Tapped Density Formula: ρ_t = m/V_t, Hausner Ratio, Examples

The tapped density formula is $\rho_t = \frac{m}{V_t}$ : weigh a powder, tap its measuring cylinder until the level stops falling, and divide the mass by that settled volume. Because tapping squeezes out some of the air between grains, the tapped density is always equal to or higher than the loose bulk density.

The comparison between the two numbers is what makes tapped density useful. Pharmacy students, food technologists and materials engineers use it to judge how well a powder will flow from a hopper or fill a capsule. Everything here rests on the plain density formula, one of the matter results gathered on the fluids hub page.

Quick facts

Formula
$\rho_t = \frac{m}{V_t}$
SI unit
kg/m³; labs quote g/mL (1 g/mL = 1000 kg/m³)
Dimensional formula
[M¹ L⁻³ T⁰]
Hausner ratio
$H = \frac{\rho_t}{\rho_b}$
Compressibility index
$C = \frac{\rho_t - \rho_b}{\rho_t} \times 100\%$
Rule
ρₜ ≥ ρ_b for the same powder

What goes into the tapped density formula

Terms in ρₜ = m / Vₜ
Symbol	Meaning	Unit in lab work
$\rho_t$	Tapped density	g/mL
$m$	Mass of powder poured into the cylinder	g
$V_t$	Volume read after tapping has finished	mL

The mass does not change during the test; only the volume shrinks. Start by pouring the powder gently into a graduated cylinder and reading the untapped volume V₀. That reading gives the bulk density, m / V₀. Then the cylinder is lifted and dropped a fixed small height many times, by hand or on a tapping machine, until two readings agree. The final reading is Vₜ.

Steps for a tapped density test

Weigh a clean, dry graduated cylinder, add the powder and weigh again to get m.
Read the loose volume V₀ without pressing the powder.
Tap the cylinder in sets, for example 500 taps then a further 750, reading the volume after each set.
Stop when the change between sets is very small; record that volume as Vₜ.
Calculate ρₜ = m / Vₜ and, from V₀, the bulk density.

Exact tap counts and drop heights follow the method your lab uses. The calculator above works in kg and m³; to use grams and millilitres, multiply the g/mL answer by 1000 to compare.

Worked example with Hausner ratio and compressibility

Example: a tablet powder

A 50 g sample of powder fills 100 mL when poured loosely. After tapping, the volume settles at 80 mL. Find the bulk density, the tapped density, the Hausner ratio and the compressibility index.

Bulk density: $\rho_b = \frac{50}{100} = 0.50\ \text{g/mL}$

Tapped density: $\rho_t = \frac{50}{80} = 0.625\ \text{g/mL} = 625\ \text{kg/m}^3$

Hausner ratio: $H = \frac{0.625}{0.50} = 1.25$

Compressibility index: $C = \frac{0.625 - 0.50}{0.625} \times 100\% = 20\%$

Both values place this powder in the fair flow band of the table below.

Example 2: tapped volume from a known density

A powder has a tapped density of 0.72 g/mL. What tapped volume will 36 g of it occupy?

$V_t = \frac{m}{\rho_t} = \frac{36}{0.72} = 50\ \text{mL}$

Use the same mass for both densities. Only the volume reading changes between the loose and tapped states.

Reading flow from the two ratios

Usual flow ratings for powders
Compressibility index (%)	Hausner ratio	Flow character
10 or less	1.00 to 1.11	excellent
11 to 15	1.12 to 1.18	good
16 to 20	1.19 to 1.25	fair
21 to 25	1.26 to 1.34	passable
26 to 31	1.35 to 1.45	poor
32 to 37	1.46 to 1.59	very poor
38 or more	1.60 or more	very, very poor

The two ratios carry the same information, since H = 100 / (100 − C). A powder that settles a lot on tapping has many loose gaps and sticky, irregular grains, so it tends to bridge and clog. A powder that hardly settles flows almost like sand.

Common mistakes

Dividing by the loose volume. That is bulk density; tapped density uses the settled volume.
Stopping too early. Keep tapping until the reading stops changing.
Turning the ratios upside down. The Hausner ratio is tapped over bulk, so it is never less than 1.
Mixing units. g/mL and kg/m³ differ by a factor of 1000.

Frequently asked questions

What is the formula for tapped density?

Tapped density = mass of powder / tapped volume, ρₜ = m / Vₜ, where Vₜ is the volume after repeated tapping has settled the powder.

What is the difference between bulk and tapped density?

Bulk density uses the loose, poured volume. Tapped density uses the smaller volume after tapping, so it is the higher of the two.

How is the Hausner ratio calculated?

Divide tapped density by bulk density: H = ρₜ / ρ_b. Values close to 1 mean the powder flows freely.

What unit is tapped density given in?

Labs usually quote g/mL or g/cm³. The SI unit is kg/m³, and 1 g/mL equals 1000 kg/m³.