Density and Pressure — Question 7

Question 7

Density changes can reveal a leak during compression

A gas sample initially has volume $2.0\,\mathrm{L}$ and density $1.5\,\mathrm{kg/m^3}$. After a process its volume is $0.80\,\mathrm{L}$ and density is measured as $3.0\,\mathrm{kg/m^3}$. Do not assume a gas equation of state.

1. Find the initial and final masses and the mass fraction lost.

2. What density would the final sample have if compression alone occurred in a sealed system? Can pressure be inferred from the given density data alone?

Question 7 - Solution

Density changes can reveal a leak during compression

(a) Derivation and calculation

$$m_i=\rho_iV_i=(1.50)(2.0\times10^{-3})=.00300\,\mathrm{kg}.$$$$m_f=\rho_fV_f=(3.00)(.80\times10^{-3})=.00240\,\mathrm{kg}.$$$$\boxed{\Delta m_{\mathrm{lost}}=.600\,\mathrm g},\qquad\boxed{\frac{m_i-m_f}{m_i}=.200=20.0\%}.$$

(b) Result and interpretation

If the sample had remained sealed,$$\boxed{\rho_{f,\mathrm{sealed}}=\frac{m_i}{V_f}=\frac{.00300}{.00080}=3.75\,\mathrm{kg/m^3}}.$$$$\rho_{f,\mathrm{measured}}<\rho_{f,\mathrm{sealed}}\Rightarrow\text{mass loss under the stated measurements}.$$Density alone cannot determine pressure without an equation of state and additional thermodynamic information.

Physical check

Compression alone conserves mass; the observed decrease in $\rho V$ is the evidence for loss.