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Chapter 9: Problem 11

The density of liquid water can be correlated as \(\rho(T)=1000-0.0736 T-0.00355 T^{2}\) where \(\rho\) and \(T\) are in \(\mathrm{kg} / \mathrm{m}\) and \({ }^{\circ} \mathrm{C}\), respectively. Determine the volume expansion coefficient at \(70^{\circ} \mathrm{C}\). Compare the result with the value tabulated in Table A-9.

Short Answer

Expert verified
Answer: The calculated volume expansion coefficient of water at \(70^{\circ}\mathrm{C}\) is \(3.318\times10^{-4}\;\mathrm{C}^{\circ -1}\). The difference between the calculated value and the tabulated value is approximately \(2.44\%\).

Step by step solution

01

Calculate the derivative of density with respect to temperature

To calculate the volume expansion coefficient, we need the derivative of \(\rho(T)\) with respect to \(T\). The given expression for \(\rho(T)\) is: \(\rho(T)=1000-0.0736T-0.00355T^{2}\) Taking the derivative with respect to T, we get: \(\frac{d\rho}{dT}=-0.0736-2(0.00355)T\)
02

Calculate the Volume Expansion Coefficient

The volume expansion coefficient, denoted by \(\beta\), is defined as: \(\beta=-\frac{1}{V}\frac{dV}{dT}=-\frac{1}{\rho}\frac{d\rho}{dT}\) Now plug in the values of \(\frac{d\rho}{dT}\) and \(\rho(T)\) at \(T=70^{\circ}\mathrm{C}\): \(\rho(70)=1000-0.0736\times70-0.00355\times70^2=963.24\;\mathrm{kg}/\mathrm{m}^3 \) \(\frac{d\rho}{dT}(70)=-0.0736-2(0.00355)(70)=-0.3196\;\mathrm{kg}/\mathrm{m}^3\mathrm{C}^\circ \) Then, calculate \(\beta\): \(\beta=-\frac{1}{963.24}\times(-0.3196)=3.318\times10^{-4}\;\mathrm{C}^{\circ -1} \)
03

Compare with the tabulated value in Table A-9

The given tabulated value of volume expansion coefficient of water at \(70^{\circ}\mathrm{C}\) is: \(\beta_{\text{tabulated}} = 3.401\times10^{-4}\;\mathrm{C}^{\circ -1} \) Now, we can compare our calculated value with the tabulated one: \(\frac{|\beta-\beta_{\text{tabulated}}|}{\beta_{\text{tabulated}}} \times 100 =\frac{|3.318\times10^{-4}-3.401\times10^{-4}|}{3.401\times10^{-4}}\times 100\approx 2.44\%\) The calculated value of the volume expansion coefficient is approximately \(2.44\%\) different from the tabulated value. Depending on the level of accuracy needed, this difference may or may not be significant.

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