Density and Pressure — Question 8

Question 8

A two-fill measurement separates tare from density

A vessel has unknown empty mass. With $120\,\mathrm{cm^3}$ of a uniform liquid, total mass is $246\,\mathrm{g}$; with $280\,\mathrm{cm^3}$ of the same liquid, total mass is $374\,\mathrm{g}$. The marked volumes and masses are exact for this model.

1. Infer the liquid density and vessel mass without assuming the first mass is a tare.

2. Predict the total mass at $200\,\mathrm{cm^3}$. A third measured mass of $315\,\mathrm{g}$ is reported; is it consistent with the model?

Question 8 - Solution

A two-fill measurement separates tare from density

(a) Derivation and calculation

$$M_1=M_0+\rho V_1,\quad M_2=M_0+\rho V_2.$$$$\boxed{\rho=\frac{374-246}{280-120}=.800\,\mathrm{g/cm^3}=800\,\mathrm{kg/m^3}}.$$$$\boxed{M_0=246-.800(120)=150\,\mathrm g}.$$

(b) Result and interpretation

$$\boxed{M(200)=150+.800(200)=310\,\mathrm g}.$$$$r=M_{\mathrm{observed}}-M_{\mathrm{predicted}}=315-310=\boxed{5\,\mathrm g}.$$The third reading is inconsistent with the exact fixed-tare, constant-density model. For experimental data, compare this residual with measurement uncertainty.

Physical check

Both original points are reproduced by $M=150+.800V$ in grams and cubic centimeters.