Eq. \((12-4)\) as: (a) Starting with Eq. \((12-3),\) substitute \(T=T_{\mathrm{C}}+273.15 .\) (b) Apply the binomial approximation to the square root (see Appendix A.5) and simplify.

Short answer

Question: Derive Eq. (12-4) using the given expression for T and applying the binomial approximation to simplify the square root. Answer: Eq. (12-4) is derived as follows: \(V(T) \approx V_0 \left(1-\frac{1}{2}\frac{T_\mathrm{C}}{T_0 + 273.15}\right)\).

Step-by-step solution

Substitute the Given Expression for T in Eq. (12-3)

Let's start by substituting the given expression for T in Eq. (12-3):

Eq. (12-3)

\[ V(T) = V_0 \sqrt{\frac{T}{T_0}} \] Now, let's substitute \(T = T_\mathrm{C} + 273.15\).

Eq. (12-4)

\[ V(T) = V_0 \sqrt{\frac{T_\mathrm{C} + 273.15}{T_0}} \] Now we have the initial form of Eq. (12-4). The next step is to apply the binomial approximation.

Apply the Binomial Approximation and Simplify

Using the binomial approximation formula \((1+x)^n \approx 1+nx\), where \(|x| \ll 1\) and \(n=-\frac{1}{2}\), we can approximate the square root in Eq. (12-4): Let's rewrite Eq. (12-4) as follows: \[ V(T) = V_0 \left(\frac{T_\mathrm{C} + 273.15}{T_0}\right)^{-\frac{1}{2}} \] Let \(x = \frac{T_\mathrm{C}}{T_0 + 273.15}\), then: \[ V(T) = V_0 \left(1+x\right)^{-\frac{1}{2}} \] Applying the binomial approximation: \[ V(T) \approx V_0 \left(1-\frac{1}{2}x\right) \] Now, substitute back \(x=\frac{T_\mathrm{C}}{T_0 + 273.15}\): \[ V(T) \approx V_0 \left(1-\frac{1}{2}\frac{T_\mathrm{C}}{T_0 + 273.15}\right) \] This is the final form of Eq. (12-4) after applying the binomial approximation and simplifying.