Mass Density

1. Definition and symbols

Definition: \[ \rho \equiv \frac{m}{V} \] where \(m\) is mass and \(V\) is volume.
SI units: kg/m\(^3\). Common alternates: g/cm\(^3\) (1 g/cm\(^3\) = 1000 kg/m\(^3\)).
Specific volume: \[ v \equiv \frac{1}{\rho} \quad \text{(m}^3\text{/kg)}. \]

Specific weight (weight density): \[ \gamma = \rho g \quad \text{(N/m}^3\text{)}, \qquad g \approx 9.81\ \text{m/s}^2. \]
Relative density (specific gravity): \[ \text{SG} = \frac{\rho}{\rho_{\text{ref}}}. \] For liquids, \(\rho_{\text{ref}}\) is often water at 4°C (1000 kg/m\(^3\)); for gases, dry air at STP is sometimes used.
Ideal gas relation: \[ \rho = \frac{p}{R T}, \] with \(p\) absolute pressure, \(T\) absolute temperature, and \(R\) the specific gas constant.

Thermal expansion (solids/liquids, small \(\Delta T\)): \[ \rho(T) \approx \frac{\rho_0}{1+\beta\,(T - T_0)} \;\approx\; \rho_0\,[1 - \beta\,(T-T_0)], \] where \(\beta\) is the volumetric thermal expansion coefficient.
Compressibility (isothermal bulk modulus \(K_T\)): \[ \frac{\Delta \rho}{\rho} \approx \frac{\Delta p}{K_T}, \qquad K_T = -V\left(\frac{\partial p}{\partial V}\right)_T. \] For most liquids \(K_T\) is large, so \(\rho\) is weakly dependent on \(p\).
Gases: Use an equation of state; ideal gas gives \(\rho \propto p/T\). Real-gas compressibility factor \(Z\) refines it: \[ \rho = \frac{p}{Z R T}. \]

Two-component mixture (no volume change on mixing, ideal approximation): \[ \rho_{\text{mix}} \approx \frac{1}{\sum_i \frac{y_i}{\rho_i}}, \] where \(y_i\) are volume fractions; for gas mixtures, ideal gas: \[ \rho_{\text{mix}} = \frac{p}{\bar{R} T}, \quad \bar{R} = \sum_i x_i R_i = \frac{R_u}{\sum_i \frac{x_i}{M_i}}, \] with mole fractions \(x_i\), molar masses \(M_i\), and universal gas constant \(R_u\).
Slurries/suspensions (mass fractions \(w_i\)): \[ \frac{1}{\rho_{\text{mix}}} \approx \sum_i \frac{w_i}{\rho_i} \quad \text{(if partial volumes add)}. \]
Two-phase (e.g., gas–liquid) bulk density: \[ \rho_{\text{bulk}} = \alpha\,\rho_g + (1-\alpha)\,\rho_\ell, \] where \(\alpha\) is gas void fraction.

Gravimetric (direct): Measure mass \(m\) and volume \(V\) (by displacement or dimensional metrology), \(\rho = m/V\).
Hydrometer / pycnometer: Compare buoyant forces or use calibrated volume flasks for liquids.
Oscillating U-tube: Resonant frequency relates to fluid density with high precision.
Gas density: Infer via \(p,\,T\) and equation of state; or use vibrating-element densitometers.

Continuity equation (integral form, single-phase): \[ \frac{d}{dt}\int_V \rho\, dV + \oint_{\partial V} \rho\,\vec{u}\cdot \vec{n}\, dA = 0. \]
Differential form: \[ \frac{\partial \rho}{\partial t} + \nabla\cdot(\rho \vec{u}) = 0. \] For incompressible flow, \(\rho = \text{const}\) and \(\nabla\cdot\vec{u}=0\).
Hydrostatics (barotropic fluid): \[ \frac{dp}{dz} = -\rho g. \] With \(\rho\) constant, \(p(z)=p_0 - \rho g (z-z_0).\)

Water at 4°C: \(1\ \text{g/cm}^3 = 1000\ \text{kg/m}^3\). Aluminum (2.70 g/cm\(^3\)) is \(2700\ \text{kg/m}^3\).

Given: \(p=101{,}325\ \text{Pa},\ T=300\ \text{K},\ R_{\text{air}}=287\ \text{J/(kg·K)}\).
Result: \[ \rho = \frac{p}{R T} = \frac{101{,}325}{287\times 300} \approx 1.176\ \text{kg/m}^3. \]

Given: 40% ethanol (0.789 g/cm\(^3\)) and 60% water (0.998 g/cm\(^3\)) by volume.
Result: \[ \frac{1}{\rho} \approx \frac{0.40}{789} + \frac{0.60}{998} \quad \text{(m}^3\text{/kg, using kg/m}^3), \] \[ \rho \approx \frac{1}{0.000507 + 0.000601} \approx 919\ \text{kg/m}^3. \] Note: real mixtures may deviate due to non-ideal mixing volumes.

Given: Seawater \(\rho \approx 1025\ \text{kg/m}^3\).
Compute: \[ \gamma = \rho g \approx 1025 \times 9.81 \approx 10{,}055\ \text{N/m}^3. \]

Report \(\rho\) with the corresponding \(T\) and \(p\); density is state dependent, especially for gases.
Incompressible assumptions are valid for most liquids under moderate \(p,T\) changes; verify when precision matters.
Porous materials: distinguish between true, skeletal, and bulk (apparent) density depending on voids and saturation.