Mohs Hardness Relative Scaling Equation, Derivation & Rearrangements

Core idea

Overview

This empirical formula provides a mathematical approximation for converting the qualitative Mohs scale of mineral hardness into a quantitative absolute value. It illustrates that mineral hardness increases exponentially rather than linearly as the scale rank progresses.

When to use: Apply this formula when you need to estimate relative quantitative hardness from a qualitative Mohs rank. It is useful for material science comparisons where precise laboratory testing like Vickers or Knoop is unavailable.

Why it matters: The Mohs scale is not linear; for example, the hardness gap between Corundum (9) and Diamond (10) is much larger than between Talc (1) and Gypsum (2). This formula helps engineers understand the exponential energy and wear requirements for processing harder minerals.

Symbols

Variables

$H_{ab s}$ = Approx Absolute Hardness, M = Mohs Scale Rank

H_{ab s}

Approx Absolute Hardness

Variable

M

Mohs Scale Rank

Variable

Walkthrough

Derivation

Understanding the Mohs Hardness Scale

The Mohs scale ranks mineral hardness from 1 (talc) to 10 (diamond), but the scale is non-linear in terms of absolute hardness.

The empirical approximation $H_{a}$ bs ≈ 2^(M/2) estimates relative absolute hardness.
Scratch resistance is used as the test.

1

The Mohs scale is ordinal, not linear:

Each step means one mineral can scratch the one below, but the hardness jumps are unequal.

1 (talc) \to 2 (gypsum) \to \dots \to 10 (diamond)

2

Approximate absolute hardness:

This exponential approximation captures the non-linear spacing. Diamond (M = 10) is vastly harder than corundum (M = 9) in absolute terms.

H_{abs} \approx 2^{M /2}

Note: Practical tests: fingernail ≈ 2.5, copper coin ≈ 3.5, glass ≈ 5.5, steel file ≈ 6.5.

Result

H_{abs} \approx 2^{M /2}

Source: A-Level Geology — Mineralogy

Visual intuition

Graph

Graph unavailable for this formula.

The graph displays an exponential curve where the independent variable is plotted on the X-axis and the approximate hardness (HABS) on the Y-axis. Because the variable appears in the exponent, the curve rises at an increasing rate as the X-axis value grows. The function has a positive Y-intercept at 1 and no horizontal asymptotes.

Graph type: exponential

Why it behaves this way

Intuition

A staircase where each step is significantly taller than the one before it, representing how the physical resistance to scratching accelerates as you move toward diamond.

H_{ab s}

Quantitative absolute hardness value

The actual physical resistance of a material to permanent deformation or scratching, typically measured in units like kg/mm2.

M

Mohs scale ordinal rank

A qualitative ranking from 1 to 10 that determines which mineral can scratch another.

Signs and relationships

M/2: The positive exponent indicates that absolute hardness grows exponentially with the Mohs rank, reflecting that the physical effort to scratch a mineral increases drastically at higher levels.

Free study cues

Insight

Canonical usage

This equation converts a dimensionless Mohs hardness rank into a dimensionless relative absolute hardness value.

Common confusion

A common mistake is to assume the Mohs scale is linear or to mistake the dimensionless output of this formula for a physical hardness value with units (e.g., Vickers or Knoop hardness in MPa or kgf/mm2).

Dimension note

Both the input Mohs hardness (M) and the output estimated absolute hardness ( $H_{ab s}$ ) are dimensionless quantities in this formula, representing relative indices rather than physical units.

Unit systems

$M$ dimensionless · Represents a rank on the Mohs scale (typically 1-10).

$H_{ab s}$ dimensionless · Represents a relative quantitative value derived from the Mohs scale, not a physical hardness with units like MPa or kgf/mm2.

Ballpark figures

Quantity:

One free problem

Practice Problem

Quartz has a Mohs hardness of 7. Using the relative scaling formula, calculate its approximate absolute hardness (HABS).

Mohs Scale Rank7

Solve for: HABS

Hint: Divide the Mohs rank by 2 and then use that result as the exponent for the base 2.

The full worked solution stays in the interactive walkthrough.

Where it shows up

Real-World Context

In a chemistry investigation involving Mohs Hardness Relative Scaling, Mohs Hardness Relative Scaling is used to calculate Approx. Hardness from Mohs Scale Rank. The result matters because it helps connect measured amounts to reaction yield, concentration, energy change, rate, or equilibrium.

Study smarter

Tips

Divide the Mohs rank by two before calculating the exponent.
Use the base-2 logarithm to reverse the formula when starting with absolute hardness.
Treat results as approximations, as the Mohs scale is inherently an ordinal ranking system.

Avoid these traps

Common Mistakes

Assuming Mohs 8 is 'twice as hard' as Mohs 4. It's an ordinal rank, not a linear scale.
Convert units and scales before substituting, especially percentages, time units, or powers of ten.
Interpret the answer with its unit and context; a percentage, rate, ratio, and physical quantity do not mean the same thing.

Keep going

Related Formulas

Common questions

Frequently Asked Questions

The Mohs scale ranks mineral hardness from 1 (talc) to 10 (diamond), but the scale is non-linear in terms of absolute hardness.

Apply this formula when you need to estimate relative quantitative hardness from a qualitative Mohs rank. It is useful for material science comparisons where precise laboratory testing like Vickers or Knoop is unavailable.

The Mohs scale is not linear; for example, the hardness gap between Corundum (9) and Diamond (10) is much larger than between Talc (1) and Gypsum (2). This formula helps engineers understand the exponential energy and wear requirements for processing harder minerals.

Assuming Mohs 8 is 'twice as hard' as Mohs 4. It's an ordinal rank, not a linear scale. Convert units and scales before substituting, especially percentages, time units, or powers of ten. Interpret the answer with its unit and context; a percentage, rate, ratio, and physical quantity do not mean the same thing.

In a chemistry investigation involving Mohs Hardness Relative Scaling, Mohs Hardness Relative Scaling is used to calculate Approx. Hardness from Mohs Scale Rank. The result matters because it helps connect measured amounts to reaction yield, concentration, energy change, rate, or equilibrium.

Divide the Mohs rank by two before calculating the exponent. Use the base-2 logarithm to reverse the formula when starting with absolute hardness. Treat results as approximations, as the Mohs scale is inherently an ordinal ranking system.

References

Sources

Britannica: Mohs scale
Wikipedia: Mohs scale of mineral hardness
Klein & Hurlbut: Manual of Mineralogy
Manual of Mineralogy by Cornelis Klein and Barbara Dutrow (23rd Edition)
Earth Materials: Introduction to Mineralogy and Petrology by Cornelis Klein and Anthony Philpotts (2nd Edition)
A-Level Geology — Mineralogy

Mohs Hardness Relative Scaling