2.2 Conversion of units

Unit Conversions

Unit conversions let you express a physical quantity in different units without changing its value. In chemical engineering, you'll constantly move between unit systems (SI, CGS, English/Imperial), so getting comfortable with conversions now saves you from errors in every course that follows.

This guide covers how conversion factors work, how to chain them together using dimensional analysis, and how to apply them to typical chemical engineering problems.

Unit Conversions

Converting Between Different Units of the Same Physical Quantity

A single physical quantity can be measured in many units. Length, for example, might appear as meters, centimeters, feet, or inches depending on the source. To convert between them, you multiply by a conversion factor: a fraction that equals 1, with the desired unit on top and the given unit on the bottom.

Because the conversion factor equals 1, multiplying by it doesn't change the physical value. It only changes the unit label.

Steps for a single conversion:

1. Write down the quantity you're starting with, including its unit.
2. Find the relationship between the given unit and the desired unit (e.g., 1 ft = 0.3048 m).
3. Build a fraction with the desired unit in the numerator and the given unit in the denominator.
4. Multiply. The given unit cancels, leaving only the desired unit.

Example: Convert 15.0 feet to meters.

$15.0 \; \text{ft} \times \frac{0.3048 \; \text{m}}{1 \; \text{ft}} = 4.572 \; \text{m}$

The "ft" in the numerator and denominator cancel, and you're left with meters.

Dimensional Analysis and the Factor-Label Method

Dimensional analysis (also called the factor-label method) extends this idea to multi-step conversions. You line up a chain of conversion factors so that every unwanted unit cancels, and only the target unit survives.

This technique does two things for you:

• It keeps your math organized, even when a problem requires three or four conversion steps.
• It acts as a built-in error check. If the units don't cancel cleanly, something is set up wrong.

Example: Convert 60.0 miles per hour to meters per second.

$60.0 \; \frac{\text{mi}}{\text{hr}} \times \frac{1609.34 \; \text{m}}{1 \; \text{mi}} \times \frac{1 \; \text{hr}}{3600 \; \text{s}} = 26.8 \; \frac{\text{m}}{\text{s}}$

Notice how "mi" cancels with "mi" and "hr" cancels with "hr," leaving $\text{m/s}$.

Conversion Factors for Simplification

Deriving Conversion Factors from Known Relationships

A conversion factor is just a ratio of two equivalent quantities expressed in different units. You can build one from any known equivalence:

• 1 in = 2.54 cm (exact, by definition)
• 1 kg = 1000 g
• 1 hr = 3600 s
• 1 atm = 101,325 Pa
• 1 L = 0.001 m³

From any of these, you can write two fractions (e.g., $\frac{2.54 \; \text{cm}}{1 \; \text{in}}$ or $\frac{1 \; \text{in}}{2.54 \; \text{cm}}$). Choose whichever version puts the unit you want to get rid of in the position where it will cancel.

Combining Conversion Factors for Complex Unit Conversions

When no single conversion factor bridges the gap between your starting and target units, chain several together.

Example: Convert 5.00 cubic feet ($\text{ft}^3$) to liters.

There's no single "cubic feet to liters" factor you need to memorize. Instead, break it into steps:

1. Convert feet to meters: $1 \; \text{ft} = 0.3048 \; \text{m}$
2. Because the unit is cubed, cube the conversion factor: $(0.3048)^3 = 0.02832 \; \text{m}^3/\text{ft}^3$
3. Convert cubic meters to liters: $1 \; \text{m}^3 = 1000 \; \text{L}$

$5.00 \; \text{ft}^3 \times \frac{0.02832 \; \text{m}^3}{1 \; \text{ft}^3} \times \frac{1000 \; \text{L}}{1 \; \text{m}^3} = 141.6 \; \text{L}$

A common mistake with area and volume conversions is forgetting to raise the linear conversion factor to the appropriate power (squared for area, cubed for volume). Always check your exponents.

Unit Conversions in Chemical Engineering

Identifying Given and Desired Units

Chemical engineering problems routinely mix unit systems. A flow rate might be reported in gallons per minute while a reactor design equation expects $\text{m}^3/\text{s}$. A pressure gauge might read in psi while your ideal gas law calculation uses atmospheres or pascals.

Before you start any calculation:

1. List every quantity you're given, with its units.
2. Identify the units your answer needs to be in.
3. Plan the conversion path by noting which conversion factors bridge the gap.

Doing this up front prevents the frustrating experience of getting a numerical answer and then realizing your units are inconsistent.

Applying Unit Conversions in Problem Solving

Here are some conversions you'll see frequently in chemical engineering coursework:

Flow rate: Convert 50.0 gallons per minute (gpm) to liters per second.

$50.0 \; \frac{\text{gal}}{\text{min}} \times \frac{3.785 \; \text{L}}{1 \; \text{gal}} \times \frac{1 \; \text{min}}{60 \; \text{s}} = 3.15 \; \frac{\text{L}}{\text{s}}$

Pressure: Convert 2.50 atm to pascals.

$2.50 \; \text{atm} \times \frac{101{,}325 \; \text{Pa}}{1 \; \text{atm}} = 253{,}313 \; \text{Pa} \approx 2.53 \times 10^5 \; \text{Pa}$

Concentration: For dilute aqueous solutions at standard conditions, 1 mg/L is numerically equivalent to 1 ppm (by mass). This shortcut works because the density of dilute aqueous solutions is approximately $1 \; \text{kg/L}$. Be careful applying it to non-aqueous or concentrated solutions where the density differs significantly.

Temperature: Temperature conversions are not simple multiplication by a conversion factor. Going from Celsius to Kelvin uses an offset:

$T(\text{K}) = T(°\text{C}) + 273.15$

This is different from converting a temperature difference, where $\Delta T$ of 1 °C equals $\Delta T$ of 1 K (no offset needed). Mixing these up is a common source of error in energy balance problems.

General approach for any problem:

1. Write the given value with its full units.
2. Set up a chain of conversion factors so all unwanted units cancel.
3. Multiply across the numerators and denominators.
4. Check that only the desired unit remains.
5. Verify the result makes physical sense (order of magnitude, direction of change).

Getting into this habit early will make material and energy balance courses much smoother.