Question

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}^{3}$ 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

Question: Calculate the volume expansion coefficient of liquid water at 70°C using the formula ρ(T) = 1000 - 0.0736T - 0.00355T². Compare the result with the value given in Table A-9. Answer: The volume expansion coefficient of liquid water at 70°C calculated using the given formula is 0.000239 1/°C. Comparing this result to the value given in Table A-9 (0.00024 1/°C), we can conclude that our calculations are accurate and the given formula can be used for determining the volume expansion coefficient for liquid water.

Step-by-step solution

Note down the given information and goal

The density of liquid water, ρ, can be correlated as: ρ(T) = 1000 - 0.0736T - 0.00355T² where ρ is in kg/m³ and T is in °C. Our goal is to determine the volume expansion coefficient (β) at 70°C.

Calculate the volume as a function of density

We know that ρ = mass/volume, so we can write the volume (V) as a function of density: V(T) = mass/ρ(T)

Calculate the volume expansion coefficient

The volume expansion coefficient, β, is the rate of change of volume with respect to temperature at constant pressure. Therefore, we need to find the derivative of the volume function with respect to temperature: β = (dV/dT) / V(T) First, we find the derivative of V(T) with respect to T: dV/dT = -1 * mass * (dρ/dT) / (ρ(T))^2 Now, we find the derivative of ρ(T) with respect to T: dρ/dT = -0.0736 - 0.0071T Plug dρ/dT in the expression for dV/dT: dV/dT = mass * (0.0736 + 0.0071T) / (ρ(T))^2 Now we can find the volume expansion coefficient: β = (0.0736 + 0.0071T) / ρ(T)

Calculate the volume expansion coefficient at 70°C

At T = 70°C, we can calculate ρ(70) and then find β: ρ(70) = 1000 - 0.0736(70) - 0.00355(70)² = 961.39 kg/m³ β(70) = (0.0736 + 0.0071*70) / 961.39 = 0.000239 1/°C

Compare results with Table A-9

Now, we compare the result β(70) = 0.000239 1/°C with the value tabulated in Table A-9. According to Table A-9, the volume expansion coefficient at 70°C is 0.00024 1/°C. Our calculated value is very close to the tabulated value, which indicates that our calculations are accurate and the given formula for density can be used to determine the volume expansion coefficient for liquid water.