6.3 Estimate mixture gas constant and specific heat | University of Alberta MEC E 340: Applied Thermodynamics

University of Alberta MEC E 340: Applied Thermodynamics

Gas Mixtures and Real Gases

MEC E 340 study topic 6.3

Consider a fixed quantity of a gas mixture in a closed system, or the gas flowing through a control volume. The mixture is the working fluid; label any states needed for an energy calculation, such as state 1 before heating and state 2 after heating. For the estimates in this lesson, assume the components are well mixed, behave as ideal gases, and do not react. Component composition is unchanged between the states. These are stated modeling assumptions, not property data. The first-law background is that an energy change can be related to heat and work; here, the main task is to estimate the mixture properties needed to evaluate that change. No cycle or device diagram is needed for the composition calculations.

What you will learn

Distinguish mole fractions from mass fractions and select the appropriate basis for a mixture property.
Estimate a mixture gas constant from component gas constants or molar masses.
Estimate mixture specific heat from component data at a stated temperature and on a stated basis.
Use the estimates consistently in a simple ideal-gas energy calculation.

1. Choose a composition basis

A mixture may be described by mole fractions or mass fractions. The mole fraction of component

i

is its amount of substance divided by the total amount of substance. The mass fraction is its mass divided by total mixture mass. Both sets of fractions sum to one, but they are not interchangeable: a light gas can have a large mole fraction without having the same mass fraction.

Use mole fractions with molar quantities, such as molar mass and molar specific heat. Use mass fractions with mass-based quantities, such as the specific gas constant and mass specific heat. Before substituting numbers, write down which basis the supplied composition uses.

\sum_i y_i=1,\qquad \sum_i w_i=1

Mole fractions are denoted $y_i$ ; mass fractions are denoted $w_i$ .
Each composition basis sums to one.
Convert the composition if the property equation requires the other basis.

2. Estimate the mixture gas constant

For ideal-gas components, the mixture behaves as an ideal gas when written using its mixture gas constant:

p v=R_{mix}T

, where

p

is absolute pressure,

v

is specific volume per unit mixture mass, and

T

is absolute temperature. This relation is useful only with consistent units and the ideal-gas assumption.

If component specific gas constants are known, combine them using mass fractions. If the data are instead mole fractions and component molar masses, first find the mixture molar mass, then divide the universal gas constant by that molar mass. The two routes give the same answer when the data and units are consistent.

For a mixture with total mass

m

, the mass fraction is

w_i=m_i/m

. If starting with mole fractions, the component mass contribution is proportional to

y_i M_i

, where

M_i

is component molar mass. This gives a direct conversion to mass fractions without guessing which component dominates the mixture mass.

R_{mix}=\sum_i w_iR_i=\frac{R_u}{M_{mix}},\qquad M_{mix}=\sum_i y_iM_i

Use mass fractions to average component specific gas constants.
Use mole fractions to calculate mixture molar mass.
The mixture gas constant has units of energy per mass per kelvin, such as $\mathrm{kJ/(kg\cdot K)}$ .

3. Estimate mixture specific heat

Specific heat depends on temperature. To estimate a mixture value, use component specific heats that correspond to the same stated temperature or the same stated temperature range. For an ideal-gas mixture with fixed composition, the mass-based mixture specific heat at constant pressure is the mass-fraction-weighted sum of the component values.

If component data are provided on a molar basis, first average the molar specific heats using mole fractions, then divide by mixture molar mass to obtain a mass-based value. Keep the units visible: a molar specific heat and a mass specific heat are different quantities.

The same weighted-average idea applies to constant-volume specific heat. Under the ideal-gas model, the mass-based values are related by

c_p-c_v=R_{mix}

. This relation is a useful consistency check; it does not replace the need to match the supplied specific-heat data to the temperature of interest.

When specific heat changes appreciably over a temperature interval, a single value is an estimate. Use a value explicitly supplied for the interval or temperature in the problem. Do not silently treat a room-temperature value as exact at a much higher temperature.

c_{p,mix}=\sum_i w_ic_{p,i}=\frac{\sum_i y_i\bar c_{p,i}}{M_{mix}}

State whether the requested value is $c_p$ or $c_v$ .
Average on a consistent basis and at a consistent temperature.
Use $c_p-c_v=R_{mix}$ as an ideal-gas consistency check.

4. Carry the estimate into an energy calculation

For a closed, rigid vessel containing a fixed-composition ideal-gas mixture, there is no boundary work because the volume does not change. If the only energy transfer is heat, the first law gives

Q=\Delta U

. With a constant estimated

c_v

, the internal-energy change is approximated by

m c_v(T_2-T_1)

. Thus, mixture-property estimation supplies the value needed for the energy balance; it does not change the balance itself.

For a steady-flow component, use the appropriate control-volume mass and energy balances and the property basis required by those equations. This lesson does not assume a particular device or invent its heat or work directions. Always identify the system, states, assumptions, and sign convention before inserting mixture properties.

After calculating, check that composition fractions sum to one, the gas constant is positive, the units reduce correctly, and the mixture property lies between the smallest and largest component values when it is an ordinary weighted average with positive fractions.

Q=\Delta U\approx m c_{v,mix}(T_2-T_1)

A mixture property estimate supports the balance; it does not replace it.
For the rigid-vessel example, use mass specific heat with mass and temperature difference.
A weighted-average result should be bounded by the component values.

Worked example

Mass-fraction estimate for a two-gas mixture

A nonreacting ideal-gas mixture contains 0.70 kg nitrogen and 0.30 kg oxygen. For this estimate, use the supplied values

R_{N_2}=0.2968\ \mathrm{kJ/(kg\cdot K)}

and

R_{O_2}=0.2598\ \mathrm{kJ/(kg\cdot K)}

. Estimate the mixture gas constant.

Identify the basis
The supplied component amounts are masses, so they directly give mass fractions. The mixture is treated as ideal and nonreacting, and the supplied component gas constants are the data source.
$w_{N_2}=0.70,\qquad w_{O_2}=0.30$
Apply the mass-weighted estimate
Weight each component specific gas constant by its share of the total mass. The fractions sum to one, so this is a direct weighted average.
$R_{mix}=(0.70)(0.2968)+(0.30)(0.2598)=0.2857\ \mathrm{kJ/(kg\cdot K)}$

Answer: The estimated mixture gas constant is

0.2857\ \mathrm{kJ/(kg\cdot K)}

.

Check: The result is between the two supplied component values, as required for a weighted average. The fractions sum to one, and the units remain those of a specific gas constant.

Worked example

Mole fractions to mixture gas constant

A nonreacting ideal-gas mixture has mole fractions

y_A=0.25

and

y_B=0.75

. Use the supplied molar masses

M_A=28.0\ \mathrm{kg/kmol}

and

M_B=44.0\ \mathrm{kg/kmol}

, and

R_u=8.314\ \mathrm{kJ/(kmol\cdot K)}

. Estimate the mixture molar mass and specific gas constant.

Find mixture molar mass
Molar mass is averaged using mole fractions because each fraction counts a share of the total amount of substance.
$M_{mix}=(0.25)(28.0)+(0.75)(44.0)=40.0\ \mathrm{kg/kmol}$
Convert to mass-specific gas constant
Divide the universal gas constant by mixture molar mass. The kmol units cancel, leaving energy per mixture mass per kelvin.
$R_{mix}=\frac{8.314\ \mathrm{kJ/(kmol\cdot K)}}{40.0\ \mathrm{kg/kmol}}=0.2079\ \mathrm{kJ/(kg\cdot K)}$

Answer: The mixture molar mass is

40.0\ \mathrm{kg/kmol}

, and its estimated gas constant is

0.2079\ \mathrm{kJ/(kg\cdot K)}

.

Check: The mixture molar mass lies between the two component molar masses. The gas constant is positive and has the correct mass-based units.

Worked example

Specific heat and rigid-vessel heating estimate

A 2.00 kg ideal-gas mixture in a rigid vessel contains mass fractions

w_A=0.60

and

w_B=0.40

. At the temperature range of interest, use supplied constant values

c_{v,A}=0.70\ \mathrm{kJ/(kg\cdot K)}

and

c_{v,B}=0.90\ \mathrm{kJ/(kg\cdot K)}

. The mixture is heated from

300\ \mathrm{K}

to

350\ \mathrm{K}

. Estimate the required heat transfer, assuming negligible kinetic and potential energy changes and no work other than boundary work.

Estimate the mixture specific heat
The composition and component values are mass-based, so use a mass-fraction average. The stated constant values apply to the stated temperature range.
$c_{v,mix}=(0.60)(0.70)+(0.40)(0.90)=0.78\ \mathrm{kJ/(kg\cdot K)}$
Apply the closed-system energy balance
A rigid vessel has no boundary work. With the given assumptions, heat transfer equals the internal-energy increase, estimated using the mixture constant-volume specific heat.
$Q\approx(2.00\ \mathrm{kg})(0.78\ \mathrm{kJ/(kg\cdot K)})(350-300\ \mathrm{K})=78\ \mathrm{kJ}$

Answer: The estimated mixture constant-volume specific heat is

0.78\ \mathrm{kJ/(kg\cdot K)}

, and the heat supplied is

78\ \mathrm{kJ}

.

Check: The temperature rises, so the estimated internal-energy increase and required heat are positive. The specific heat lies between the two component values, and the final heat unit is kJ.

Common mistakes and how to avoid them

Using mole fractions to average mass-based component specific gas constants.

Correction: Use mass fractions for mass-specific gas constants, or convert through mixture molar mass using mole fractions.

Mixing molar specific heat units with mass specific heat units.

Correction: Keep the basis explicit and divide a mole-based mixture specific heat by mixture molar mass before using it with mixture mass.

Using a component specific heat without checking its temperature basis.

Correction: Use component values supplied for the stated temperature or interval; identify a constant-value result as an estimate.

Treating the ideal-gas mixture relation as valid without stating an assumption.

Correction: State that the mixture is modeled as an ideal gas and use the resulting estimate only within that model.

Lesson summary

Composition must be stated on a mole or mass basis before estimating a mixture property.
The ideal-gas mixture gas constant is a mass-fraction average of component gas constants, or the universal gas constant divided by mixture molar mass.
Mixture specific heat is estimated by a weighted average on a consistent basis and at the stated temperature.
Use the estimated property in the appropriate energy balance, carry SI units, and check bounds and signs.

Check your understanding

Question 1

A mixture's mass fractions are 0.25 and 0.75. Which weighting is appropriate for combining mass-specific component heat capacities?

A mole-fraction average, even if mole fractions are not provided
A mass-fraction average
An unweighted sum
The product of the two component values

Show answer and explanation

A mass-fraction average

Mass-specific heat capacities are combined using mass fractions. The fractions should sum to one.

Question 2

For an ideal-gas mixture, what information is needed to calculate its gas constant directly from molar data?

Mole fractions and component molar masses
Only the total pressure
Only the vessel volume
The heat-transfer direction

Show answer and explanation

Mole fractions and component molar masses

Mole fractions and component molar masses give mixture molar mass; dividing the universal gas constant by it gives the mass-specific mixture gas constant.

Question 3

A positive-mass mixture is estimated using positive mass fractions summing to one. What should be true of its mass-weighted specific heat?

It must exceed every component value
It must be less than every component value
It must lie between the smallest and largest component values
It must equal the universal gas constant

Show answer and explanation

It must lie between the smallest and largest component values

A weighted average with nonnegative fractions summing to one lies between the smallest and largest values being averaged.

Key terms

Mole fraction: The amount of a mixture component divided by the total amount of substance in the mixture.
Mass fraction: The mass of a mixture component divided by total mixture mass.
Mixture molar mass: The total mixture mass divided by total amount of substance; for a mixture it is the mole-fraction-weighted component molar mass.
Specific heat: The energy needed per unit mass and per kelvin of temperature change under the stated condition, such as constant pressure or constant volume.

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6.3 · Estimate mixture gas constant and specific heat

University of Alberta MEC E 340: Applied Thermodynamics

MEC E 340 study topic 6.3

What you will learn

1. Choose a composition basis

2. Estimate the mixture gas constant

3. Estimate mixture specific heat

4. Carry the estimate into an energy calculation

Mass-fraction estimate for a two-gas mixture

Mole fractions to mixture gas constant

Specific heat and rigid-vessel heating estimate

Common mistakes and how to avoid them

Lesson summary

Check your understanding

Question 1

Question 2

Question 3

Key terms

Continue through MEC E 340

About this lesson and its review